Infinite Turing Bifurcations in Chains of Van der Pol Systems

Sergey Kashchenko ()
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Sergey Kashchenko: Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia

Mathematics, 2022, vol. 10, issue 20, 1-10

Abstract: A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. In a number of cases, normalized parabolic equations with two spatial variables arise while considering the most important diffusion type couplings. It has been established, for example, that for the considered systems with a large number of elements, the dynamics change significantly with a slight change in the number of such elements.

Keywords: Van der Pol equation; asymptotics; stability; normal form; bifurcation; critical cases (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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