Ming-Chia Li

We study the hyperbolic dynamics of three-dimensional quadratic maps with constant Jacobian the inverse of which are again quadratic maps (the so-called 3D Hénon maps). We consider two classes of such maps having applications to the nonlinear dynamics and find certain sufficient conditions under which the maps possess hyperbolic nonwandering sets topologically conjugating to the Smale horseshoe. We apply the so-called Shilnikov’s crossmap for proving the existence of the horseshoes and show the existence of horseshoes of various types: (2,1)- and (1,2)-horseshoes (where the first (second) index denotes the dimension of stable (unstable) manifolds of horseshoe orbits) as well as horseshoes of saddle and saddle-focus types.

We consider piecewise monotone (not necessarily, strictly) piecewise $C^2$ maps on the interval with positive topological entropy. For such a map $f$ we prove that its topological entropy $h_{top}(f)$ can be approximated (with any required accuracy) by restriction on a compact strictly $f$-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.

In this paper, we study the family of Arneodo–Coullet–Tresser maps $F(x,y,z)=(ax-b(y-z)$, $bx+a(y-z)$, $cx-dxk+e z)$ where $a$, $b$, $c$, $d$, $e$ are real parameters with $bd \ne 0$ and $k>1$ is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of $F$ to these sets is conjugate to the full shift on two or three symbols.