American Institute of Mathematical Sciences

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

Coupled-expanding maps under small perturbations

Xu Zhang 1, , Yuming Shi 1, and  Guanrong Chen 2,

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.
Keywords: coupled-expanding, structural stability., perturbation, Chaos.
Mathematics Subject Classification: Primary: 37D45; Secondary: 37B1.
Citation: Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
References:
[1]

A. A. Andronov and C. E. Chaikin, "Theory of Oscillations" (translated and adapted by S. Lefschetz), Princeton Univ. Press, Princeton, 1949.  Google Scholar

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. 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., Singapore, 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), 332-334. doi: 10.2307/2324899.  Google Scholar

[5]

G. D. Birkhoff, "Dynamical Systems," Amer. Math. Soc., United States of America, 1927.  Google Scholar

[6]

L. Block and W. Coppel, "Dynamics in One Dimension, Lecture Notes in Math. Vol. 1513," Springer-Verlag, Berlin/Heidelberg, 1992.  Google Scholar

[7]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimentional maps, in "Global Theory of Dynamical Systems, Lecture Notes in Math. Vol. 819," Springer-Verlag, Berlin, (1980), 18-34.  Google Scholar

[8]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Addison-Wesley, New York, 1989.  Google Scholar

[9]

M. Fečkan, "Topological Degree Approach to Bifurcation Problems," Springer, New York, 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), 81-137. 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), 753-772. doi: 10.2307/2154651.  Google Scholar

[12]

B. P. Kitchens, "Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Shifts," Springer-Verlag, New York, 1998.  Google Scholar

[13]

T. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. 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, London, 1992.  Google Scholar

[15]

R. Mane, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210. doi: 10.1007/BF02698931.  Google Scholar

[16]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167-169.  Google Scholar

[17]

J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. Pure Math., 14 (1970), 223-231.  Google Scholar

[18]

K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications," Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[19]

M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222. doi: 10.2307/1970100.  Google Scholar

[20]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 2 (1962), 101-120. 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), 1-270. doi: 10.1007/BF02392506.  Google Scholar

[22]

H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3," Gauthiers-Villars, Paris, 1892, 1893, 1899; English translation edited by D. Goroff, Amer. Institute of Physics, New York, 1993.  Google Scholar

[23]

J. Robbin, A structural stability theorem, Ann. of Math., 94 (1971), 447-493. doi: 10.2307/1970766.  Google Scholar

[24]

C. Robinson, Structural stability of $C^1$ flows, in "Lecture Notes in Math. Vol. 468," Springer-Verlag, Berlin/Heidelberg, (1975), 262-277.  Google Scholar

[25]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8.  Google Scholar

[26]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos," CRC Press, Florida, 1999.  Google Scholar

[27]

Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces, Chaos Solit. Fract., 22 (2004), 555-571. doi: 10.1016/j.chaos.2004.02.015.  Google Scholar

[28]

Y. Shi and G. Chen, Discrete chaos in Banach spaces, Science in China, Ser. A: Mathematics, Chinese version: 34 (2004), 595-609; English version: 48 (2005), 222-238.  Google Scholar

[29]

Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps, in "Proc. 1st IFAC Conference on Analysis and Control of Chaotic Systems," Reims, France, June 28-30, (2006), 157-162. Google Scholar

[30]

Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solit. Fract., 39 (2009), 2138-2149. 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), 1165-1180. doi: 10.1016/j.chaos.2005.08.162.  Google Scholar

[32]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[33]

S. Wiggins, "Chaotic Transport in Dynamical Systems," Springer-Verlag, New York, 1992.  Google Scholar

[34]

X. Yang and Y. Tang, Horseshoes in piecewise continuous maps, Chaos Solit. Fract., 19 (2004), 841-845. 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, Beijing, 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, Princeton, 1949.  Google Scholar

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251. 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., Singapore, 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), 332-334. doi: 10.2307/2324899.  Google Scholar

[5]

G. D. Birkhoff, "Dynamical Systems," Amer. Math. Soc., United States of America, 1927.  Google Scholar

[6]

L. Block and W. Coppel, "Dynamics in One Dimension, Lecture Notes in Math. Vol. 1513," Springer-Verlag, Berlin/Heidelberg, 1992.  Google Scholar

[7]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimentional maps, in "Global Theory of Dynamical Systems, Lecture Notes in Math. Vol. 819," Springer-Verlag, Berlin, (1980), 18-34.  Google Scholar

[8]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Addison-Wesley, New York, 1989.  Google Scholar

[9]

M. Fečkan, "Topological Degree Approach to Bifurcation Problems," Springer, New York, 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), 81-137. 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), 753-772. doi: 10.2307/2154651.  Google Scholar

[12]

B. P. Kitchens, "Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Shifts," Springer-Verlag, New York, 1998.  Google Scholar

[13]

T. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. 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, London, 1992.  Google Scholar

[15]

R. Mane, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210. doi: 10.1007/BF02698931.  Google Scholar

[16]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167-169.  Google Scholar

[17]

J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. Pure Math., 14 (1970), 223-231.  Google Scholar

[18]

K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications," Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[19]

M. Peixoto, On structural stability, Ann. of Math., 69 (1959), 199-222. doi: 10.2307/1970100.  Google Scholar

[20]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 2 (1962), 101-120. 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), 1-270. doi: 10.1007/BF02392506.  Google Scholar

[22]

H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3," Gauthiers-Villars, Paris, 1892, 1893, 1899; English translation edited by D. Goroff, Amer. Institute of Physics, New York, 1993.  Google Scholar

[23]

J. Robbin, A structural stability theorem, Ann. of Math., 94 (1971), 447-493. doi: 10.2307/1970766.  Google Scholar

[24]

C. Robinson, Structural stability of $C^1$ flows, in "Lecture Notes in Math. Vol. 468," Springer-Verlag, Berlin/Heidelberg, (1975), 262-277.  Google Scholar

[25]

C. Robinson, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8.  Google Scholar

[26]

C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos," CRC Press, Florida, 1999.  Google Scholar

[27]

Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces, Chaos Solit. Fract., 22 (2004), 555-571. doi: 10.1016/j.chaos.2004.02.015.  Google Scholar

[28]

Y. Shi and G. Chen, Discrete chaos in Banach spaces, Science in China, Ser. A: Mathematics, Chinese version: 34 (2004), 595-609; English version: 48 (2005), 222-238.  Google Scholar

[29]

Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps, in "Proc. 1st IFAC Conference on Analysis and Control of Chaotic Systems," Reims, France, June 28-30, (2006), 157-162. Google Scholar

[30]

Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solit. Fract., 39 (2009), 2138-2149. 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), 1165-1180. doi: 10.1016/j.chaos.2005.08.162.  Google Scholar

[32]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[33]

S. Wiggins, "Chaotic Transport in Dynamical Systems," Springer-Verlag, New York, 1992.  Google Scholar

[34]

X. Yang and Y. Tang, Horseshoes in piecewise continuous maps, Chaos Solit. Fract., 19 (2004), 841-845. 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, Beijing, 2003. Google Scholar

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