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 & Continuous Dynamical Systems - A, 2011, 30 (1) : 243-252. doi: 10.3934/dcds.2011.30.243
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K. Burns and H. Weiss, A geometric criterion for positive topological entropy,, Comm. Math. Phys., 172 (1995), 95. doi: 10.1007/BF02104512.

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M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori,, J. Differential Equations, 193 (2003), 49. doi: 10.1016/S0022-0396(03)00065-2.

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M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems-II,, J. Differential Equations, 202 (2004), 59. doi: 10.1016/j.jde.2004.03.014.

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J. Juang, M.-C. Li and M. Malkin, Chaotic difference equations in two variables and their multidimensional perturbations,, Nonlinearity, 21 (2008), 1019. doi: 10.1088/0951-7715/21/5/007.

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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. 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. 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. 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. doi: 10.1142/S021812740100281X.

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C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999).

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J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach Science Publishers, (1969).

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L.-S. Young, Chaotic phenomena in three settings: Large, noisy and out of equilibrium,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/11/T04.

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

[13]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243. 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. 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. doi: 10.1007/BF02104512.

[2]

M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invariant tori,, J. Differential Equations, 193 (2003), 49. 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. 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. 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. 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. 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. 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. doi: 10.1142/S021812740100281X.

[9]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999).

[10]

J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach Science Publishers, (1969).

[11]

L.-S. Young, Chaotic phenomena in three settings: Large, noisy and out of equilibrium,, Nonlinearity, 21 (2008). 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.

[13]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243. 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. doi: 10.1016/j.jde.2004.03.013.

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