Strange invariant sets and multistability in implicit discrete dynamical system

Abstract

This work is devoted to the study of fractal-generating complex maps, arising as an iterative processes of the numerical solution of the nonlinear equations: algebraic - by the Newton method, ordinal differential - by the Euler method. Such a research clarifies in a certain way the question, whether chaotic dynamics and fractal invariant sets in phase space are the numerical artifacts of ODE discretization or not. On the other hand, relevant in modern nonlinear theory is an approach, when new interesting models are generated by Euler discretization of a flow system. Models obtained via such an approach can demonstrate phenomena not inherent in an original system - the numerical artifacts mentioned above. This approach can be used to justify studies of a new class of systems - the implicit systems. Such systems have an evolution operator, which can be multi-valued not only in backward time, but also in forward time, in contrast to traditional noninvertible systems. Posessing some special symmetry, implicit maps satisfy the generalized unitarity condition and demonstrate features of area-preserving maps. So, despite beeing artificial, implicit maps show phenomena, associated with realistic systems, namely, strong multistability. In the present work we study coexisting periodic orbits and the process of formation of the strange invariant sets in the implicit maps.

The work is supported by Russian Science Foundation, Grant No. 21-12-00121, https://rscf.ru/project/21-12-00121/

Speaker

Olga Isaeva
Kotel'nikov Institute of Radio-Engineering and Electronics of RAS, Saratov branch
Russia

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