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Numerical Continuation, And Computation Of Normal Forms
Abstract and Figures
Contents 1 Introduction 3 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Continuation of Equilibria and Cycles 4 2.1 Parameter continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Pseudo-arclength continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The bordering algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Cycle continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Locating Codimension-1 Bifurcations 9 3.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Locating codimension-1 equilibrium bifurcations . . . . . . . . . . . . . . . . . . 9 3.2.1 Folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.2 Hopf points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Locatin
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... , n). Of these n multipliers, one is the trivial multiplier, λ n = 1, which arises due to solution periodicity [21]. ...
... (C. 21) In 2D, the transformation to polar co-ordinates is given by dr = rdθdr, and k · r = kr cos θ, where |k| = k and |r| = r. Then the 2D Fourier transform is given by appearing in the expression for g(k; σ, ρ) must be evaluated. ...
... 21 Exotic patterns in the dynamic threshold model . . . 134 ...
The brain is the central hub regulating thought, memory, vision, and many other processes occurring within the body. Neural information transmission occurs through the firing of billions of connected neurons, giving rise to a rich variety of complex patterning. Mathematical models are used alongside direct experimental approaches in understanding the underlying mechanisms at play which drive neural activity, and ultimately, in understanding how the brain works. This thesis focuses on network and continuum models of neural activity, and computational methods used in understanding the rich patterning that arises due to the interplay between non-local coupling and local dynamics. It advances the understanding of patterning in both cortical and sub-cortical domains by utilising the neural field framework in the modelling and analysis of thalamic tissue – where cellular currents are important in shaping the tissue firing response through the post-inhibitory rebound phenomenon – and of cortical tissue. The rich variety of patterning exhibited by different neural field models is demonstrated through a mixture of direct numerical simulation, as well as via a numerical continuation approach and an analytical study of patterned states such as synchrony, spatially extended periodic orbits, bumps, and travelling waves. Linear instability theory about these patterns is developed and used to predict the points at which solutions destabilise and alternative emergent patterns arise. Models of thalamic tissue often exhibit lurching waves, where activity travels across the domain in a saltatory manner. Here, a direct mechanism, showing the birth of lurching waves at a Neimark-Sacker-type instability of the spatially synchronous periodic orbit, is presented. The construction and stability analyses carried out in this thesis employ techniques from non-smooth dynamical systems (such as saltation methods) to treat the Heaviside nature of models. This is often coupled with an Evans function approach to determine the linear stability of patterned states. With the ever-increasing complexity of neural models that are being studied, there is a need to develop ways of systematically studying the non-trivial patterns they exhibit. Computational continuation methods are developed, allowing for such a study of periodic solutions and their stability across different parameter regimes, through the use of Newton-Krylov solvers. These techniques are complementary to those outlined above. Using these methods, the relationship between the speed of synaptic transmission and the emergent properties of periodic and travelling periodic patterns such as standing waves and travelling breathers is studied. Many different dynamical systems models of physical phenomena are amenable to analysis using these general computational methods (as long as they have the property that they are sufficiently smooth), and as such, their domain of applicability extends beyond the realm of mathematical neuroscience.
... (4.14)-(4. 16). @(X 1 f ⇢ ⇢ ...
... A method for predicting the next member of the family is necessary -this involves a continuation algorithm. Two forms of continuation are used in this dissertation: natural parameter continuation (NPC) [16], and pseudo-arclength continuation (PAC) [5,19,113]. ...
... Pseudo-arclength continuation (PAC) is a slightly more complex option for generating a next guess along a periodic orbit family, but it is very robust [19,113,75,16,5]. In the PAC process, the direction tangent to the current family is determined via calculation of the orthonormal basis for the null space of DF , represented asn DF , and a small step, whose size is determined by a user-specified parameter s, is taken in this direction. ...
Outer planets such as Jupiter and Saturn are ripe with moons that contain all the ingredients for life as we know it (water, organic molecules, energy, and time), however, these destinations are very difficult to reach, especially land on. Located deep within the gravitational wells of their host planets and lacking sufficient atmospheres to allow for the use of parachutes, these moons are a unique challenge in mission design. Any spacecraft intending to land on one of these moons must approach with a low relative energy, but motion at low energies in the three-body problem is chaotic and dominated by local dynamical structures. The field of applied dynamical systems theory which studies the utility of these structures, such as periodic orbits and invariant manifolds, has grown much over recent decades, but there has been a limited focus on using these tools to help design landing trajectories.
In this dissertation, a framework for understanding low-energy landing trajectories to the secondary body in the three-body problem is established. Bounds of possible motion and their relationship to nearby dynamical structures are explored. The utility of using the invariant mani- folds of common families of periodic orbits is analyzed, and maps of feasible landing locations are generated. A study of periodic orbit bifurcations results in a publicly accessible database of over 70 families near Jupiter’s moon Europa, and over 400 bifurcations points are identified. Certain members from the various families are labeled as possible “abort solutions” due to their desirable stability and landing-opportunity repeatability. Many of the identified solutions are studied under the lens of a perturbed three-body problem that accounts for zonal harmonics. The z-axis symmetry of these perturbations allows the problem to remain autonomous, making them ideal for higher-fidelity study of the three-body problem without a significant increase in model complexity.
... The response is reciprocal if and only if R = 0. We use and develop numerical continuation techniques through the software package AUTO [19,20] to compute the steady-state response of Eq. (1) as a family of periodic orbits [21,22] that satisfies a suitable two-point boundary value problem. This formulation allows for simultaneous computation of all the relevant parameters and norms, including the nonreciprocal phase shift, thus eliminating the need for lengthy post-processing of data. ...
Circumventing the reciprocity invariance has posed an interesting challenge in the design of modern devices for wave engineering. In passive devices, operating the device in the nonlinear response regime is a common means for realizing nonreciprocity. Because mirror-symmetric systems are trivially reciprocal, breaking the mirror symmetry is a necessary requirement for nonreciprocal dynamics to exist in nonlinear systems. However, the response of an asymmetric nonlinear system is not necessarily nonreciprocal. In this work, we report on the existence of stable, steady-state nonlinear reciprocal dynamics in coupled asymmetric systems subject to external harmonic excitation. We restore reciprocity in the asymmetric system by tuning two symmetry-breaking parameters simultaneously. We identify response regimes in the vicinity of the primary resonances of the system where the steady-state left-to-right transmission characteristics are identical to the right-to-left characteristics in terms of frequency, amplitude and phase. We interpret these regimes of reciprocal dynamics in the context of phase nonreciprocity, wherein incident waves undergo a nonreciprocal phase shift depending on their direction of travel. We hope these findings help design devices with new functionalities for controlling and steering of elastic waves.
... In this paper we adopt a complementary point of view. Thanks to advances in two-point boundary value problem (2PBVP) continuation techniques [38], we compute relevant invariant manifolds [20] and their interactions in a three-dimensional vector field introduced by Sandstede [34], which exhibits a homoclinic flip bifurcation of case C. In this way, we are able to present the overall dynamics near this bifurcation, beyond considering intersection sets on a Poincaré section. More specifically, our focus is on the role of global invariant manifolds of saddle equilibria and saddle periodic orbits in the organization of the three-dimensional phase space. ...
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward in understanding the generic cases of homoclinic flip bifurcations, which started with the study of the simpler cases \textbf{A} and \textbf{B}. In a three-dimensional vector field due to Sandstede, we compute relevant bifurcation curves in the two-parameter bifurcation diagram near the central codimension-two bifurcation in unprecedented detail. We present representative images of invariant manifolds, computed with a boundary value problem setup, both in phase space and as intersection sets with a suitable sphere. In this way, we are able to identify infinitely many cascades of homoclinic bifurcations that accumulate on specific codimension-one heteroclinic bifurcations between an equilibrium and various saddle periodic orbits. Our computations confirm what is known from theory but also show the existence of bifurcation phenomena that were not considered before. Specifically, we identify the boundaries of the Smale--horseshoe region in the parameter plane, one of which creates a strange attractor that resembles the R\"{o}ssler attractor. The computation of a winding number reveals a complicated overall bifurcation structure in the wider parameter plane that involves infinitely many further homoclinic flip bifurcations associated with so-called homoclinic bubbles.
In order to address stringent standards, energy efficiency objectives or also cost reduction imperatives, the optimal design of complex industrial structures is an important concern. In this context, parametric optimization offers a powerful tool for engineers. Taking into account nonlinear behavior in the models allows performing high-fidelity numerical simulations and thus reducing safety margins. However, in vibration dynamics, the use of classical global optimization methods on industrial-scale structures with nonlinearities is not affordable due to too many required solver calls to localize the optimum. This work proposes an approach to achieve global constrained optimization of structures with local nonlinearities in vibration. The strategy is based on a Bayesian Optimization process relying on two tools: (i) a Gaussian Process as a surrogate model and (ii) a dedicated nonlinear mechanical solver based on the Harmonic Balance Method. These two tools are presented in detail. Their performance and characteristics are presented and analyzed on academic and industrial-scale examples. The whole strategy is finally applied on a Duffing oscillator without optimization’s constraint and a gantry crane to illustrate the efficiency on constrained optimization. In addition, many discussions are made relative to the number of initial sample points. The results show that the proposed approach is able to find the global optimum with a limited number of solver calls, demonstrating its ability to be integrated into an actual industrial design process.
To be able to design gas-liquid separation equipment, the efficiency of
separation units is necessary to collect. Packings efficiency is a critical
parameter because not only the equipment capital cost are limited but even
production cost can increase and plant capacity decrease in the case of poor
design. The efficiency depends on: packings geometry and type, relative
volatility, vapor and liquid load, physical properties (density, viscosity,
surface tension and diffusivity), water occurrence in the system, liquid
distribution and column diameter.
Several tests with Neopentyl glycol (NPG) – Water binary mixture have
been performed in two different pilot distillation columns filled with Sulzer
BX gauze stainless steel structured packing. For the efficiency calculations,
commercial software Chemcad has been used. In Chemcad, the theoretical
models used for HETP evaluation are: Bravo, Rocha and Fair and Billet and
Schultes. System NPG– Water is a non-ideal and UNIFAC has been chosen
in calculation of VLE and then compared with NRTL with modified binary
parameters. Reported deviations, in literature, in prediction of mass transfer
efficiency for HETP models used in this work are ± 50% which could give
uncertainty in column up-scaling. In this work, NRTL parameters have been
modified in order to minimize predicted components concentrations at the
top and bottom of the column. A good consistency with experimental values
has been shown using Bravo, Rocha and Fair model. That approach has
been applied in an existing plant column filled by Sulzer MELLAPAK
250Y packings showing rather good predicted results in comparison with
collected samples for the given conditions. ody text – Arial 12 pt., leftjustified - margins: 4 cm
The reciprocity theorem in elastic materials states that the response of a linear, time-invariant system to an external load remains invariant with respect to interchanging the locations of the input and output. In the presence of nonlinear forces within a material, circumventing the reciprocity invariance requires breaking the mirror symmetry of the medium, thus allowing different wave propagation characteristics in opposite directions along the same transmission path. This work highlights the application of numerical continuation methods for exploring the steady-state nonreciprocal dynamics of nonlinear periodic materials in response to external harmonic drive. Using the archetypal example of coupled oscillators, we apply continuation methods to analyze the influence of nonlinearity and symmetry on the reciprocity invariance. We present symmetry-breaking bifurcations for systems with and without mirror symmetry, and discuss their influence on the nonreciprocal dynamics. Direct computation of the reciprocity bias allows the identification of response regimes in which nonreciprocity manifests itself as a phase shift in the output displacements. Various operating regimes, bifurcations and manifestations of nonreciprocity are identified and discussed throughout the work.
In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus’ matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.
Periodic orbits and their invariant manifolds are known to be useful for transportation in space, but a large portion of the related research goes toward a small number of periodic orbit families that are relatively simple to compute. In this study, motivated by a search for new and lesser-known families of useful periodic orbits, the bifurcation diagram near Europa is explored and 400 bifurcation points are found. Families are generated for 74 of these and provided in a publicly accessible database. Of these 74 generated families, those that also appear to exist in a model perturbed by certain zonal harmonics of Jupiter and Europa are identified. Differential corrections techniques are discussed, and a new method for natural parameter continuation in the three-body problem is presented. Periodic orbits with particularly useful geometric and stability properties for science purposes are highlighted.
A numerical method for the detection and continuation of homoclinic and heteroclinic orbits is developed for the case of biparametric dynamical systems in two and three dimensions. We formulate a continuation problem for which the regularity conditions are studied. The numerical method is applied to several systems, some of them well-known.
Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
A systematic method for locating and computing branches of connecting orbits developed by the authors is outlined. The method is applied to the sine-Gordon and Hodgkin-Huxley equations.
We consider nonlinear evolution equations of the form
$$
\frac{{du}}{{dt}} = F(u,\lambda, \gamma)\;\quad F:\text{R}^n \times \text{R} \to \text{R}^n
$$ (1)
where λ and γ denote real parameters. In order to obtain a qualitative description of the solution set of (1) as function of the parameters, one can determine the “bifurcation set” by computing paths of turning points, Hopf bifurcation points and (in some cases) regular bifurcation points. This can be done by a continuation procedure applied to a suitable determining system, starting from a known branching point for the one-parameter problem
\(
du/dt = F(u,\lambda,\gamma ^f ) (\gamma ^f \text{fixed)}
\)
(see e.g. [9,17]).
An approach which combines recent results on local bifurcation theory with the continuation method is discussed. This approach has been implemented in the LINLBF program. The program provides an advanced continuation strategy to analyze bifurcations of equilibrium points of ODEs up to codimension three singularities. We discuss the main principles, mathematics and numerics of the program, and illustrate this on a three-dimensional chemical kinetic model depending upon seven parameters. A new interactive version of the program for IBM PC compatibles is also described.
A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong λ-lemma, and Lyapunov-Schmidt reduction for the Poincaré maps under Sil’nikov variables. The method is based on the center manifold theory, the contraction mapping principle, and the implicit function theorem.