Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity

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

Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity

Ming-Chia Li ^1, and Ming-Jiea Lyu ^1,

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

Abstract
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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 $0$.

Keywords: topological crossing, homoclinicity., topological entropy, Multidimensional perturbation.

Mathematics Subject Classification: Primary: 37D45, 37B30; Secondary: 37B10, 37B40, 37J4.

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:

[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.

show all references

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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