Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries
It has been established by Gavrilov and Shilnikov (Math USSR Sb 17:467–485, 1972) that at the bifurcation boundary, separating Morse–Smale systems from systems with complicated dynamics, there are systems with homoclinic tangencies. Moreover, when crossing this boundary, infinitely many periodic orbits appear immediately, just by “explosion.” Newhouse and Palis (Asterisque 31:44–140, 1976) have shown that in this case, there are infinitely many intervals of values of the splitting parameter corresponding to hyperbolic systems. In the present chapter, we show that such hyperbolicity intervals have natural bifurcation boundaries, so that the phenomenon of homoclinic Ω-explosion gains, in a sense, complete description in the case of 2D diffeomorphisms.
KeywordsInvariant Manifold Unstable Manifold Homoclinic Orbit Symbolic Dynamic Global Bifurcation
The authors thank D. Turaev for fruitful discussions. This research was carried out within the framework of the Russian Federation Government grant, contract No.11.G34.31.0039. The paper was supported also in part by grants of RFBR No.10-01-00429, No.11-01-00001, and No.11-01-97017-povoljie.
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