Lyapunov dimension formulas for Lorenz-like systems

New classes of Lyapunov-type functions are suggested for the estimation of Lyapunov dimension of Lorenz-like systems. Lyapunov dimension formulas for Lorenz and Tigan systems are obtained.

[1]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

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

[3]  A. Eden.  Local estimates for the Hausdorff dimension of an attractor , 1990 .

[4]  R. Temam,et al.  Local and Global Lyapunov exponents , 1991 .

[5]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

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

[7]  Gennady A. Leonov,et al.  The dimension formula for the Lorenz attractor , 2011 .

[8]  Gennady A. Leonov,et al.  Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system , 2013 .

[9]  G. Leonov,et al.  Lyapunov’s direct method in estimates of topological entropy , 1998 .

[10]  J. Yorke,et al.  Chaotic behavior of multidimensional difference equations , 1979 .

[11]  Nikolay V. Kuznetsov,et al.  Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu-Morioka System , 2015, Entropy.

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

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

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

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

[16]  James A. Yorke,et al.  The Lyapunov dimension of a nowhere differentiable attracting torus , 1984, Ergodic Theory and Dynamical Systems.

[17]  Gennady A. Leonov,et al.  Fishing principle for homoclinic and heteroclinic trajectories , 2014 .

[18]  G. A. Leonov,et al.  Lyapunov dimension formula for the global attractor of the Lorenz system , 2015, Commun. Nonlinear Sci. Numer. Simul..

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

[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]  A. Thomas.  Dimension de Hausdorff , 1974 .

[22]  Charles R. Doering,et al.  On the shape and dimension of the Lorenz attractor , 1995 .

[23]  E. Dinaburg.  ON THE RELATIONS AMONG VARIOUS ENTROPY CHARACTERISTICS OF DYNAMICAL SYSTEMS , 1971 .

[24]  Gennady A. Leonov,et al.  Erratum to “The dimension formula for the Lorenz attractor” [Phys. Lett. A 375 (8) (2011) 1179] , 2012 .

[25]  Roy L. Adler,et al.  Topological entropy , 2008, Scholarpedia.

[26]  Nikolay V. Kuznetsov,et al.  Time-Varying Linearization and the Perron Effects , 2007, Int. J. Bifurc. Chaos.

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

[28]  Ya. G. Sinai,et al.  On the Notion of Entropy of a Dynamical System , 2010 .

[29]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[30]  N. Kuznetsov,et al.  The Lyapunov dimension and its estimation via the Leonov method , 2016, 1602.05410.

[31]  N. A. Izobov.  Lyapunov exponents and stability , 2012 .

[32]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[33]  L. Barreira,et al.  Dimension estimates in smooth dynamics: a survey of recent results , 2010, Ergodic Theory and Dynamical Systems.

[34]  F. Ledrappier,et al.  Some relations between dimension and Lyapounov exponents , 1981 .

[35]  M. Kunze,et al.  Non-Smooth Dynamical Systems: An Overview , 2001 .

[36]  Gennady A. Leonov,et al.  Lyapunov functions in the attractors dimension theory , 2012 .

[37]  G. A. Leonov,et al.  Lyapunov dimension formula of attractors in the Tigan and Yang systems , 2015, 1510.01492.

[38]  Lyapunov dimension formula for the attractor of the Glukhovsky–Dolzhansky system , 2016 .