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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.
Reviewer: Wang Duo (Beijing)

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
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