Dynamics of a one-parameter family of Hénon maps

Abstract

Abstract. We consider a one-parameter family of Hénon maps on R 2 given by fa(x, y) = (y, y 2 +ax) where 0 < a < 1, and provide a complete description of the dynamics of fa. In particular, we show that each fa has precisely two periodic points α and p, where α is an attracting fixed point, and p is a saddle fixed point. Moreover, the basin boundary of α coincides with the stable manifold of p. As a consequence, we obtain that each fa is a Morse-Smale diffeomorphism.

Diffeomorphisms with Stable Manifolds as Basin Boundaries

“…If a diffeomorphism of R 2 has exactly two periodic points and one is attracting while the other one is a saddle point, it is of interest to find out when the boundary of the basin of attraction is the stable manifold of the saddle. Only in a few special cases of Hénon maps (see [4]) this has been proved, although in the standard literature on Hénon maps (see for example the books [1,5]) it is often said to be true based on computer experiments. In this paper we study C 2 -diffeomorphisms F : R 2 → R 2 defined by F (x, y) = (g(x) + h(y), h(x)),…”

Hopf Bifurcation of the Higher Dimensional Hénon Map

“…The original Hénon map is a two dimensional discrete time dynamical system 2 . The various properties of this map have been extensively studied 8,9,15,13 and the generalized form of the original map also has been studied 1,10 . The chaotic and hyperchaotic behavior for certain parameter values and initial conditions of higher dimensional generalized Hénon map have been studied by Richter 11 .…”

Filled Julia set of some class of Hénon-like map

“…In particular, when f is a Hénon map defined by f (x, y) = (y, P (y) + c − ax), for all (x, y) ∈ C 2 , where P is a complex polynomial function and a, c are fixed complex numbers, many topological and dynamical properties of the forward and backward Julia set associated to f have been proved ( [1], [9]). For example, in [11] the authors considered the map H a : R 2 −→ R 2 defined by H a (x, y) = (y, y 2 + ax) where 0 < a < 1 is given and they proved that K(H a ) = {α, p} ∪ [W s (α) ∩ W u (p)], where α = (0, 0) is the attracting fixed point of H a , p = (1 − a, 1 − a) is the repelling fixed point of H a and K(H a ) := K + (H a ) ∩ K − (H a ) is the Julia set associated H a .…”

“…, by relation (11) there exists −1 < z 1 < z 0 such that z 0 = g(z 1 ). Continuing in this way, we construct a decreasing sequence (z n ) n such that −1 < z n = g(z n+1 ) < z 0 , for all n ≥ 1.…”

“…Let g : R −→ R be the function defined by g(x) = x(x 2 + c) + c. Since the critical points of g are − −c 3 and −c 3 , it is not hard to prove that (11) g…”