Existence and characterization of strange nonchaotic attractors in nonlinear systems

Evidence has accumulated in recent years of the occurence, in certain nonlinear systems, of strange nonchaotic attractors, that is attractors whose geometrical character is not simple, but on and near to which the exponential divergence of trajectories, characteristic of chaotic behaviour, does not occur. This behaviour has implications for predictability; small errors in initial conditions grow much more slowly than in a chaotic system. Such attractors occur commonly in quasiperiodically forced nonlinear oscillators, where their range of existence in parameter space is substantial; we describe two particular cases, one restricted to mechanics, the other to chemistry. Long nonchaotic transients occur in other system. Most evidence for strange nonchaotic attractors arises from numerical experiments, and certain spectral features have been proposed [F. Romeiras and E. Ott, Phys. Rev.A35, 4404 (1987)] as distinguishing characteristics. Some analytical methods are also indicated which give plausible and, as compared with numerical results, quite accurate bounds in parameter space for their existence.

[1]  Grebogi,et al.  Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic. , 1989, Physical review. A, General physics.

[2]  S. Scott,et al.  On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme , 1987 .

[3]  Ott,et al.  Experimental observation of a strange nonchaotic attractor. , 1990, Physical review letters.

[4]  Route to chaos via strange non-chaotic attractors , 1990 .

[5]  Tomasz Kapitaniak,et al.  Chaotic Oscillations in Mechanical Systems , 1991 .

[6]  Ott,et al.  Quasiperiodically forced damped pendula and Schrödinger equations with quasiperiodic potentials: Implications of their equivalence. , 1985, Physical review letters.

[7]  Tomasz Kapitaniak,et al.  On strange nonchaotic attractors and their dimensions , 1991 .

[8]  Mingzhou Ding,et al.  Dimensions of strange nonchaotic attractors , 1989 .

[9]  Tomasz Kapitaniak,et al.  A note on randomness and strange behaviour , 1991 .

[10]  Thomas M. Antonsen,et al.  Quasiperiodically forced dynamical systems with strange nonchaotic attractors , 1987 .

[11]  Peter Gray,et al.  A new model for oscillatory behaviour in closed systems: the autocatalator , 1986 .

[12]  E. Sel'kov,et al.  Self-oscillations in glycolysis. 1. A simple kinetic model. , 1968, European journal of biochemistry.

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

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

[15]  Alexander Wolf,et al.  Progress in Computing Lyapunov Exponents from Experimental Data , 1984 .

[16]  J. Yorke,et al.  Strange attractors that are not chaotic , 1984 .

[17]  Ott,et al.  Strange nonchaotic attractors of the damped pendulum with quasiperiodic forcing. , 1987, Physical review. A, General physics.

[18]  S. Scott,et al.  Oscillatory chemical reactions in closed vessels , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[19]  F. Takens,et al.  On the nature of turbulence , 1971 .

[20]  Tomasz Kapitaniak,et al.  ANALYTIC PREDICTORS FOR STRANGE NON-CHAOTIC ATTRACTORS , 1991 .

[21]  Tomasz Kapitaniak,et al.  Analytical conditions for strange chaotic and nonchaotic attractors of the quasiperiodically forced van der pol equation , 1991 .