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 book develops a unified treatment of
deterministic and stochastic systems that extends the applicability
of the Melnikov method to physically realizable stochastic planar
systems with additive, state-dependent, white, colored, or
dichotomous noise. The extended Melnikov method yields the novel
result that motions with transitions are chaotic regardless of
whether the excitation is deterministic or stochastic. It explains
the role in the occurrence of transitions of the characteristics of
the system and its deterministic or stochastic excitation, and is a
powerful modeling and identification tool.
The book is designed primarily for readers interested in
applications. The level of preparation required corresponds to the
equivalent of a first-year graduate course in applied mathematics.
No previous exposure to dynamical systems theory or the theory of
stochastic processes is required. The theoretical prerequisites and
developments are presented in the first part of the book. The
second part of the book is devoted to applications, ranging from
physics to mechanical engineering, naval architecture,
oceanography, nonlinear control, stochastic resonance, and
neurophysiology.
General
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