Impact Factor

Ming-Chia Li

1001 Ta Hsueh Road, 300, Hsinchu, Taiwan
Department of Applied Mathematics & Center of Mathematical Modelling and Scientific Computing, National Chiao Tung University


Gonchenko S. V., Li M.
Shilnikov’s cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps
2010, vol. 15, no. 2-3, pp.  165-184
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.
Keywords: quadratic map, Smale horseshoe, hyperbolic set, symbolic dynamics, saddle, saddlefocus
Citation: Gonchenko S. V., Li M.,  Shilnikov’s cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 165-184
Li M., Malkin M. I.
Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations
2010, vol. 15, no. 2-3, pp.  210-221
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.
Keywords: chaotic dynamics, difference equations, one-dimensional maps, topological entropy, hyperbolic orbits
Citation: Li M., Malkin M. I.,  Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 210-221
Du B., Li M., Malkin M. I.
Topological horseshoes for Arneodo–Coullet–Tresser maps
2006, vol. 11, no. 2, pp.  181-190
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.
Keywords: topological horseshoe, full shift, polynomial maps, generalized Hénon maps, nonwandering set, inverse limit, topological entropy
Citation: Du B., Li M., Malkin M. I.,  Topological horseshoes for Arneodo–Coullet–Tresser maps , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 181-190
DOI: 10.1070/RD2006v011n02ABEH000344

Back to the list