# American Institute of Mathematical Sciences

July  2011, 29(3): 1291-1307. doi: 10.3934/dcds.2011.29.1291

## Coupled-expanding maps under small perturbations

 1 Department of Mathematics, Shandong University, Jinan, Shandong 250100, China, China 2 Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong S.A.R.

Received  March 2010 Revised  August 2010 Published  November 2010

This paper studies the $C^1$-perturbation problem of strictly $A$-coupled-expanding maps in finite-dimensional Euclidean spaces, where $A$ is an irreducible transition matrix with one row-sum no less than $2$. It is proved that under certain conditions strictly $A$-coupled-expanding maps are chaotic in the sense of Li-Yorke or Devaney under small $C^1$-perturbations. It is shown that strictly $A$-coupled-expanding maps are $C^1$ structurally stable in their chaotic invariant sets under certain stronger conditions. One illustrative example is provided with computer simulations.
Citation: Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
##### References:
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Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller),, Taylor & Francis, (1992). Google Scholar [15] R. Mane, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161. doi: 10.1007/BF02698931. Google Scholar [16] M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167. Google Scholar [17] J. Palis and S. Smale, Structural stability theorems,, Global Analysis, 14 (1970), 223. Google Scholar [18] K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer Academic Publishers, (2000). Google Scholar [19] M. Peixoto, On structural stability,, Ann. of Math., 69 (1959), 199. doi: 10.2307/1970100. Google Scholar [20] M. Peixoto, Structural stability on two dimensional manifolds,, Topology, 2 (1962), 101. doi: 10.1016/0040-9383(65)90018-2. Google Scholar [21] H. J. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. doi: 10.1007/BF02392506. Google Scholar [22] H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3,", Gauthiers-Villars, (1892). Google Scholar [23] J. Robbin, A structural stability theorem,, Ann. of Math., 94 (1971), 447. doi: 10.2307/1970766. Google Scholar [24] C. Robinson, Structural stability of $C^1$ flows,, in, 468 (1975), 262. Google Scholar [25] C. Robinson, Structural stability of $C^1$ diffeomorphisms,, J. Differential Equations, 22 (1976), 28. doi: 10.1016/0022-0396(76)90004-8. Google Scholar [26] C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos,", CRC Press, (1999). Google Scholar [27] Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Solit. Fract., 22 (2004), 555. doi: 10.1016/j.chaos.2004.02.015. Google Scholar [28] Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Science in China, 34 (2004), 595. Google Scholar [29] Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps,, in, (2006), 28. Google Scholar [30] Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems,, Chaos Solit. Fract., 39 (2009), 2138. doi: 10.1016/j.chaos.2007.06.090. Google Scholar [31] Y. Shi and P. Yu, Study on chaos induced by turbulent maps in noncompact sets,, Chaos Solit. Fract., 28 (2006), 1165. doi: 10.1016/j.chaos.2005.08.162. Google Scholar [32] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar [33] S. Wiggins, "Chaotic Transport in Dynamical Systems,", Springer-Verlag, (1992). Google Scholar [34] X. Yang and Y. Tang, Horseshoes in piecewise continuous maps,, Chaos Solit. Fract., 19 (2004), 841. doi: 10.1016/S0960-0779(03)00202-9. Google Scholar [35] X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices,, Int. J. Bifurcation and Chaos, (). Google Scholar [36] X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets,, submitted for publication., (). Google Scholar [37] Z. Zhang, "The Princinple of Differential Dynamics,", Scientific Publishing, (2003). Google Scholar

show all references

##### References:
 [1] A. A. Andronov and C. E. Chaikin, "Theory of Oscillations" (translated and adapted by S. Lefschetz),, Princeton Univ. Press, (1949). Google Scholar [2] A. Andronov and L. Pontrjagin, Systèmes grossiers,, Dokl. Akad. Nauk. SSSR, 14 (1937), 247. Google Scholar [3] J. Awrejcewicz and M. M. Holicke, "Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods,", World Scientific Publishing Co. Pte. Ltd., (2007). doi: 10.1142/9789812709103. Google Scholar [4] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332. doi: 10.2307/2324899. Google Scholar [5] G. D. Birkhoff, "Dynamical Systems,", Amer. Math. Soc., (1927). Google Scholar [6] L. Block and W. Coppel, "Dynamics in One Dimension, Lecture Notes in Math. Vol. 1513,", Springer-Verlag, (1992). Google Scholar [7] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimentional maps,, in, 819 (1980), 18. Google Scholar [8] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Addison-Wesley, (1989). Google Scholar [9] M. Fečkan, "Topological Degree Approach to Bifurcation Problems,", Springer, (2008). doi: 10.1007/978-1-4020-8724-0. Google Scholar [10] S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$-stability and $\Omega$-stability conjectures for flows,, Ann. of Math., 145 (1997), 81. doi: 10.2307/2951824. Google Scholar [11] S. Hu, A proof of $C^1$ stability conjecture for three-dimensional flows,, Trans. Amer. Math. Soc., 342 (1994), 753. doi: 10.2307/2154651. Google Scholar [12] B. P. Kitchens, "Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Shifts,", Springer-Verlag, (1998). Google Scholar [13] T. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254. Google Scholar [14] A. M. Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller),, Taylor & Francis, (1992). Google Scholar [15] R. Mane, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161. doi: 10.1007/BF02698931. Google Scholar [16] M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167. Google Scholar [17] J. Palis and S. Smale, Structural stability theorems,, Global Analysis, 14 (1970), 223. Google Scholar [18] K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer Academic Publishers, (2000). Google Scholar [19] M. Peixoto, On structural stability,, Ann. of Math., 69 (1959), 199. doi: 10.2307/1970100. Google Scholar [20] M. Peixoto, Structural stability on two dimensional manifolds,, Topology, 2 (1962), 101. doi: 10.1016/0040-9383(65)90018-2. Google Scholar [21] H. J. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. doi: 10.1007/BF02392506. Google Scholar [22] H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3,", Gauthiers-Villars, (1892). Google Scholar [23] J. Robbin, A structural stability theorem,, Ann. of Math., 94 (1971), 447. doi: 10.2307/1970766. Google Scholar [24] C. Robinson, Structural stability of $C^1$ flows,, in, 468 (1975), 262. Google Scholar [25] C. Robinson, Structural stability of $C^1$ diffeomorphisms,, J. Differential Equations, 22 (1976), 28. doi: 10.1016/0022-0396(76)90004-8. Google Scholar [26] C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos,", CRC Press, (1999). Google Scholar [27] Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Solit. Fract., 22 (2004), 555. doi: 10.1016/j.chaos.2004.02.015. Google Scholar [28] Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Science in China, 34 (2004), 595. Google Scholar [29] Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps,, in, (2006), 28. Google Scholar [30] Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems,, Chaos Solit. Fract., 39 (2009), 2138. doi: 10.1016/j.chaos.2007.06.090. Google Scholar [31] Y. Shi and P. Yu, Study on chaos induced by turbulent maps in noncompact sets,, Chaos Solit. Fract., 28 (2006), 1165. doi: 10.1016/j.chaos.2005.08.162. Google Scholar [32] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar [33] S. Wiggins, "Chaotic Transport in Dynamical Systems,", Springer-Verlag, (1992). Google Scholar [34] X. Yang and Y. Tang, Horseshoes in piecewise continuous maps,, Chaos Solit. Fract., 19 (2004), 841. doi: 10.1016/S0960-0779(03)00202-9. Google Scholar [35] X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices,, Int. J. Bifurcation and Chaos, (). Google Scholar [36] X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets,, submitted for publication., (). Google Scholar [37] Z. Zhang, "The Princinple of Differential Dynamics,", Scientific Publishing, (2003). Google Scholar
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