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 and 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.

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251.

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[9]

M. Fečkan, "Topological Degree Approach to Bifurcation Problems," Springer, New York, 2008. doi: 10.1007/978-1-4020-8724-0.

[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.

[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.

[12]

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

[13]

T. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.

[14]

A. M. Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller), Taylor & Francis, London, 1992.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 2 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[35]

X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices, Int. J. Bifurcation and Chaos, in press.

[36]

X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets, submitted for publication.

[37]

Z. Zhang, "The Princinple of Differential Dynamics," Scientific Publishing, Beijing, 2003.

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.

[2]

A. Andronov and L. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR, 14 (1937), 247-251.

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[9]

M. Fečkan, "Topological Degree Approach to Bifurcation Problems," Springer, New York, 2008. doi: 10.1007/978-1-4020-8724-0.

[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.

[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.

[12]

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

[13]

T. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.

[14]

A. M. Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller), Taylor & Francis, London, 1992.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 2 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[35]

X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices, Int. J. Bifurcation and Chaos, in press.

[36]

X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets, submitted for publication.

[37]

Z. Zhang, "The Princinple of Differential Dynamics," Scientific Publishing, Beijing, 2003.

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