https://www.intechopen.com/books/chaos-theory/bifurcation-theory-of-dynamical-chaos
Bifurcation Theory of Dynamical Chaos
By Nikolai A. Magnitskii
Submitted: June 23rd 2017Reviewed: September 13th 2017Published: March 28th 2018
DOI: 10.5772/intechopen.70987
Sections
chapter and author info
1.Introduction2.Dynamical chaos in nonlinear dissipative systems of ordinary differential equations3.Dynamical chaos in Hamiltonian and conservative systems4.Spatio-temporal chaos in nonlinear partial differential equations5.ConclusionAcknowledgmentsAbstract
The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
Keywordsnonlinear systemsdynamical chaosbifurcationssingular attractorsFShM theorychapter and author info
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1. Introduction
Well-known, that chaotic dynamics is inherent practically in all nonlinear mappings and systems of differential equations having irregular attractors, distinct from stable fixed and singular points, limit cycles and tori. However, many years there was no clear understanding of that from itself represent irregular attractors and how they are formed. In this connection it was possible to find in the literature more than 20 various definitions of irregular attractors: stochastic, chaotic, strange, hyperbolic, quasiattractors, attractors of Lorenz, Ressler, Chua, Shilnikov, Chen, Sprott, Magnitskii and many others. It was considered that there are differences between attractors of autonomous and nonautonomous nonlinear systems, systems of ordinary differential equations and the equations with partial derivatives, and that the chaos in dissipative systems essentially differs from chaos in conservative and Hamiltonian systems. There was also an opinion which many outstanding scientists adhered, including Nobel prize winner I.R. Prigogine, that irregular attractors of complex nonlinear systems cannot be described by trajectory approach, that are systems of differential equations. And only in twenty-first century it has been proved and on numerous examples it was convincingly shown, that there is one universal bifurcation scenario of transition to chaos in nonlinear systems of mappings and differential equations: autonomous and nonautonomous, dissipative and conservative, ordinary, with private derivatives and with delay argument (see, for example, [, , , , , , , , ]). It is bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario, beginning with the Feigenbaum cascade of period-doubling bifurcations of stable cycles or tori and continuing from the Sharkovskii subharmonic cascade of bifurcations of stable cycles or tori of an arbitrary period up to the cycle or torus of the period three, and then proceeding to the Magnitskii homoclinic or heteroclinic cascade of bifurcations of stable cycles or tori. All irregular attractors born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
