Lyapunov functions in the attractors dimension theory

Abstract The effectiveness of constructing Lyapunov functions in the attractors dimension theory is theory of the dimension demonstrated. Formulae for the Lyapunov dimension of the Lorenz, Henon and Chirikov attractors are derived and proved. A hypothesis regarding the formula for the dimension of the Rossler attractor is formulated.

[1]  N. Rouche,et al.  Stability Theory by Liapunov's Direct Method , 1977 .

[2]  G. Leonov.  Strange attractors and classical stability theory , 2006 .

[3]  G. Leonov,et al.  Das RÖSSLER‐System ist nicht dissipativ im Sinne von LEVINSON , 1986 .

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

[5]  V. Boichenko,et al.  Dimension theory for ordinary differential equations , 2005 .

[6]  Gennady A. Leonov,et al.  Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors , 1992 .

[7]  Brian R. Hunt,et al.  Maximum local Lyapunov dimension bounds the box dimension of chaotic attractors , 1996 .

[8]  G. Leonov,et al.  Frequency-Domain Methods for Nonlinear Analysis: Theory and Applications , 1996 .

[9]  O. Ladyzhenskaya,et al.  A dynamical system generated by the Navier-Stokes equations , 1975 .

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

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

[12]  J Yeh.  Lectures on Real Analysis , 1999 .

[13]  K. A. Semendyayev,et al.  Handbook of mathematics , 1985 .

[14]  Gennady A. Leonov,et al.  Hausdorff and Fractal Dimension Estimates for Invariant Sets of Non-Injective Maps , 1998 .

[15]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[16]  O. Rössler.  CONTINUOUS CHAOS—FOUR PROTOTYPE EQUATIONS , 1979 .

[17]  Otto E. Rössler,et al.  Different Types of Chaos in Two Simple Differential Equations , 1976 .

[18]  Sergei Yu. Pilyugin.  Introduction to Structurally Stable Systems of Differential Equations , 1992 .

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

[20]  R. A. Smith,et al.  Some applications of Hausdorff dimension inequalities for ordinary differential equations , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[22]  A. Fomenko.  Variational Principles in Topology , 1990 .

[23]  Mikhail I. Rabinovich,et al.  Stochastic self-oscillations and turbulence , 1978 .