A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems
- PDF / 1,002,545 Bytes
- 44 Pages / 439.37 x 666.142 pts Page_size
- 24 Downloads / 157 Views
A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems Jan Bouwe van den Berg1 · Chris Groothedde1 · Jean-Philippe Lessard2
Received: 21 April 2020 / Revised: 10 August 2020 / Accepted: 14 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values. Keywords Delay differential equations · Periodic solutions · Computer–assisted proofs · Fourier series · Mackey–Glass equation · Contraction mapping
1 Introduction In many biological phenomena and engineering applications the dynamics of the system is determined in part by a feedback loop. When this feedback is delayed significantly compared to the time scale of the dynamics, such systems are often described by Delay Differential Equations (DDEs). The analysis of DDEs is considerably more difficult than that of ordinary differential equations (ODEs), since the phase space of the dynamics of DDEs is effectively infinite dimensional. Much progress has been made in studying DDEs, and we refer to [12,15,22–24,33,34,41,42,55] for overviews and highlights. Nevertheless, it is fair to say
B
Jean-Philippe Lessard [email protected] Jan Bouwe van den Berg [email protected] Chris Groothedde [email protected]
1
Department of Mathematics, VU Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
2
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC H3A 0B9, Canada
123
Journal of Dynamics and Differential Equations
that even for the study of relatively simple dynamic structures such as periodic solutions a great desire for new flexible, generally applicable techniques remains. In this paper we develop computer-assisted techniques for finding (and proving) periodic solutions of DDEs. We will, in particular, focus on DDEs with finitely many discrete delays: u (t) = F u(t − τ1 ), . . . , u(t − τq ) . (1) The unknown u : R → R p is a periodic function with a priori unknown period. The delays q are {τ j } j=1 ⊂ R, and the nonlinearity F maps (R p )q to R p . Ever since the first proof of the universality of the Feigenbaum constant [27], great strides have been made in the application of computer-assisted proofs in dynamical systems. By far the most progress has been achieved in ODE problems, with the existence of chaos in the Lorenz system [37,48,
Data Loading...
