Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.

[1]  G. Floquet,et al.  Sur les équations différentielles linéaires à coefficients périodiques , 1883 .

[2]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[3]  J. J. Stoker.  Nonlinear Vibrations in Mechanical and Electrical Systems , 1950 .

[4]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[5]  On the existence and stability of the periodic solution of the second kind of a certain mechanical system , 1959 .

[6]  I. G. Malkin,et al.  Some problems in the theory of nonlinear oscillations , 1959 .

[7]  L. Cesari,et al.  Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations , 1960 .

[8]  S. Fomin,et al.  Elements of the Theory of Functions and Functional Analysis , 1961 .

[9]  R. Reissig.  L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 16). VII + 271 S. m. 37 Abb. Berlin/Göttingen/Heidelberg 1959. Springer‐Verlag. Preis geh. 68,— DM , 1960 .

[10]  P. Hartman,et al.  On global asymptotic stability of solutions of differential equations. , 1962 .

[11]  L. Pontryagin,et al.  Ordinary differential equations , 1964 .

[12]  R. Kálmán.  LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Lamberto Cesari,et al.  Asymptotic Behavior and Stability Problems in Ordinary Differential Equations , 1963 .

[14]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[15]  P. Hartman.  Ordinary Differential Equations , 1965 .

[16]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[17]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[18]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[19]  Frequency conditions of existence of two almost periodic solutions in a nonlinear control system , 1975 .

[20]  J. Eckmann.  Roads to turbulence in dissipative dynamical systems , 1981 .

[21]  E. Ott.  Strange attractors and chaotic motions of dynamical systems , 1981 .

[22]  Statistical description of an attractor , 1981 .

[23]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  Philip Holmes,et al.  Strange Attractors and Chaos in Nonlinear Mechanics , 1983 .

[26]  H. Schuster.  Deterministic chaos: An introduction , 1984 .

[27]  Takashi Matsumoto,et al.  A chaotic attractor from Chua's circuit , 1984 .

[28]  P. Bergé,et al.  L'ordre dans le chaos. , 1984 .

[29]  Leon O. Chua,et al.  The double scroll , 1985 .

[30]  L. Chua,et al.  The double scroll family , 1986 .

[31]  The structure of a strange attractor , 1986 .

[32]  Leon O. Chua,et al.  The dual double scroll equation , 1987 .

[33]  Gennady A. Leonov,et al.  Attraktoreingrenzung für nichtlineare Systeme , 1987 .

[34]  G. Leonov,et al.  Attraktorlokalisierung des Lorenz-Systems , 1987 .

[35]  G. Leonov,et al.  NONLOCAL REDUCTION METHOD IN DIFFERENTIAL EQUATIONS THEORY , 1989 .

[36]  F. Brauer,et al.  The Qualitative Theory of Ordinary Differential Equations: An Introduction , 1989 .

[37]  Maciej Ogorzalek,et al.  Order and chaos in a third-order RC ladder network with nonlinear feedback , 1989 .

[38]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[39]  Zur Eingrenzung des Lorenz‐Attraktors durch die Anwendung der nichtlokalen Reduktionsmethode und mit Hilfe von Vergleichssystemen zweiter Ordnung , 1990 .

[40]  L. Chua,et al.  Canonical realization of Chua's circuit family , 1990 .

[41]  L. Chua,et al.  GLOBAL BIFURCATION ANALYSIS OF THE DOUBLE SCROLL CIRCUIT , 1991 .

[42]  Tomasz Kapitaniak,et al.  Chaotic Oscillations in Mechanical Systems , 1991 .

[43]  Polina S. Landa,et al.  Stochastic and Chaotic Oscillations , 1992 .

[44]  Gennady A. Leonov,et al.  Non-local methods for pendulum-like feedback systems , 1992 .

[45]  G. B. Mindlin,et al.  Periodically kicked hard oscillators. , 1993, Chaos.

[46]  Gennady A. Leonov,et al.  Localization of attractors in Multidimensional Chua's Circuit equations , 1993, J. Circuits Syst. Comput..

[47]  G. Leonov,et al.  Frequency Methods in Oscillation Theory , 1995 .