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Homoclinic twisting bifurcations and cusp horseshoe maps. (English) Zbl 0782.34042
It is known that the homoclinic orbits involved in the chaos generating mechanisms of the Lorenz and Shilnikov types are hyperbolic or nondegenerate but in the contrast the Chay-Rinzel-Terman type homoclinic orbit is degenerate and it occurs when the orbit approaches the equilibrium point from its strong stable manifold. In this paper the author considers another type of degeneracy at which the strong inclination property fails along the homoclinic orbit. The main results of this paper are: Chaotic dynamics arises when the unstable manifold of a hyperbolic equilibrium point changes its twist type along a homoclinic orbit as some generic parameter is varied; Such bifurcation points occur naturally in singularly perturbed systems; Some quotient symbolic systems induced from the Bernoulli symbolic system on two symbols are proved to be characteristic for this new mechanism of chaos generation; Combination of geometrical and analytical methods is proved to be more fruitful. Some open problems are presented at the end of the paper.

34C23Bifurcation (ODE)
37-99Dynamic systems and ergodic theory (MSC2000)
34C37Homoclinic and heteroclinic solutions of ODE
34D30Structural stability of ODE and analogous concepts
37D45Strange attractors, chaotic dynamics
37G99Local and nonlocal bifurcation theory
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