On the topological entropy, nonwandering set and chaos of monotone and competitive dynamical systems

Abstract In this paper, we prove that for strongly monotone and competitive dynamical systems, the topological entropy is equal to the supremum of the topological entropy on invariant and unordered submanifolds with codimension one, and the nonwandering set of a strongly monotone system is the union of the nonwandering sets on invariant and unordered submanifolds with codimension one. Using these conclusions, we prove that a two-dimensional strongly monotone or competitive system has positive topological entropy iff the system has a periodic orbit whose period is not a power of two, and that the depth of centre of two-dimensional monotone systems is not more than two. Finally, we generalize many well-known results about one-dimensional systems to two-dimensional strongly monotone systems.

[1]  Jiang Jifa,et al.  Stable Cycles for Attractors of Strongly Monotone Discrete-Time Dynamical Systems , 1996 .

[2]  P. Polácik,et al.  Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems , 1992 .

[3]  A. Blokh.  On the limit behaviour of one-dimensional dynamical systems , 1982 .

[4]  E. N. Dancer,et al.  Stability of fixed points for order-preserving discrete-time dynamical systems. , 1991 .

[5]  M. Hirsch.  Stability and convergence in strongly monotone dynamical systems. , 1988 .

[6]  Hal L. Smith.  Complicated dynamics for low-dimensional strongly monotone maps , 1997 .

[7]  R. Bowen.  Entropy for group endomorphisms and homogeneous spaces , 1971 .

[8]  M. Misiurewicz,et al.  Periodic points and topological entropy of one dimensional maps , 1980 .

[9]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[10]  Hal L. Smith.  Planar competitive and cooperative difference equations , 1998 .

[11]  M. Hirsch.  The dynamical systems approach to differential equations , 1984 .

[12]  M. Hirsch.  Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[13]  Jifa Jiang,et al.  The General Properties of Discrete-Time Competitive Dynamical Systems , 2001 .

[14]  Horst R. Thieme,et al.  Quasi convergence and stability for strongly order-preserving semiflows , 1990 .

[15]  P. Polácik,et al.  Convergence in smooth strongly monotone flows defined by semilinear parabolic equations , 1989 .

[16]  E. Coven,et al.  Non-wandering sets of the powers of maps of the interval , 1981, Ergodic Theory and Dynamical Systems.

[17]  Horst R. Thieme,et al.  Convergence for strongly order-preserving semiflows , 1991 .

[18]  M. Hirsch.  Systems of differential equations that are competitive or cooperative. VI: A local Cr Closing Lemma for 3-dimensional systems , 1985, Ergodic Theory and Dynamical Systems.

[19]  Peter Takáč,et al.  Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems , 1992 .

[20]  Attractors for discrete-time monotone dynamical systems in strongly ordered spaces , 1985 .

[21]  P. Takáč.  A construction of stable subharmonic orbits in monotone time-periodic dynamical systems , 1993 .

[22]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[23]  M. Hirsch.  Systems of di erential equations which are competitive or cooperative I: limit sets , 1982 .

[24]  W. A. Coppel.  Maps of an interval , 1983 .

[25]  Jiang Jifa,et al.  Attractors in strongly monotone flows , 1991 .

[26]  P. Takáč.  Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. , 1992 .