Figure 6 - uploaded by Kevin E. M. Church

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# The black dots are the eigenvalues of the finite (matrix) part of the operator˜Moperator˜ operator˜M N1,N2 . Observe that −1 is a (numerical) eigenvalue, which is consistent with the discussion following (103). The radial sector R λ * (r, ω) from the proof of Theorem 8.2.1 appears in blue. The unit circle is provided for relative scale, and is delineated by a black dashed curve. The gap between the radial sector and the unit circle is very small and difficult to resolve in the figure: in the radial direction the length of the gap is bounded below by 3.4 × 10 −3 .

Source publication

We develop validated numerical methods for the computation of Floquet multipliers of equilibria
and periodic solutions of delay differential equations, as well as impulsive delay differential equations.
Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc
centered at zero or the number of multipliers c...

## Contexts in source publication

**Context 1**

... our observations above concerning the spectral mapping theorem and the relationship between the eigenvalues of M and˜M and˜ and˜M , we conclude the unstable manifold of the periodic solution (x * (t), y * (t)) is at least two-dimensional. See Figure 6 for the validation structure and approximate eigenvalues. ...

**Context 2**

... our observations above concerning the spectral mapping theorem and the relationship between the eigenvalues of M and˜M and˜ and˜M , we conclude the unstable manifold of the periodic solution (x * (t), y * (t)) is at least two-dimensional. See Figure 6 for the validation structure and approximate eigenvalues. ...

## Citations

... Bifurcation theory for periodic solutions of impulsive functional differential equations has undergone some new developments in recent years [5,7], but such results are only useful if one has computed a periodic solution to begin with. This is a main motivation for considering here the problem of computation and continuation of periodic solutions for impulsive delay differential equations. ...

... Methods for computer-assisted proofs of stability for linear impulsive delay differential equations has recently been accomplished [7] using Chebyshev spectral collocation techniques, and the approach we take here shares some similarities. The idea is as follows. ...

... Converting problems in nonlinear dynamics into zero-finding problems in sequence spaces is not a new idea. In the context of sequence spaces representing Chebyshev series coefficients, see [2,7,15,20] for a few recent applications. One solitary application of Chebyshev expansions in nonlinear impulsive dynamical systems we could find appears in [24], where it was used to generate a simplified approximation of an optimal control problem involving impulsive integrodifferential equations. ...

We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state.

... Bifurcation theory for periodic solutions of impulsive functional differential equations has undergone some new developments in recent years [5,7], but such results are only useful if one has computed a periodic solution to begin with. This is a main motivation for considering here the problem of computation and continuation of periodic solutions for impulsive delay differential equations. ...

... Methods for computer-assisted proofs of stability for linear impulsive delay differential equations has recently been accomplished [7] using Chebyshev spectral collocation techniques, and the approach we take here shares some similarities. The idea is as follows. ...

... Converting problems in nonlinear dynamics into zero-finding problems in sequence spaces is not a new idea. In the context of sequence spaces representing Chebyshev series coefficients, see [2,7,15,20] for a few recent applications. One solitary application of Chebyshev expansions in nonlinear impulsive dynamical systems we could find appears in [24], where it was used to generate a simplified approximation of an optimal control problem involving impulsive integrodifferential equations. ...

We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state.

... To accompany this publication we have implemented the numerical discretization of the monodromy operator in MATLAB. It can be found at the author's GitHub [9]. The implementation can handle arbitrary period p and delay q, provided these are integers. ...

... The code is general-purpose, taking as input the matrix-valued functions A(t) and B(t) appropriately pre-processed, impulse matrices C1 and C2 and various other user-specified data. The code can be found at [9]. INTLAB is required for rigorous proof, but is not required for monodromy operator discretization, so the latter is suitable for eigenvalue (Floquet multiplier) estimation. ...

... The number of points in this mesh has a significant impact on computation time, and this number is generally different for each proof. See the associated MATLAB code [9] for documentation. ...

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes.

We consider nonlinear impulsive systems on Banach spaces subjected to disturbances and look for dwell-time conditions guaranteeing the ISS property. In contrary to many existing results our conditions cover the case where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov type methods are used for this purpose. The effectiveness of our approach is illustrated on a rather nontrivial example, which is feedback connection of an ODE and a PDE system.

In this chapter, we develop theoretical and computational aspects of Floquet theory for periodic linear systems.