# 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, 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
 [1] Jean Lerbet, Noël Challamel, François 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] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409 [4] 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 [5] 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 [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, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763 [9] Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021089 [10] 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, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997 [11] Jean-Baptiste Bardet, Bastien Fernandez. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 669-684. doi: 10.3934/dcds.2011.31.669 [12] 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 [13] Wancheng Sheng, Tong Zhang. Structural stability of solutions to the Riemann problem for a scalar nonconvex CJ combustion model. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 651-667. doi: 10.3934/dcds.2009.25.651 [14] José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363 [15] Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67 [16] 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 [17] 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 [18] 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 [19] 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, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809 [20] Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076

2020 Impact Factor: 1.392