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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

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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos," 2nd edition, CRC Press, Boca Raton, FL, 1999.  Google Scholar

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J. T. Schwartz, "Nonlinear Functional Analysis," Gordon and Breach Science Publishers, New York, 1969.  Google Scholar

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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.  Google Scholar

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P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos, Topological Methods in Nonlinear Analysis, 8 (1996), 169-177.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos," 2nd edition, CRC Press, Boca Raton, FL, 1999.  Google Scholar

[10]

J. T. Schwartz, "Nonlinear Functional Analysis," Gordon and Breach Science Publishers, New York, 1969.  Google Scholar

[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.  Google Scholar

[12]

P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos, Topological Methods in Nonlinear Analysis, 8 (1996), 169-177.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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