Algebraically Simple Chaotic Flows

It came as a surprise to most scientists when Lorenz in 1963 discovered chaos in a simple system of three autonomous ordinary differential equations with two quadratic nonlinearities. This paper reviews efforts over the subsequent years to discover even simpler examples of chaotic flows. There is reason to believe that the algebraically simplest examples of chaotic flows with quadratic and piecewise linear nonlinearities have now been identified. The properties of these and other simple systems will be described.

[1]  J. C. Sprotta.  Some simple chaotic jerk functions , 1997 .

[2]  Hans Peter Gottlieb,et al.  What is the Simplest Jerk Function that Gives Chaos , 1996 .

[3]  A. Iserles.  Numerical recipes in C—the art of scientific computing , by W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling. Pp 735. £27·50. 1988. ISBN 0-521-35465-X (Cambridge University Press) , 1989, The Mathematical Gazette.

[4]  S. Schot,et al.  Jerk: The time rate of change of acceleration , 1978 .

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

[6]  Baron,et al.  The Conservative case , 1959 .

[7]  A. Baglin,et al.  A dynamical instability as a driving mechanism for stellar oscillations , 1985 .

[8]  Analyse et synthèse de systèmes à dynamique chaotique en termes de circuits de rétroaction (feedback) , 1996 .

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

[10]  René Thomas,et al.  DETERMINISTIC CHAOS SEEN IN TERMS OF FEEDBACK CIRCUITS: ANALYSIS, SYNTHESIS, "LABYRINTH CHAOS" , 1999 .

[11]  Julien Clinton Sprott,et al.  A new class of chaotic circuit , 2000 .

[12]  J.-M. Malasoma.  What is the simplest dissipative chaotic jerk equation which is parity invariant , 2000 .

[13]  Julien Clinton Sprott,et al.  On the probability of chaos in large dynamical systems: A Monte Carlo study , 1999 .

[14]  William H. Press,et al.  Numerical recipes in C , 2002 .

[15]  D. W. Moore,et al.  A Thermally Excited Non-Linear Oscillator , 1966 .

[16]  M. Komuro.  Birth and death of the double scroll , 1985, IEEE Conference on Decision and Control.

[17]  A. Tamasevicius,et al.  Double scroll in a simple '2D' chaotic oscillator , 1996 .

[18]  J. Sprott,et al.  Some simple chaotic flows. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Ivars Peterson,et al.  Books-Received - Newton's Clock - Chaos in the Solar System , 1995 .

[20]  Jack Heidel,et al.  Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case , 1999 .

[21]  Ralf Eichhorn,et al.  Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows , 1998 .

[22]  Alain Arneodo,et al.  Oscillators with chaotic behavior: An illustration of a theorem by Shil'nikov , 1982 .

[23]  L. P. Šil'nikov,et al.  A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE , 1970 .

[24]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[25]  Ahmed S. Elwakil,et al.  Two Modified for Chaos Negative Impedance Converter Op Amp Oscillators with Symmetrical and Antisymmetrical Nonlinearities , 1998 .

[26]  Stefan J. Linz,et al.  Nonlinear dynamical models and jerky motion , 1997 .

[27]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

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

[29]  O. Rössler.  An equation for continuous chaos , 1976 .

[30]  Julien Clinton Sprott,et al.  Simplest dissipative chaotic flow , 1997 .

[31]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[32]  Hoover.  Remark on "Some simple chaotic flows" , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[34]  Hans Christian von Baeyer.  All Shook Up , 1998 .

[35]  S. J. Linz.  No-chaos criteria for certain jerky dynamics , 2000 .

[36]  D. D. Dixon,et al.  Continuous “chaotic” dynamics in two dimensions , 1993 .

[37]  J. Sprott.  Strange Attractors: Creating Patterns in Chaos , 1993 .

[38]  Julien Clinton Sprott,et al.  Predicting the dimension of strange attractors , 1994 .

[39]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[40]  Alain Arneodo,et al.  Possible new strange attractors with spiral structure , 1981 .

[41]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[42]  Alain Arneodo,et al.  Transition to stochasticity for a class of forced oscillators , 1979 .

[43]  James A. Yorke,et al.  Numerical solution of a generalized eigenvalue problem for even mappings , 1979 .

[44]  Jack Heidel,et al.  ERRATUM: Non-chaotic behaviour in three-dimensional quadratic systems , 1997 .

[45]  Julien Clinton Sprott,et al.  How common is chaos , 1993 .

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

[47]  Julien Clinton Sprott,et al.  Automatic generation of strange attractors , 1993, Comput. Graph..

[48]  J. Barrow-Green.  Poincare and the Three Body Problem , 1996 .

[49]  Julien Clinton Sprott,et al.  Elementary chaotic flow , 1999 .

[50]  R. Baker.  All shook up , 1990, Nature.

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

[52]  Sebastian Fischer,et al.  KLEINER-ATTRACTOR IN A PIECEWISE-LINEAR C1-SYSTEM , 1999 .

[53]  Vesely,et al.  Canonical dynamics of the Nosé oscillator: Stability, order, and chaos. , 1986, Physical review. A, General physics.

[54]  Julien Clinton Sprott,et al.  Simple chaotic systems and circuits , 2000 .

[55]  Colin Sparrow,et al.  Local and global behavior near homoclinic orbits , 1984 .

[56]  Ömer Morgül,et al.  Inductorless realisation of Chua oscillator , 1995 .