A Unified Theory of Deterministic and Noise-Induced Transitions: Melnikov Processes and Their Application in Engineering, Physics and Neuroscience
For a class of deterministic systems chaotic dynamics entails irregular 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 some of the applications of Melnikov processes reported in the literature in the fields of oceanography, mechanical engineering, nonlinear open-loop control, stochastic resonance, experimental physics, and neurophysiology.
Stochastic and Chaotic Dynamics in the Lakes, Conference
A 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, Conference, Cumbria, UK, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=860242
(Accessed November 28, 2023)