Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Application of Melnikov Method in Engineering, Physics, and Neuroscience
The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This paper describes and illustrates a unified treatment of deterministic and stochastic systems that extends the applicability of the classical Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, colored, or dichotomous noise. The extended method yields the novel result that motions with transitions are chaotic for either deterministic or stochastic excitation, explains the role in the occurrence of transitions of the system and excitiation characteristics, and is a powerful modeling and identification tool.