American Institute of Mathematical Sciences

Advanced
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 - A, 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, (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

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

[1]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010
[2]

M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743
[3]

Sondes khabthani, Lassaad Elasmi, François Feuillebois. Perturbation solution of the coupled Stokes-Darcy problem. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 971-990. doi: 10.3934/dcdsb.2011.15.971
[4]

M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems & Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219
[5]

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997
[6]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
[7]

Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure & Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861
[8]

Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763
[9]

Jean-Baptiste Bardet, Bastien Fernandez. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 669-684. doi: 10.3934/dcds.2011.31.669
[10]

Ramon Quintanilla. Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 463-470. doi: 10.3934/dcdsb.2001.1.463
[11]

Wancheng Sheng, Tong Zhang. Structural stability of solutions to the Riemann problem for a scalar nonconvex CJ combustion model. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 651-667. doi: 10.3934/dcds.2009.25.651
[12]

José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363
[13]

Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67
[14]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
[15]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[16]

Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010
[17]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[18]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045
[19]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-27. doi: 10.3934/dcdss.2020076
[20]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences