Global Bifurcations and Chaos: Analytical Methods, Volume 73
Global Bifurcations and Chaos: Analytical Methods is unique in the literature of chaos in that it not only defines the concept of chaos in deterministic systems, but it describes the mechanisms which give rise to chaos (i.e., homoclinic and heteroclinic motions) and derives explicit techniques whereby these mechanisms can be detected in specific systems. These techniques can be viewed as generalizations of Melnikov's method to multi-degree of freedom systems subject to slowly varying parameters and quasiperiodic excitations. A unique feature of the book is that each theorem is illustrated with drawings that enable the reader to build visual pictures of global dynamcis of the systems being described. This approach leads to an enhanced intuitive understanding of the theory.
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0's and l's admissible string bifurcation Cantor set chaotic component compute consider construction contained coordinate corresponding countable infinity cross-section curve defined Definition denote diffeomorphism dimensional set dimensional stable dimensional unstable manifold discuss dynamical systems eigenvalues example exists flow function geometry given global graph Hamiltonian systems Holmes homeomorphism homoclinic orbit horizontal slabs hyperbolic fixed point infinite integrable intersect transversely invariant manifold invariant set invariant torus iteration iv-vertical Lemma linearized matrix Melnikov vector neighborhood normally hyperbolic invariant open set orbit structure orbits homoclinic ordinary differential equation overflowing invariant parameter pendulum periodic orbit perturbed system phase space Poincare map PROOF Proposition rectangles resp respectively satisfy Section sequence of 0's set of points Silnikov slices Smale horseshoe solution stable and unstable sufficiently small surface tangent space Theorem topologically conjugate tori transverse homoclinic unstable manifolds vector field vertical boundaries vertical slabs Wf xq Wu(Me zero