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Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Rotor type singular points of nonautonomous systems of differential equations and their role in... Differential Equations, Vol. 40, No. 11, 2004, pp. 1579–1593. Translated from Differentsial'nye Uravneniya, Vol. 40, No. 11, 2004, pp. 1500–1514. Original Russian Text Copyright c 2004 by Magnitskii, Sidorov. ORDINARY DIFFERENTIAL EQUATIONS Rotor Type Singular Points of Nonautonomous Systems of Di erential Equations and Their Role in the Generation of Singular Attractors of Nonlinear Autonomous Systems N. A. Magnitskii and S. V. Sidorov Institute for Systems Analysis, Russian Academy of Sciences, Moscow, Russia Received May 27, 2004 1. INTRODUCTION It was shown in [1{5] that the passage to chaos under variation of a system parameter in a wide class of three-dimensional autonomous nonlinear dissipative systems of ordinary di erential equations, including all classical chaotic systems such as Lorenz, Ressler, and Chua systems etc., occurs in accordance with a unique scenario of passage to chaos. This scenario begins with a cascade of Feigenbaum period doubling bifurcations of stable cycles and then continues by a subharmonic cascade of Sharkovskii bifurcations (that is, the cascade of generation of stable cycles of an arbitrary period) and, if the system has a saddle{focus separatrix loop, by a homoclinic cascade of bifurcations of stable cycles converging to a homoclinic contour. Therefore, neither the presence of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Magnitskii, N.; Sidorov, S.
Differential Equations , Volume 40 (11) – Feb 26, 2004
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References (1)

  • N.A. Magnitskii (2001)

    1494

    Differents. Uravn., 37

Publisher
Springer Journals
Copyright
Copyright © 2004 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1007/PL00021827
Publisher site
See Article on Publisher Site

Abstract

Differential Equations, Vol. 40, No. 11, 2004, pp. 1579–1593. Translated from Differentsial'nye Uravneniya, Vol. 40, No. 11, 2004, pp. 1500–1514. Original Russian Text Copyright c 2004 by Magnitskii, Sidorov. ORDINARY DIFFERENTIAL EQUATIONS Rotor Type Singular Points of Nonautonomous Systems of Di erential Equations and Their Role in the Generation of Singular Attractors of Nonlinear Autonomous Systems N. A. Magnitskii and S. V. Sidorov Institute for Systems Analysis, Russian Academy of Sciences, Moscow, Russia Received May 27, 2004 1. INTRODUCTION It was shown in [1{5] that the passage to chaos under variation of a system parameter in a wide class of three-dimensional autonomous nonlinear dissipative systems of ordinary di erential equations, including all classical chaotic systems such as Lorenz, Ressler, and Chua systems etc., occurs in accordance with a unique scenario of passage to chaos. This scenario begins with a cascade of Feigenbaum period doubling bifurcations of stable cycles and then continues by a subharmonic cascade of Sharkovskii bifurcations (that is, the cascade of generation of stable cycles of an arbitrary period) and, if the system has a saddle{focus separatrix loop, by a homoclinic cascade of bifurcations of stable cycles converging to a homoclinic contour. Therefore, neither the presence of

Journal

Differential Equations – Springer Journals

Published: Feb 26, 2004

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