Time averages for unpredictable orbits of deterministic systems

In many cases, the orbits of deterministic systems displaying highly irregular oscillations yield smoothly converging time averages. It may happen, however, that these time averages do not converge and themselves display wild oscillations. This is analyzed for heteroclinic attractors and hyperbolic strange attractors.

[1]  William A. Brock,et al.  Chaos and Complexity in Economic and Financial Science , 1990 .

[2]  K. Sigmund,et al.  Ergodic Theory on Compact Spaces , 1976 .

[3]  Josef Hofbauer,et al.  On the dynamics of asymmetric games , 1991 .

[4]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[5]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[6]  Alain Arneodo,et al.  Occurence of strange attractors in three-dimensional Volterra equations , 1980 .

[7]  Karl Sigmund,et al.  On the Space of Invariant Measures for Hyperbolic Flows , 1972 .

[8]  M. Eigen,et al.  The Hypercycle: A principle of natural self-organization , 2009 .

[9]  Michael C. Mackey,et al.  From Clocks to Chaos , 1988 .

[10]  Murray Z. Frank,et al.  CHAOTIC DYNAMICS IN ECONOMIC TIME‐SERIES , 1988 .

[11]  Otto E. Rössler,et al.  Chaos and chemistry , 2022, Blackstar Theory.

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

[13]  Andrea Gaunersdorfer,et al.  Time averages for heteroclinic attractors , 1992 .

[14]  M Nowak,et al.  Oscillations in the evolution of reciprocity. , 1989, Journal of theoretical biology.

[15]  Karl Sigmund,et al.  Generic properties of invariant measures for AxiomA-diffeomorphisms , 1970 .

[16]  Jean-Pierre Eckmann,et al.  Addendum: Ergodic theory of chaos and strange attractors , 1985 .

[17]  G. Kirlinger,et al.  Permanence in Lotka-Volterra equations: Linked prey-predator systems , 1986 .

[18]  K. H. Homann.  C. Vidal, A. Pacault: (Eds.): Nonlinear Phenomena in Chemical Dynamics; Proceedings of an International Conference, Bordeaux, France; Vol. 12 aus: Springer Series in Synergetics, Springer-Verlag, Berlin, Heidelberg, New York 1981. 280 Seiten, Preis: DM 70,— , 1983 .

[19]  P. Cvitanović.  Universality in Chaos , 1989 .

[20]  Franz Hofbauer.  Generic properties of invariant measures for continuous piecewise monotonic transformations , 1988 .

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

[22]  D. Ruelle.  Chaotic evolution and strange attractors , 1989 .

[23]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[24]  J. C. Oxtoby.  Measure and Category , 1971 .

[25]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[26]  D. Stoffer,et al.  On the Definition of Chaos , 1989 .

[27]  P. Holmes,et al.  Structurally stable heteroclinic cycles , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[28]  Stephen Smale,et al.  The mathematics of time , 1980 .

[29]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[30]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[31]  Hermann Haken,et al.  Chaos and Order in Nature , 1981 .

[32]  Karl Sigmund,et al.  On the time evolution of statistical states for Anosov systems , 1974 .

[33]  P. Schuster,et al.  On $\omega $-Limits for Competition Between Three Species , 1979 .