Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numerics
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Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numerics Kevin E. M. Church1
Received: 29 April 2020 / Revised: 11 September 2020 / Accepted: 24 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 computerassisted 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. Keywords Impulsive delay differential equations · Floquet multipliers · Chebyshev series · Rigorous numerics · Computer-assisted proofs
1 Introduction Linearized growth and decay rates near steady states and invariant manifolds play a central role in the analysis of dynamical systems. When these manifolds have simple descriptions such as fixed points or periodic orbits, the computation of these growth rates is equivalent to an eigenvalue problem. For delay differential equations (or more generally, retarded functional
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Kevin E. M. Church [email protected] Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC H3A 0B9, Canada
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Journal of Dynamics and Differential Equations
differential equations), several authors have proposed solutions to the “delay eigenvalue problem”. For autonomous linear equations, these include methods based on discretization of the associated infinitesimal generator [3,4,18,33] and the solution operator [15]. In the scope of equations with periodic coefficients, there are several results concerning discretization and characteristic matrices [15,16,26,29,30]. Discretization schemes can provide strong convergence properties, but these still may not be able to provide mathematical proof concerning one or more approximate eigenvalues. For example, the spectral accuracy of the i
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