Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization
journal contributionposted on 2022-01-13, 13:36 authored by Natalia JansonNatalia Janson, Christopher J. Marsden
Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well “potential” function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.
DTA - Loughborough University
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- Mathematical Sciences
Published inChaos: An Interdisciplinary Journal of Nonlinear Science
- VoR (Version of Record)
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Publisher statementThis is an Open Access Article. It is published by AIP under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/