Solution to the boundary values problem in chaotic flows and maps

Fluctuational escape via an unstable limit cycle is investigated in stochastic flows and maps. A new topological method is suggested for analysis of the corresponding boundary value problems when the action functional has multiple local minima along the escape trajectories and the search for the global minimum is otherwise impossible. The method is applied to the analysis of the escape problem in the inverted Van der Pol oscillator and in the Henon map. An application of this technique to solution of the escape problem in chaotic maps with fractal boundaries, and in maps with chaotic saddles embedded within the basin of attraction, is discussed.

[1]  Robert S. Maier,et al.  A scaling theory of bifurcations in the symmetric weak-noise escape problem , 1996 .

[2]  R. Maier,et al.  Noise-activated escape from a sloshing potential well. , 2000, Physical review letters.

[3]  Bernard J. Matkowsky,et al.  A direct approach to the exit problem , 1990 .

[4]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[5]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[6]  Reimann,et al.  Surmounting oscillating barriers , 2000, Physical review letters.

[7]  Woerdman,et al.  Anatomy of a polarization switch of a vertical-cavity semiconductor laser , 2000, Physical review letters.

[8]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[9]  Peter Talkner,et al.  Mean first passage time and the lifetime of a metastable state , 1987 .

[10]  Vadim N. Smelyanskiy,et al.  Optimal Control of Large Fluctuations. , 1997 .

[11]  Beale.  Noise-induced escape from attractors in one-dimensional maps. , 1989, Physical review. A, General physics.

[12]  Stein,et al.  Optimal paths and the prehistory problem for large fluctuations in noise-driven systems. , 1992, Physical review letters.

[14]  Ulrike Feudel,et al.  Enhancement of noise-induced escape through the existence of a chaotic saddle. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Riccardo Mannella,et al.  Stochastic resonance in perspective , 1995 .

[16]  Vadim N. Smelyanskiy,et al.  Nucleation in periodically driven electrochemical systems , 1999 .

[17]  Vadim Smelyanskiy,et al.  TIME OSCILLATIONS OF ESCAPE RATES IN PERIODICALLY DRIVEN SYSTEMS , 1999 .

[18]  Kautz.  Thermally induced escape: The principle of minimum available noise energy. , 1988, Physical review. A, General physics.

[19]  J. Vesecky,et al.  Wave propagation and scattering. , 1989 .

[20]  Surmounting oscillating barriers: path-integral approach for weak noise , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Vadim N. Smelyanskiy,et al.  Fluctuations, Escape, and Nucleation in Driven Systems: Logarithmic Susceptibility , 1997 .

[22]  Graham,et al.  Nonequilibrium potentials for dynamical systems with fractal attractors or repellers. , 1991, Physical review letters.

[23]  Peter Grassberger,et al.  Noise-induced escape from attractors , 1989 .

[24]  Maier,et al.  Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle. , 1996, Physical review letters.

[25]  Vadim N. Smelyanskiy,et al.  Observable and hidden singular features of large fluctuations in nonequilibrium systems , 1994 .

[26]  Robert S. Maier,et al.  Limiting Exit Location Distributions in the Stochastic Exit Problem , 1994, SIAM J. Appl. Math..

[27]  Graham,et al.  Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems. , 1985, Physical review. A, General physics.

[28]  M. V. Day,et al.  Recent progress on the small parameter exit problem , 1987 .

[29]  J. P. Woerdman,et al.  PHYSICAL INSIGHT INTO THE POLARIZATION DYNAMICS OF SEMICONDUCTOR VERTICAL-CAVITY LASERS , 1998 .

[30]  Peter V. E. McClintock,et al.  Analogue studies of nonlinear systems , 1998 .

[31]  Robert Graham,et al.  Existence of a potential for dissipative dynamical systems , 1984 .

[32]  Zhukov,et al.  Dynamics of activated escape and its observation in a semiconductor laser , 2000, Physical review letters.

[33]  F. Jülicher,et al.  Modeling molecular motors , 1997 .

[34]  V. N. Chizhevsky,et al.  Optimal timing for targeting periodic orbits in a loss-driven CO2 laser☆ , 1997 .

[35]  D. Ludwig.  Persistence of Dynamical Systems under Random Perturbations , 1975 .

[36]  D. Luchinsky,et al.  Corrals and Critical Behavior of the Distribution of Fluctuational Paths. , 1996, Physical review letters.

[37]  J. A. Crowther.  Reports on Progress in Physics , 1941, Nature.