Unified Theory of Deterministic and Noise-Induced Transitions: Melnikov Processes and Their Application in Engineering, Physics and Neuroscience.
Unified Theory of Deterministic and Noise-Induced
Transitions: Melnikov Processes and Their Application
in Engineering, Physics and Neuroscience.
Stochastic and Chaotic Dynamics in the Lakes: STOCHAOS.
CP502, American Institute of Physics, Broomhead, D. S.;
Luchinskaya, E. A.; McClintock, P. V. E.; Mullin, T.,
Editor(s)(s), 266-271 pp, 2000.
noise (sound); dynamics
For a class of deterministic systems chaotic dynamics
entails irrugular transitions between motions within a
potential well (librations) and motions across a
potential barrier (rotations). The necessary condition
for the occurrence of chaos - and transitions - is that
the system's Melnikov function have simple zeros. The
behavior of those systems' stochastic counterparts,
including their chaotic behavior, is similarly
characterized by their Melnikov processes. The
application of the Melnikov method shows that
deterministic and stochastic excitations play similar
roles in the promotion of chaos, meaning that stochastic
systems exhibiting transitions between librations and
rotations have chaotic behavior, including sensitivity
to initial conditions, just like their deterministic
counterparts. We briefly review the Melnikov method and
its use to obtain: criteria guaranteeing the
non-occurrence of transitions in systems excited by
bounded processes; upper bounds for the probability that
transitions can occur during a specified time interval
in systems excited by unbounded processes; and
assessments of the influence of the excitation's
spectral density on the transition rate. We also
briefly review applications of Melnikov processes.