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How do hyperbolic homoclinic classes collide at heterodimensional cycles. (English) Zbl 1130.37026

In this paper a one-parameter family of local diffeomorphisms \(\{f_t;\,t\in [-\tau,\tau]\}\) defined on the cube \(R= [-1,1]\times [-1,5]\times [-1,1]\) on \(\mathbb{R}^3\) is investigated. The definition of this family is done in three steps: (1) semi-local product partially hyperbolic dynamics, (2) existence and unfolding of the heterodimensional cycle, and (3) semi-global hypotheses (existence of a filtration). More detailed: in the cube \(R\), for every \(t\in [-\tau,\tau]\), it holds \(f_t= f\), where \(f(x,y,z)= (\lambda_s x, F(y), \lambda u z)\), \(0< \lambda_s< 1< \lambda_u\), where \(F\) is a strictly increasing function such that: \(F(y)=\beta y\), if \(y\in [-1,1/\beta]\) and \(F(y)= \lambda(y- 4)= \lambda(y- 4)+ 4\), if \(y\in [3,5]\), where \(1/2<\lambda< 1\) and \(\beta= 1/\lambda\), and it is supposed that \(\lambda_s< F'(y)< \lambda_u\) for every \(y\in[-1, 5]\). The authors illustrate on this model heterodimensional cycles as a mechanism leading to the collision of hyperbolic homoclinic classes and thereafter to the persistence of heterodimensional cycles. The collisions are associated to secondary (saddle-node) bifurcations appearing in the unfolding of the initial cycle.

MSC:

37G25 Bifurcations connected with nontransversal intersection in dynamical systems
37G30 Infinite nonwandering sets arising in bifurcations of dynamical systems
37D30 Partially hyperbolic systems and dominated splittings
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