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Heterodimensional tangencies on cycles leading to strange attractors. (English) Zbl 1213.37054

The paper is devoted to the study of a two-parameter family \(\{\varphi_{\mu,\nu}\}\) of three-dimensional diffeomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. It is shown that such a codimension-2 bifurcation generates a quadratic homoclinic tangency associated to one of the saddle continuations which unfolds generically with respect to some one-parameter subfamily of \(\{\varphi_{\mu,\nu}\}\). The main result is as follows.
Theorem. Let \(M\) be a 3-dimensional \(C^2\) manifold, and let \(\{\varphi_{\mu,\nu}\}\) be a two-dimensional family of \(C^2\) diffeomorphisms \(\varphi_{\mu,\nu}:M\to M\) which \(C^2\) depends on \((\mu,\nu)\) and has continuations of saddle fixed points \(p_{\mu ,\nu}\) and \(q_{\mu ,\nu}\) with \(\text{index}(p_{\mu ,\nu })=1\) and \(\text{index}(q_{\mu ,\nu })=2\). Suppose that the following conditions hold.
1. Each \(\varphi_{\mu,\nu}\) is locally linearizable in a neighborhood of \(q_{\mu ,\nu }\).
2. \(\varphi_{0,0}\) has a heterodimensional cycle containing the fixed points \(p=p_{0,0}\), \(q=q_{0,0}\), a nondegenerate heterodimensional tangency \(r\), a quasi-transverse intersection \(s\in W^s(q)\cap W^u(p)\).
3. \(\{\varphi_{\mu,\nu}\}\) satisfies some generic conditions.
Then, for a sufficiently small \(\varepsilon>0\) and any \(\mu\) in either \((0,\varepsilon)\) or \((-\varepsilon,0)\), there exist infinitely many \(\nu\) such that \(\varphi_{\mu,\nu}\) has a quadratic homoclinic tangency associated to \(p_{\mu ,\nu }\) which unfolds generically with respect to the \(\nu\)-parameter family \(\{\varphi_{\mu\text{(fixed)},\nu}\}\).
From this result together with some well-known facts follow some nonhyperbolic phenomena (the existence of nonhyperbolic strange attractors and the \(C^2\)-robust tangencies) arbitrarily close to the codimension-2 bifurcation.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G25 Bifurcations connected with nontransversal intersection in dynamical systems
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