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Pseudospectral Discretization of Nonlinear Delay Equations: New Prospects for Numerical Bifurcation Analysis

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Abstract

We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations.

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... In [3] the DDE is reformulated as an abstract differential equation and a pseudospectral discretization is applied [4], yielding a system of ordinary differential equations (ODEs); LEs are then computed by using the standard discrete QR method (henceforth DQR) for ODEs proposed in [5,6]. In [2], instead, the problem is tackled directly: the DDE is posed in an infinite-dimensional Hilbert space as the state space, the associated family of evolution operators is discretized and the DQR is adapted and applied to the finite-dimensional approximation; for the error analysis, the DQR is raised to infinite dimension and compared to the approximated DQR used for the computations. ...
... The above summary was taken mainly from [3], where the DQR is applied to the ODE obtained from the pseudospectral collocation of a given DDE (see Section 3), thus following the original approach of [4] to also address the study of chaotic dynamics. As anticipated in Section 1, the aim of the present work is to extend this procedure to more general classes of delay equations, such as REs and coupled equations. ...
... In this section, we illustrate the use of pseudospectral collocation to reduce delay equations to ODEs, in view of the application of the DQR described in Section 2. For the reader's convenience, we first present, separately, the discretization of an RE in Section 3.1 and that of a DDE in Section 3.2, summarizing from, respectively, [8] and [3] the main aspects for the present objective (for a full treatment, see again [3,8] and also [4]). Eventually, we combine the two approaches in Section 3.3 for a coupled equation. ...
Article
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We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists of the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
... These approaches provide an ODE approximation of the RFDE. To analyze that ODE is also the core idea, e.g., in [25]- [27]. The involved ODE can be obtained by various methods. ...
... The discretization scheme used in the proposed ODE-based approach should be stability preserving in the following sense. Chebyshev collocation has successfully been applied in various fields [22], [25], [27], [28], [52] where Cond. 6.1 is also desirable. ...
... cf. [25]. The upper part of A C y that is given by 2 h (ℓ ′ k (θ j )) j∈{0,...,N −1},k∈{0,...,N } ⊗ I n requires the first N rows of the (N + 1) × (N + 1) differentiation matrix (ℓ ′ k (θ j )) j,k∈{0,...,N } . ...
Article
The article proposes an approach to complete-type and related Lyapunov-Krasovskii functionals that neither requires knowledge of the delay Lyapunov matrix function nor does it involve linear matrix inequalities. The approach is based on ordinary differential equations (ODEs) that approximate the time-delay system. The ODEs are derived via spectral methods, e.g., the Chebyshev collocation method (also called pseudospectral discretization) or the Legendre tau method. A core insight is that the Lyapunov-Krasovskii theorem resembles a theorem for Lyapunov-Rumyantsev partial stability in ODEs. For the linear approximating ODE, only a Lyapunov equation has to be solved to obtain a partial Lyapunov function. The latter approximates the Lyapunov-Krasovskii functional. Results are validated by applying Clenshaw-Curtis and Gauss quadrature to a semi-analytical result of the functional, yielding a comparable finite-dimensional approximation. In particular, the article provides a formula for a tight quadratic lower bound, which is important in applications. Examples confirm that this new bound is significantly less conservative than known results.
... A common approach to study numerically the infinite-dimensional system is to reduce it to a finitedimensional system of ODEs, whose properties can be studied with well-established software, see e.g. [5] and references therein. In some cases this reduction can be done exactly, e.g., via the linear chain trick [9,14,15]. ...
... Among the various techniques available for this reduction [3,23,37], pseudospectral discretization (PSD), also known as spectral collocation, has been successfully applied in the last decade both to linear(ized) DEs for the stability analysis of equilibria, and to nonlinear ones for the numerical bifurcation in various contexts, including DEs with bounded delay and PDEs with nonlocal boundary conditions [1,4,5,6,7,10,34,35]. For DEs, the main advantage of using PSD compared to other approximation techniques is the fast (exponential) order of convergence for the stability of the equilibria, which follows from the fact that the eigenfunctions of the infinitesimal generator of the linear(ized) equation are exponential, hence analytic. ...
... To treat the unbounded delay, several approaches can be used in the literature, including domain truncation or suitable change of variables [3,23,37], or even quadrature rules to approximate an iDDE with a DDE with a finite number of point delays to be studied via established methods (e.g. [5]) or software like DDE-BIFTOOL [21]. However, these approaches do not exploit some of the fundamental features of iDEs, in particular the "exponentially fading memory", which is the major advantage of the PSD proposed here. ...
Preprint
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We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by introducing a unifying abstract framework and derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, via integration we consider a reformulation in a space of absolutely continuous functions that ensures that point evaluation is well defined. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations, which ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. This result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations.
... Note that (8) is actually of a more general type than (2). 9 However, from the point of view of the implementation, treating the terms with A X (t) and B (k) X (t) is no more difficult than treating the others, while considering them in (8) has the advantage of allowing the application of the method to neutral REs, which are the object of ongoing research (see [14] for a first example). ...
... 5.2 we use the piecewise approach for a DDE with two discrete delays and a periodic solution computed via DDE-BIFTOOL. The example in 5.3 is a coupled equation with terms of all the supported kinds (current time, discrete delays, distributed delays) and a periodic solution computed via MatCont and [9]. Finally, we compute in Sect. ...
... where all parameters are positive andā < τ. We consider a periodic solution (b,S) of (12) computed using MatCont 13 [22] discretizing the equation as a system of ordinary differential equations according to [9], with β = 2,ā = 3, r = 0.3, K = 1, γ = 1 and τ = 4; the computed period is ω = 23.133253862004800. The software repository for eigTMNpw contains the script test_logisticdaphniadelayed_sol_mc.m computing the solution and the file test_logisticdaphniadelayed_sol_db.mat containing the result of the computation. ...
Preprint
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In recent years we provided numerical methods based on pseudospectral collocation for computing the Floquet multipliers of different types of delay equations, with the goal of studying the stability of their periodic solutions. The latest work of the series concerns the extension of these methods to a piecewise approach, in order to take the properties of numerically computed solutions into account. In this chapter we describe the MATLAB implementation of this method and provide practical usage examples.
... Chebyshev collocation has successfully been applied in various fields [24], [28], [55], [56], and Condition 6.1 is always a fundamental requirement. Eigenvalues of A C y converge to the characteristic roots of the RFDE, i.e., to the solutions s of det(sI n − A 0 − e −sh A 1 ) = 0, or, equivalently, to the eigenvalues of the infinitesimal generator A of the C 0 -semigroup {T (t)} t≥0 [28]. ...
... For linear RFDEs, M 2 is an appropriate state space. It brings some simplifications as the product space M 2 becomes a Hilbert space with the natural inner product ·, · M2 defined in (55). Similar to the well-known V (x) = x, P x R n = x P x in the finite dimensional ODE setting, the LK functional for ψ = (x t , x(t)) ∈ M 2 can be described via V M2 (ψ) = ψ, Pψ M2 with a self-adjoint operator P from an operator-valued Lyapunov equation. ...
... cf. [55]. The upper part of A C y that is given by 2 h ( k (θ j )) j∈{0,...,N −1},k∈{0,...,N } ⊗ I n requires the first N rows of the (N + 1) × (N + 1) differentiation matrix ( k (θ j )) j,k∈{0,...,N } . ...
Preprint
Full-text available
The article proposes an approach to complete-type and related Lyapunov-Krasovskii functionals that neither requires knowledge of the delay-Lyapunov matrix function nor does it involve linear matrix inequalities. The approach is based on ordinary differential equations (ODEs) that approximate the time-delay system. The ODEs are derived via spectral methods, e.g., the Chebyshev collocation method (also called pseudospectral discretization) or the Legendre tau method. A core insight is that the Lyapunov-Krasovskii theorem resembles a theorem for Lyapunov-Rumyantsev partial stability in ODEs. For the linear approximating ODE, only a Lyapunov equation has to be solved to obtain a partial Lyapunov function. The latter approximates the Lyapunov-Krasovskii functional. Results are validated by applying Clenshaw-Curtis and Gauss quadrature to a semi-analytical result of the functional, yielding a comparable finite dimensional approximation. In particular, the article provides a formula for a tight quadratic lower bound, which is important in applications. Examples confirm that this new bound is significantly less conservative than known results.
... For instance, a Floquet theory for REs which allows to study the local asymptotic stability of their periodic solutions through the principle of linearized stability was established only recently in [12]. Concerning the existing relevant computational tools, the pseudospectral method proposed in [9] allows to perform a numerical bifurcation analysis of nonlinear delay equations, and is based on approximating the equations via collocation, meaning to study the resulting ODE system through numerical packages for stability and bifurcation analyses for ODEs (e.g., the widespread continuation-based software package MatCont [2]). This is what makes it an instance of what is called pragmatic-pseudospectral approach in [10], where the validity of the method has been substantiated through its application on a class of REs. ...
... This is what makes it an instance of what is called pragmatic-pseudospectral approach in [10], where the validity of the method has been substantiated through its application on a class of REs. However, while the convergence of the method in [9] concerning the approximation of equilibria and their stability has been also proved theoretically, that concerning periodic solutions was only conjectured, both in [9] and [10]. The pragmatic-pseudospectral approach is opposed to the so-called expert-pseudospectral one, first used in [10] for REs, which relies more directly on the Principle of Linearized Stability: it requires to compute (an approximation of) the sought solution and then linearize the system around it in order to investigate its local stability. ...
... This is what makes it an instance of what is called pragmatic-pseudospectral approach in [10], where the validity of the method has been substantiated through its application on a class of REs. However, while the convergence of the method in [9] concerning the approximation of equilibria and their stability has been also proved theoretically, that concerning periodic solutions was only conjectured, both in [9] and [10]. The pragmatic-pseudospectral approach is opposed to the so-called expert-pseudospectral one, first used in [10] for REs, which relies more directly on the Principle of Linearized Stability: it requires to compute (an approximation of) the sought solution and then linearize the system around it in order to investigate its local stability. ...
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We describe a piecewise collocation method for computing periodic solutions of renewal equations, obtained as an extension of the corresponding method in [K. Engelborghs et al., SIAM J. Sci. Comput., 22 (2001), pp. 1593--1609] for retarded functional differential equations. Then, we rigorously prove its convergence under the abstract framework proposed in [S. Maset, Numer. Math., 133 (2016), pp. 525--555], as previously done in [A.A. and D.B., SIAM J. Numer. Anal., 58 (2020), pp. 3010--3039] for general retarded functional differential equations. Finally, we show some numerical experiments on models from populations dynamics which confirm the order of convergence obtained theoretically, as well as a few applications in view of bifurcation analysis.
... In [4] the idea is launched to first reduce the infinite dimensional dynamical system corresponding to a delay equation to a finite dimensional one by pseudospectral approximation, and next use tools for ordinary differential equations (ODE) in order to perform a numerical bifurcation analysis. Several examples illustrate that this approach is promising (also see [5,6,7,8,9,10,11]). ...
... The fact that, for RE, the rule for extension does specify the value, rather than the derivative, makes its incorporation in the ODE less straightforward. In [4,5] an ad hoc method was employed: the value in the current time point was computed from the (approximate) history and the right-hand side of the RE by way of a numerical solver. The aim of the present paper is to introduce a much more natural and elegant alternative, which also improves the efficiency of the numerical method. ...
... An important advantage of the current method compared to the approach proposed in [4] is the remarkable reduction of computational costs in all the simulations considered here (see for instance Figure 5 and Table 1 below). The inversion of the nonlinear condition with a numerical solver is indeed the main bottleneck of the method in [4]. ...
Article
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.
... In [4] the idea is launched to first reduce the infinite dimensional dynamical system corresponding to a delay equation to a finite dimensional one by pseudospectral approximation, and next use tools for ordinary differential equations (ODE) in order to perform a numerical bifurcation analysis. Several examples illustrate that this approach is promising (also see [5,6,7,8,9,10,11]). ...
... The fact that, for RE, the rule for extension does specify the value, rather than the derivative, makes its incorporation in the ODE less straightforward. In [4,5] an ad hoc method was employed: the value in the current time point was computed from the (approximate) history and the right-hand side of the RE by way of a numerical solver. The aim of the present paper is to introduce a much more natural and elegant alternative, which also improves the efficiency of the numerical method. ...
... An important advantage of the current method compared to the approach proposed in [4] is the remarkable improvement in computation costs in all the simulations considered here (see for instance Figure 5 and Table 1 below). The inversion of the nonlinear condition with a numerical solver is indeed the main bottleneck of the method in [4]. ...
Preprint
Full-text available
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system is ten times more efficient than the one originally proposed in [Breda et al, SIAM Journal on Applied Dynamical Systems, 2016], as it avoids the numerical inversion of an algebraic equation.
... The resolution of these external problems is often the computational bottleneck when continuation techniques are applied to compute equilibria or periodic solutions, analyze their stability and detect relevant bifurcations, which are all common targets. This is the case of the approach proposed very recently in [6]. Its underlying idea is to reduce the original model to a system of ODEs, but when it comes to apply standard continuation tools for ODEs to such system the role of the external problems in determining the computational cost emerges rather clearly. ...
... We describe Daphnia in Section 3, highlighting the external problems and the technicalities involved with it. In this work we concentrate on the continuation of its equilibria as a starting point, thus the rest of Section 3 focuses on how to approach this problem, either with the general method proposed in [6] or with more specific tools as proposed in [32]. In Section 4 we explain our internal approach in full detail. ...
... The pseudo-arclength continuation is independent of the chosen parameterization. Often the original problem (1) is described with an explicit parameter λ ∈ R and hence it is rather formulated as G(u, λ) = 0 (6) for G : R n × R → R n and u ∈ R n the true unknown vector. In this case in (1) we set v = (u, λ), we talk about natural parameterization and we look for the solution branch (u(λ), λ) or, basically, for u(λ). ...
... We resume from the basic approach of identifying the correct delays by externally minimizing the reconstruction error of SINDy, first improving by using PO instead of BF or BO in terms of the number of calls to SINDy by the external optimizer. Then, in order to overcome the problem of dealing with intermediate multiple delays (i.e., those besides the maximum one), we propose a novel approach based on reducing the underlying DDE to an ODE via pseudospectral collocation, following Breda et al. (2016a). This leads to a pragmatic tool that asks to externally optimize only the maximum delay, thus resorting to univariate optimization rather than a demanding multivariate one necessary to optimize all the intermediate delays. ...
... We name this new approach P-SINDy, where P stands now for "Pragmatic", following the terms coined in Breda et al. (2016b). Indeed, opposite to an "expert" approach that deals directly with a DDE exploiting some prior knowledge, this new methodology relies on first reducing the DDE to a finite-dimensional system of approximating ODEs via pseudospectral collocation as originally proposed in Breda et al. (2016a). The reduction procedure is consolidated, easy to implement and acting on ODEs goes back to the standard SINDy approach of Brunton et al. (2016). ...
Conference Paper
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We present a pragmatic approach to the sparse identification of nonlinear dynamics for systems with discrete delays. It relies on approximating the underlying delay model with a system of ordinary differential equations via pseudospectral collocation. To minimize the reconstruction error, the new strategy avoids optimizing all possible multiple unknown delays, identifying only the maximum one. The computational burden is thus greatly reduced, improving the performance of recent implementations that work directly on the delay system.
... Following the ideas of [5,6,40], the method of [12] was based on the reformulation of the delay equation as an abstract differential equation, discretized via a pseudospectral collocation, yielding an ordinary differential equation (ODE) whose LEs were computed with the standard discrete QR (DQR) method [23,21]. The resulting method was experimentally effective, exhibiting the expected convergence properties with respect to the final time of the DQR method and providing results compatible with known properties of the test examples. ...
... We prove the thesis for (4) with (5), which implies the thesis for (1) with x s = ϕ through (3) and (2). Thanks to Theorems 2.1, 2.2, and 2.3, given (5), (4) has a unique solution x = x(·; 0, ϕ) on J := [0, τ ] given by (6); moreover, from the proof of Theorem 2.1, K is a Volterra kernel of type L 1 and of type L ∞ on J with a resolvent R of type L 1 and of type L ∞ on J, which in particular implies that |||R||| L ∞ (J 2 ) < +∞. Observe also that 7 ∥R∥ L 1 (J 2 ) ≤ τ |||R||| L ∞ (J 2 ) < +∞, which implies that R ∈ L 1 (J 2 , R d×d ). ...
... However, one could be interested in the error on x t = j −1 u. As for Euler's method (6), by applying j −1 to both sides of (24), we get ...
... Finally, we observe that since ExpRK-methods were originally formulated for ODEs, these methods can alternatively be applied to delay equations after having priorly discretized the latter to finite-dimensional systems of ODEs, e.g., via pseudospectral discretization [6]. ...
... We resume from the basic approach of identifying the correct delays by externally minimizing the reconstruction error of SINDy, first improving by using PO instead of BF or BO in terms of the number of calls to SINDy by the external optimizer. Then, in order to overcome the problem of dealing with intermediate multiple delays (i.e., those besides the maximum one), we propose a novel approach based on reducing the underlying DDE to an ODE via pseudospectral collocation, following Breda et al. (2016a). This leads to a pragmatic tool that asks to externally optimize only the maximum delay, thus resorting to univariate optimization rather than a demanding multivariate one necessary to optimize all the intermediate delays. ...
... We name this new approach P-SINDy, where P stands now for "Pragmatic", following the terms coined in Breda et al. (2016b). Indeed, opposite to an "expert" approach that deals directly with a DDE exploiting some prior knowledge, this new methodology relies on first reducing the DDE to a finite-dimensional system of approximating ODEs via pseudospectral collocation as originally proposed in Breda et al. (2016a). The reduction procedure is consolidated, easy to implement and acting on ODEs goes back to the standard SINDy approach of Brunton et al. (2016). ...
Preprint
We present a pragmatic approach to the sparse identification of nonlinear dynamics for systems with discrete delays. It relies on approximating the underlying delay model with a system of ordinary differential equations via pseudospectral collocation. To minimize the reconstruction error, the new strategy avoids optimizing all possible multiple unknown delays, identifying only the maximum one. The computational burden is thus greatly reduced, improving the performance of recent implementations that work directly on the delay system.
... The approximation scheme introduced here can be considered as a special case of the pseudo-spectral approximation of DDEs by ODEs proposed in [51]. Nevertheless, this spectral approximation presents no advantage over the low-order approximation (24), since the history segment is only differentiable once with Lipschitz continuous derivative whenever a discontinuity boundary (i.e. ...
... Because the studied system (14) with DFC (15) is piecewise linear in all four regimes, trajectory segments are all known analytically such that one could in principle determine periodic orbits as solutions of exact finite systems of nonlinear algebraic systems of equations. However, determining the linear stability of these periodic orbits requires numerical discretisation of the orbit segments (in our case pseudospectral discretisation (23) [51]), which we also use for bifurcation analysis. For all periodic orbits we detected and tracked, our numerical discretisation and event detection methods employ well-studied numerical methods fully within their regime of convergence, such that numerical discretisation errors are small compared to the uncertainties of our model. ...
Article
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The vibro-impact capsule system is a self-propelled mechanism that has abundant coexisting attractors and moves rectilinearly under periodic excitation when overcoming environmental resistance. In this paper, we study the control of coexisting attractors in this system by using a delayed feedback controller (DFC) with a constant delay. The aim of our control is to steer this complex system toward an attractor with preferable performance characteristics among multiple coexisting attractors, e.g., a periodically fast forward progression. For this purpose, we give an example of a feedback-controlled transition from a period-3 motion with low progression speed to a period-1 motion with high progression speed at the system parameters where both responses coexist. The effectiveness of this controller is investigated numerically by considering its convergence time and the required control energy input to achieve transition. We combine pseudo-spectral approximation of the delay, event detection for the discontinuities and path-following (continuation) techniques for non-smooth delay dynamical systems to carry out bifurcation analysis. We systematically study the dynamical performance of the controlled system when varying its control gain and delay time. Our numerical simulations show the effectiveness of DFC under a wide range of system parameters. We find that the desired period-1 motion is achievable in a range of control delays between a period-doubling and a grazing bifurcation. Therefore, two-parameter continuation of these two bifurcations with respect to the control delay and control gain is conducted to identify the delay-gain parameter region where the period-1 motion is stable. The findings of this work can be used for tuning control parameters in experiments, and similar analysis can be carried out for other non-smooth dynamical systems with a constant delay term.
... The approximation scheme introduced here can be considered as a special case of the pseudo-spectral approximation of DDEs by ODEs proposed in [51]. Nevertheless, this spectral approximation presents no advantage over the low-order approximation (24), since the history segment is only differentiable once with Lipschitz continuous derivative whenever a discontinuity boundary (i.e. ...
... Because the studied system (14) with DFC (15) is piecewise linear in all four regimes, trajectory segments are all known analytically such that one could in principle determine periodic orbits as solutions of exact finite systems of nonlinear algebraic systems of equations. However, determining the linear stability of these periodic orbits requires numerical discretisation of the orbit segments (in our case pseudo-spectral disretisation (23) [51]), which we also use for bifurcation analysis. For all periodic orbits we detected and tracked, our numerical discretisation and event detection methods employ well-studied numerical methods fully within their regime of convergence, such that numerical discretization errors are small compared to the uncertainties fo our model. ...
Preprint
Full-text available
The vibro-impact capsule system is a self-propelled mechanism that has abundant coexisting attractors and moves rectilinearly under periodic excitation when overcoming environmental resistance. In this paper, we study the control of coexisting attractors in this system by using a delayed feedback controller (DFC) with a constant delay. The aim of our control is to steer this complex system toward an attractor with preferable performance characteristics among multiple coexisting attractors, e.g., a periodically fast forward progression. For this purpose, we give an example of a feedback-controlled transition from a period-3 motion with low progression speed to a period-1 motion with high progression speed at the system parameters where both responses coexist. The effectiveness of this controller is investigated numerically by considering its convergence time and the required control energy input to achieve transition. We combine pseudo-spectral approximation of the delay, event detection for the discontinuities and path-following (continuation) techniques for non-smooth delay dynamical systems to carry out bifurcation analysis. We systematically study the dynamical performance of the controlled system when varying its control gain and delay time. Our numerical simulations show the effectiveness of DFC under a wide range of system parameters. We find that the desired period-1 motion is achievable in a range of control delays between a period-doubling and a grazing bifurcation. Therefore, two-parameter continuation of these two bifurcations with respect to the control delay and control gain is conducted to identify the delay-gain parameter region where the period-1 motion is stable. The findings of this work can be used for tuning control parameters in experiments, and similar analysis can be carried out for other non-smooth dynamical systems with a constant delay term.
... The approximation of the history of q by u i on the evenly spaced grid of τ i on [0, 1] with order M = 2 is a special case of the pseudospectral approximation of DDEs by ODEs, as used for bifurcation analysis in MATCONT by [48]. The spectral approximation from [48] has no advantage over the low-order approximation (19) for our problem, because the history segment is only differentiable once with Lipschitz continuous derivative, whenever the contact threshold x = e has been crossed during the previous time interval of length τ d . ...
... The approximation of the history of q by u i on the evenly spaced grid of τ i on [0, 1] with order M = 2 is a special case of the pseudospectral approximation of DDEs by ODEs, as used for bifurcation analysis in MATCONT by [48]. The spectral approximation from [48] has no advantage over the low-order approximation (19) for our problem, because the history segment is only differentiable once with Lipschitz continuous derivative, whenever the contact threshold x = e has been crossed during the previous time interval of length τ d . Thus, the proposed second-order approximation has the most suitable order M. ...
Article
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Grazing events may create coexisting attractors and cause complex dynamics in piecewise-smooth dynamical systems. This paper studies control of grazing-induced multistability in a soft impacting oscillator by using time-delayed feedback control. The control switches from one of the coexisting attractors to a desired one to suppress complex dynamics near grazing events. We use path-following (continuation) techniques for non-smooth dynamical systems to investigate robustness of the controller and the parameter dependence of the controlled system. In particular, several newly developed computational methods have been used, including a numerical method for analysing non-smooth delay equations and a method for calculating the Lyapunov exponents and the grazing point estimation. Numerical simulations demonstrate that the delayed feedback controller is effective, and a proper selection of the control gain and delay time can simplify the complex dynamics of the system near grazing.
... In [6] pseudospectral approximation is advocated as a promising approach to achieve exactly this. The aim of the present paper is to make a next step by verifying that the generic Hopf bifurcation in DDE is faithfully captured by Hopf bifurcations in the approximating ODE systems. ...
... In the following we take a famous example from mathematical biology, namely the 'Nicholson's blowflies' equation, as a testing ground to illustrate some features of the approach. However, we remark that the methodology presented here (pseudospectral approximation combined with software for bifurcation analysis of ODE) can be applied in a much more general setting: it is indeed a promising procedure to study differential equations with distributed, state-dependent, and even infinite delays [6,22,19], as well as nonlinear renewal equations [7] and first order partial differential equations [31]. The advantage of considering Nicholson's blowflies equation in this context is due to the fact that explicit comparisons are possible, both with analytically computed quantities and with alternative numerical approximations, as will become clear later on. ...
... Among the main references for delay differential equations is DDE-Biftool [1,22], where the computation of periodic solutions is based on the work [20], extending the classic piecewise orthogonal collation methods already used for the case of ordinary differential equations (see, e.g., [4,5]). But when it comes to dealing with more complicated systems, involving also renewal or Volterra integral and integro-differential equations, the lack is evident [12,13]. ...
... Nevertheless, the present work offers a first, solid background to start elaborating a succeeding strategy towards the proof of convergence. The authors plan to make this effort in the immediate future, also to substantiate the encouraging experimental results already obtained by extending the piecewise collocation to renewal equations, even coupled to RFDEs (for the target class of realistic models we have in mind see, e.g., [12,14,36]). ...
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We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math. (2016) 133(3):525-555], [S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2771--2793] and [S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2794--2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable of such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical being it directly linked to the course of time. Finally, we prove that the finite element method is convergent, while limit ourselves to comment on the unfeasibility of this approach as far as the spectral element method is concerned.
... When an exact reduction is not available, the study of relevant approximations through ODEs becomes central. In this sense, Breda, et al. [9] proposes a general numerical approach to address a broad class of functional problems. Nevertheless, in the early eighties the subject was already alive. ...
... Examples can be multimodal distributions, or distributions with unconstrained variance (recall Eq (2.9)), as well as laws of power type or Gaussian-derived kernels. Of course, also going beyond the stability of the endemic equilibrium should be considered, and in this regard we remark that efficient techniques are already available to analyze the stability of periodic behaviors of systems with delays [39,40], as well as a more complete bifurcation analysis [9]. ...
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A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents' decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.
... In particular, center manifold and normal form methods allow for the local reduction of the DDE to an ODE describing the dynamics near a bifurcation point of interest. Moreover, advanced numerical tools for simulation and bifurcation analysis of DDEs with constant delays have become available in recent years [4,5,7,17,47,72,77]. These theoretical and numerical tools have been applied very successfully in many application areas, including those mentioned above. ...
Preprint
We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincar\'e section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.
... Another interesting extension of the present work would be to explore different exponential methods other than expRK, such as exponential Rosenbrock methods, Magnus methods, linear multistep methods or general linear methods. Finally, we observe that, since ExpRK-methods were originally formulated for ODEs, these methods can alternatively be applied to delay equations after having priorly discretized the latter to finite-dimensional systems of ODEs, e.g., via pseudospectral discretization [6]. ...
Preprint
Exponential Runge-Kutta methods for semilinear ordinary differential equations can be extended to abstract differential equations, defined on Banach spaces. Thanks to the sun-star theory, both delay differential equations and renewal equations can be recast as abstract differential equations, which motivates the present work. The result is a general approach that allows us to define the methods explicitly and analyze their convergence properties in a unifying way.
... Knowing the boundaries from equation (2.18) does not reveal the stability of the system in the resulting parameter-domains. Therefore, the different parameter-domains are labelled as stabilizable/unstable by semi-discretization [33], a method of generating a set of (discretized) equations from the underlying infinite dimensional delay equation using Chebyshev polynomials [34,35]. Thus, the infinite dimensional time-delay system is approximated by a large-dimensional system of ordinary differential equations. ...
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Real-time hybrid testing is a method in which a substructure of the system is realized experimentally and the rest numerically. The two parts interact in real time to emulate the dynamics of the full system. Such experiments, however, are often difficult to realize as the actuators and sensors, needed to ensure compatibility and force–equilibrium conditions at the interface, can seriously affect the predicted dynamics of the system and result in stability and fidelity issues. The traditional approach of using feedback control to overcome the additional unwanted dynamics is challenging due to the presence of an outer feedback loop, passing interface displacements or forces to the numerical substructure. We, therefore, advocate for an alternative approach, removing the problematic interface dynamics with an iterative scheme to minimize interface errors, thus, capturing the response of the true assembly. The technique is examined by hybrid testing of a bench-top four-storey building with different interface configurations, where using conventional hybrid measurement techniques is very challenging. A case where the physical part exhibits nonlinear restoring force characteristics is also considered. These tests show that the iterative approach is capable of handling even scenarios which are theoretically infeasible with feedback control.
... As shown in [36], for r = K = γ = 1, a = 3 and a max = 4, a Hopf bifurcation occurs when β ≈ 3.0162. Starting from a periodic solution at β = 4 computed using MatCont [28] on the pseudospectral reduction to ODEs of (4.3) [37], the branch of periodic orbits is continued up to β = 5. Given the absence of an exact expression of the true solution, unlike the case (4.1), the error is computed with respect to a reference solution which is in turn computed using L = 1000 and m = 4. Figure 4.3 confirms the O(h m ) behavior when using Chebyshev collocation nodes and the O(h m+1 ) behavior when using the Gauss-Legendre nodes. ...
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We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and provide a theoretical estimate of the error. We conclude with some numerical experiments to further support the theoretical results.
... For the analysis of local stability of equilibria, pseudospectral methods have been widely used both for delay equations [10][11][12][13] and for PDE population models with one structuring variable [14][15][16]. The main advantage of pseudospectral methods is their typical spectral accuracy, by which the order of convergence of the approximation error increases with the regularity of the approximated function. ...
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The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.
... This restriction is useful as these Erlang distributed DDEs can then be reduced to an equivalent system of ordinary differential equations (ODEs) through the linear chain technique [MacDonald, 1978, Vogel, 1961. A major impediment to the implementation of the more general gamma distributed DDE is the lack of appropriate numerical techniques for their simulation [Breda et al., 2016, Diekmann et al., 20182020b]. Here, we address this impediment in two distinct manners, first, by implementing a functional continuous Runge-Kutta (FCRK) method to simulate (1.1) and, second, by deriving a finite dimensional approximation of (1.1) that is more accurate than the common Erlang approximation. ...
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Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation, but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.
... Thus numerical methods should be a powerful tool for exploring the behavior of the models, tracing asymptotic stability, Hopf bifurcations and possible chaos. At present, there are a lot of results on the numerical approximations to age-structured population models (see [1,2,6,7,8,9,3,13,22,21]), of which the numerical analysis for linearized problems is concentrate on the discretization of the unbounded linear operators based on the operator semigroup. The idea is to turn the characteristic roots approximation problem into a corresponding eigenvalue problem for a suitable matrix. ...
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In this paper, we consider a numerical threshold of a linearly implicit Euler method for a nonlinear infection-age SIR model. It is shown that the method shares the equilibria and basic reproduction number \begin{document}R0 R_0 \end{document} of age-independent SIR models for any stepsize. Namely, the disease-free equilibrium is globally stable for numerical processes when \begin{document}R0<1 R_0<1 \end{document} and the underlying endemic equilibrium is globally stable for numerical processes when \begin{document}R0>1 R_0>1 \end{document}. A natural extension to nonlinear infection-age models is presented with an initial mortality rate and the numerical thresholds, i.e., numerical basic reproduction numbers \begin{document}Rh R^h \end{document}, are presented according to the infinite Leslie matrix. Although the numerical basic reproduction numbers \begin{document}Rh R^h \end{document} are not quadrature approximations to the exact threshold \begin{document}R0 R_0 \end{document}, the disease-free equilibrium is locally stable for numerical processes whenever \begin{document}Rh<1 R^h<1 \end{document}. Moreover, a unique numerical endemic equilibrium exists for \begin{document}Rh>1 R^h>1 \end{document}, which is locally stable for numerical processes. It is much more important that both the numerical thresholds and numerical endemic equilibria converge to the exact ones with accuracy of order 1. Therefore, the local dynamical behaviors of nonlinear infection-age models are visually displayed by the numerical processes. Finally, numerical applications to the influenza models are shown to illustrate our results.
... For the analysis of local stability of equilibria, pseudospectral methods have been widely used both for delay equations [6,7,9,14] and for PDE population models with one structuring variable [4,10,34]. The main advantage of pseudospectral methods is their typical spectral accuracy, by which the order of convergence of the approximation error increases with the regularity of the approximated function. ...
Preprint
Full-text available
The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. We propose an equivalent reformulation of the problem in the space of absolutely continuous functions in the sense of Carath\'eodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.
... This exploits the prominent advantage of time-delay systems that "space" and "time" can intermingle, and delay systems are known to have rich spatio-temporal properties 32,[44][45][46] . This work significantly extends this spatio-temporal equivalence and its application while allowing the evaluation of neural networks with the tools of delay systems analysis 26,30,47,48 . In particular, we show how the transition from the time-continuous view of the physical system, i.e., the delay-differential equation, to the time-discrete feed-forward DNN can be made. ...
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Deep neural networks are among the most widely applied machine learning tools showing outstanding performance in a broad range of tasks. We present a method for folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops. This single-neuron deep neural network comprises only a single nonlinearity and appropriately adjusted modulations of the feedback signals. The network states emerge in time as a temporal unfolding of the neuron’s dynamics. By adjusting the feedback-modulation within the loops, we adapt the network’s connection weights. These connection weights are determined via a back-propagation algorithm, where both the delay-induced and local network connections must be taken into account. Our approach can fully represent standard Deep Neural Networks (DNN), encompasses sparse DNNs, and extends the DNN concept toward dynamical systems implementations. The new method, which we call Folded-in-time DNN (Fit-DNN), exhibits promising performance in a set of benchmark tasks.
... These Erlang distributed DDEs can then be reduced to an equivalent system of ordinary differential equations (ODEs) through the linear chain technique [Câmara De Souza et al., 2018;Vogel, 1961]. A major impediment to the implementation of the more general gamma distributed DDE is the lack of appropriate numerical techniques for their simulation [Breda et al., 2016;Diekmann et al., 2018Diekmann et al., , 2020b. Here, we address this by developing a functionally continuous Runge-Kutta (FCRK) method to simulate (1.1) and deriving a finite dimensional approximation that is more accurate than the common Erlang approximation. ...
Preprint
Full-text available
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations. In this work, we develop a functionally continuous Runge-Kutta method to numerically integrate the gamma distributed DDE and perform numerical tests to confirm the accuracy of the numerical method. As the functionally continuous Runge-Kutta method is not available in most scientific software packages, we then derive hypoexponential approximations of the gamma distributed DDE. Using our numerical method, we show that while using the common Erlang approximation can produce solutions that are qualitatively different from the underlying gamma distributed DDE, our hypoexponential approximations do not have this limitation. Finally, we implement our hypoexponential approximations to perform statistical inference on synthetic epidemiological data.
... This work extends this spatio-temporal equivalence and its application significantly. Moreover, such an equivalence allows for the evaluation of neural networks with the tools of delay systems analysis [26,30,48,49]. ...
Preprint
Deep neural networks are among the most widely applied machine learning tools showing outstanding performance in a broad range of tasks. We present a method for folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops. This single-neuron deep neural network comprises only a single nonlinearity and appropriately adjusted modulations of the feedback signals. The network states emerge in time as a temporal unfolding of the neuron's dynamics. By adjusting the feedback-modulation within the loops, we adapt the network's connection weights. These connection weights are determined via a modified back-propagation algorithm that we designed for such types of networks. Our approach fully recovers standard Deep Neural Networks (DNN), encompasses sparse DNNs, and extends the DNN concept toward dynamical systems implementations. The new method, which we call Folded-in-time DNN (Fit-DNN), exhibits promising performance in a set of benchmark tasks.
... The bivariate Lagrange interpolation polynomial interpolates f k ( , ) at selected points ( i , j ) in both the and directions, for i = 0, 1, 2, … , N and j = 0, 1, 2, … , N . These selected grid points are given by (4) and are called Chebyshev-Gauss-Lobatto points. 26,27 The function  i ( ) is the characteristic Lagrange cardinal polynomial based on the Chebyshev-Gauss-Lobatto grid points 26,27 ...
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In this work, three pseudospectral methods, namely bivariate spectral relaxation method, spectral local linearization, and bivariate spectral quasilinearization method are analyzed for steady nonlinear nonsimilar boundary layer partial differential equations. Their accuracy and general performance are discussed. A system of three nonlinear coupled partial differential equations that models an unsteady three-dimensional MHD-boundary-layer flow due to an impulsive motion of a stretching surface is used to demonstrate the accuracy, convergence, and general performance of the three pseudospectral methods. The general comparison of the three pseudospectral methods is presented in graphical and tabular forms. Computational times of each method is presented in tabular form, showing the skin friction and Nusselt number.
... Finally, we wish to mention that the present research originated from [12], where an embryonic study of collocation techniques for the numerical computation of R 0 is proposed (also the work [6] mentioned above is inspired from [12]). Moreover, let us also note that pseudospectral techniques have become a reference tool in the numerical treatment of infinite-dimensional dynamical systems in view of either stability and bifurcation analyses in the context of population dynamics, see, e.g., [13] or [14] and the references therein. ...
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As widely known, the basic reproduction number plays a key role in weighing birth/infection and death/recovery processes in several models of population dynamics. In this general setting, its characterization as the spectral radius of next generation operators is rather elegant, but simultaneously poses serious obstacles to its practical determination. In this work we address the problem numerically by reducing the relevant operators to matrices through a pseudospectral collocation, eventually computing the sought quantity by solving finite-dimensional eigenvalue problems. The approach is illustrated for two classes of models, respectively from ecology and epidemiology. Several numerical tests demonstrate experimentally important features of the method, like fast convergence and influence of the smoothness of the models’ coefficients. Examples of robust analysis of instances of specific models are also presented to show potentialities and ease of application.
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• A complete characterization of the bifurcation diagram for the Suarez-Schopf ENSO model is provided. It is obtained by using Galerkin–Koornwinder ODE approximations of DDEs combined with our analytic formulas for invariant manifolds. • In particular, Saddle-Node bifurcation of periodic Orbits (SNO) and homoclinic bifurcation are identified. • Stochastic tipping solution paths through SNO critical transition exhibit episodes that resemble Nino 3.4 index episodes. The interactions between noise and Unstable Periodic Orbits (UPOs) play a key role for this to happens.
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We present SSD – Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend the frequency-limited balanced reduction to delay systems first time, propose a computational algorithm and give its implementation. We collected two sets of benchmark problems on H2 norm computation and model reduction. SSD is publicly available in GitHub∗. Our reproducible paper and two benchmark collections are shared as executable notebooks. ∗https://github.com/gumussoysuat/ssd
Chapter
Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the infinitesimal generator of the semigroup of solution operators associated to the system. The focus is on both local stability of equilibria and general bifurcation analysis of nonlinear problems, for either delay differential and renewal equations.
Chapter
Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the evolution family associated to linear problems. The focus is on local stability of either equilibria and periodic orbits as well as on generic nonautonomous systems, for either delay differential and renewal equations.
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We present SSD, Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend the frequency-limited balanced reduction to delay systems first time, propose a computational algorithm and give its implementation. We collected two sets of benchmark problems on H2 norm computation and model reduction. SSD is publicly available in GitHub at https://github.com/gumussoysuat/ssd. Our reproducible paper and two benchmark collections are shared as executable notebooks.
Chapter
Stability and bifurcation analyses of delay equations represent fundamental challenges in many applications, emerging in important fields like, e.g., control theory and population dynamics. Exact approaches are usually unattainable due to the infinite dimension of the associated state space, so that resorting to numerical methods is unavoidable. This work is a survey on the use of collocation techniques to address the problems above from the numerical standpoint. As such it summarizes the main contributions in this context of the research of the authors in the last 15 years or so. Methods for the linearized stability analysis of equilibria and periodic solutions, as well as for the bifurcation analysis of general nonlinear systems are illustrated, together with an essential overview of relevant problems and potential developments.
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The growth of a population subject to maturation delay is mod-eled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and suffcient conditions are provided by analyzing the relevant char-acteristic equations. It is shown that for any choice of parameter values for which the discrete delay model presents stability switches there exists a max-imum delay variance beyond which no switch occurs for the continuous delay model: the delay variance has a stabilizing effect. Moreover, it is illustrated how, in the presence of switches, the unstable delay domain is as larger as lower is the ratio between the juveniles and the adults mortality rates.
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In [D. Breda, S. Maset, and R. Vermiglio, IMA J. Numer. Anal., 24 (2004), pp. 1--19.] and [D. Breda, The Infinitesimal Generator Approach for the Computation of Characteristic Roots for Delay Differential Equations Using BDF Methods, Research report UDMI RR17/2002, Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Udine, Italy, 2002.] the authors proposed to compute the characteristic roots of delay differential equations (DDEs) with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by Runge--Kutta (RK) and linear multistep (LMS) methods, respectively. In this work the same approach is proposed in a new version based on pseudospectral differencing techniques. We prove the "spectral accuracy" convergence behavior typical of pseudospectral schemes, as also illustrated by some numerical experiments.
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We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023-1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman-Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254-274, 1984; de Roos et al. in J Math Biol 28:609-643, 1990) and a model introduced by Gurney-Nisbet (Theor Popul Biol 28:150-180, 1985) and Jones et al. (J Math Anal Appl 135:354-368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.
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In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.
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Full-text available
First-order nonlinear differential-delay equations describing physiological control systems are studied. The equations display a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions. These results are discussed in relation to dynamical respiratory and hematopoietic diseases.
Thesis
In this dissertation, delay differential equation models from mathematical biology are studied, focusing on population ecology. In order to even begin a study of such models, one must be able to determine the linear stability of their steady states, a task made more difficult by their infinite dimensional nature. In Chapter 2, I have developed a method of reducing such questions to the problem of determining the existence or otherwise of positive real roots of a real polynomial. The method of Sturm sequences is then used to make this determination. In particular, I developed general necessary and sufficient conditions for the existence of delay-induced instability in systems of two or three first order delay differential equations. These conditions depend only on the parameters of the system, and can be easily checked, avoiding the necessity of simulations in these cases. With this tool in hand, I begin studying delay differential equations for single species, extending previously obtained results about the existence of periodic solutions, and developing a proof for a previously unproven case. Due to the infinite dimensional nature of these equations, it is quite difficult to prove the existence of periodic solutions. Nonetheless, knowledge of their existence is essential if one is to make decisions about the suitability of such models to biological situations. Furthermore, I explore the effect of delay-dependent parameters in these models, a feature whose use is becoming more common in the mathematical biology literature. Finally, I look at a delayed predator-prey model with delay dependent parameters. Although I was unable to obtain a complete proof for the existence of periodic solutions, significant progress has been made in understanding the nature of this system, and it is hoped that future work will continue to clarify this picture. This model seems to display chaotic behavior for certain parameter regimes, and thus the existence of periodic solutions may be precluded in the most general case.
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The stability of pseudospectral-Chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients. The choice of collocation points is discussed. Numerical examples are given for the case of variable coefficient hyperbolic equations.
Article
Most organisms show substantial changes in size or morphology after they become independent of their parents and have to find their own food. Furthermore, the rate at which these changes occur generally depends on the amount of food they ingest. In this book, André de Roos and Lennart Persson advance a synthetic and individual-based theory of the effects of this plastic ontogenetic development on the dynamics of populations and communities. De Roos and Persson show how the effects of ontogenetic development on ecological dynamics critically depend on the efficiency with which differently sized individuals convert food into new biomass. Differences in this efficiency--or ontogenetic asymmetry--lead to bottlenecks in and thus population regulation by either maturation or reproduction. De Roos and Persson investigate the community consequences of these bottlenecks for trophic configurations that vary in the number and type of interacting species and in the degree of ontogenetic niche shifts exhibited by their individuals. They also demonstrate how insights into the effects of maturation and reproduction limitation on community equilibrium carry over to the dynamics of size-structured populations and give rise to different types of cohort-driven cycles.
Article
This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.
Book
1 Introduction.-The Simplest Delay Equation.-Delayed Negative Feedback: A Warm-Up.- Existence of Solutions.- Linear Systems and Linearization.- Semidynamical Systems and Delay Equations.- Hopf Bifurcation.- Distributed Delay Equations and the Linear Chain Trick.- Phage and Bacteria in a Chemostat.-References.- Index.
Article
Spectral methods involve seeking the solution to a differential equation in terms of a series of known, smooth functions. They have recently emerged as a viable alternative to finite difference and finite element methods for the numerical solution of partial differential equations. The key recent advance was the development of transform methods for the efficient implementation of spectral equations. Spectral methods have proved particularly useful in numerical fluid dynamics where large spectral hydrodynamics codes are now regularly used to study turbulence and transition, numerical weather prediction, and ocean dynamics. In this monograph, we discuss the formulation and analysis of spectral methods. It turns out that several features of this analysis involve interesting extensions of the classical numerical analysis of initial value problems. This monograph is based on part of a series of lectures presented by one of us (S.A.O.) at the NSF—CBMS Regional Conference held at Old Dominion University from August 2–6, 1976. This conference was supported by the National Science Foundation. We should like to thank our colleagues M. Deville, M. Dubiner, M. Gunzburger, B. Gustaffson, D. Haidvogel, M. Israeli, and J. Ortega for helpful discussions. We are grateful to E. Cohen, A. Patera, and K. Pitman for their assistance in preparing graphs and tables. Some calculations were performed at the Computing Facility of the National Center for Atmospheric Research which is supported by the National Science Foundation. One of us (D.G.) would like to acknowledge support by the National Aeronautics and Space Administration while in residence at ICASE, NASA Langley Research Center, Hampton, Virginia. Both authors would like to acknowledge support by the Fluid Dynamics Branch of the Office of Naval Research and the Atmospheric Sciences Section of the National Science Foundation. Hampton, Virginia, Cambridge, Massachusetts September 1977
Article
The aim of this book is to teach you the essentials of spectral collocation methods with the aid of 40 short MATLAB® programs, or “M-files.”* The programs are available online at http://www.comlab.ox.ac.uk/oucl/work/nick.trefethen, and you will run them and modify them to solve all kinds of ordinary and partial differential equations (ODEs and PDEs) connected with problems in fluid mechanics, quantum mechanics, vibrations, linear and nonlinear waves, complex analysis, and other fields. Concerning prerequisites, it is assumed that the words just written have meaning for you, that you have some knowledge of numerical methods, and that you already know MATLAB. If you like computing and numerical mathematics, you will enjoy working through this book, whether alone or in the classroom—and if you learn a few new tricks of MATLAB along the way, that's OK too!
Article
This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.
Article
An approach for the numerical solution of linear delay differential equations, different from the classical step-by-step integration, was presented in (Numer. Math. 84 (2000) 351). The problem is restated as an abstract Cauchy problem (or as the advection equation with a particular nonstandard boundary condition) and then, by using a scheme of order one, it is discretized as a system of ordinary differential equations by the method of lines. In this paper we introduce a class of related schemes of arbitrarily high order and we then extend the approach to general retarded functional differential equations. An analysis of convergence, and of asymptotic stability when the numerical schemes are applied to the complex scalar equation y′(t)=ay(t)+by(t−1), is provided.
Article
Using perturbation theory for adjoint semigroups (a modification of sun-star calculus) we prove, in the case of infinite delay, the principle of linearized stability for nonlinear renewal equations, delay-differential equations and coupled systems of these two types of equations. Our results extend those of Diekmann et al. (1995) [13] and Diekmann et al. (2007) [14] to the case of infinite delay.
Chapter
This chapter is intended as a gentle introduction to models of physiologically structured populations, as developed by Metz and Diekmann (1986) (hereafter, PSP models; the origin of the adjective “physiologically structured” is described in Section 2 after some necessary definitions). You will need a basic knowledge of mathematical modeling but no familiarity with PSP models or the ensuing partial differential equations. My aim is to equip you with enough skills to be able to use moderately complex PSP models for specific biological applications. Hence, I emphasize the formulation of the models, the biological interpretation of the equations, and the tools for studying the models. The mathematical background of the modeling framework and the justification of the equations are discussed only when necessary for a better understanding of the biological aspects of the models.
Article
By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.
Article
Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.
Article
We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new.
Article
We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.
Article
The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.
Embedding of the Numerical Solution of a DDE into the Numerical Solution of a System of ODEs
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