# American Institute of Mathematical Sciences

April  2011, 30(1): 243-252. doi: 10.3934/dcds.2011.30.243

## Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity

 1 Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, Taiwan

Received  January 2010 Revised  October 2010 Published  February 2011

In this paper, we consider a one-parameter family $F_{\lambda }$ of continuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$ with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$ (ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R} ^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous and locally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having a topologically crossing homoclinic point, then $F_{\lambda }$ has positive topological entropy for all $\lambda$ close enough to $0$.
Citation: Ming-Chia Li, Ming-Jiea Lyu. Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 243-252. doi: 10.3934/dcds.2011.30.243
##### References:
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##### References:
 [1] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys., 172 (1995), 95-118. doi: 10.1007/BF02104512. [2] M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori, J. Differential Equations, 193 (2003), 49-74. doi: 10.1016/S0022-0396(03)00065-2. [3] M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems-II, J. Differential Equations, 202 (2004), 59-80. doi: 10.1016/j.jde.2004.03.014. [4] J. Juang, M.-C. Li and M. Malkin, Chaotic difference equations in two variables and their multidimensional perturbations, Nonlinearity, 21 (2008), 1019-1040. doi: 10.1088/0951-7715/21/5/007. [5] M.-C. Li and M.-J. Lyu, Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition, J. Differential Equations, 250 (2011), 799-812. doi: 10.1016/j.jde.2010.06.019. [6] M.-C Li, M.-J. Lyu and P. Zgliczyński, Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps, Nonlinearity, 21 (2008), 2555-2567. doi: 10.1088/0951-7715/21/11/005. [7] M.-C. Li and M. Malkin, Topological horseshoes for perturbations of singular difference equations, Nonlinearity, 19 (2006), 795-811. doi: 10.1088/0951-7715/19/4/002. [8] M. Misiurewicz and P. Zgliczyński, Topological entropy for multidimensional perturbations of one-dimensional maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 1443-1446. doi: 10.1142/S021812740100281X. [9] C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos," 2nd edition, CRC Press, Boca Raton, FL, 1999. [10] J. T. Schwartz, "Nonlinear Functional Analysis," Gordon and Breach Science Publishers, New York, 1969. [11] L.-S. Young, Chaotic phenomena in three settings: Large, noisy and out of equilibrium, Nonlinearity, 21 (2008), T245-T252. doi: 10.1088/0951-7715/21/11/T04. [12] P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos, Topological Methods in Nonlinear Analysis, 8 (1996), 169-177. [13] P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252. doi: 10.1088/0951-7715/10/1/016. [14] P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.
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