# American Institute of Mathematical Sciences

September  2016, 36(9): 5011-5024. doi: 10.3934/dcds.2016017

## Topological conjugacy for Lipschitz perturbations of non-autonomous systems

 1 Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300

Received  July 2015 Revised  March 2016 Published  May 2016

In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12,24,15] and the exponential dichotomy is from [2].
Citation: Ming-Chia Li, Ming-Jiea Lyu. Topological conjugacy for Lipschitz perturbations of non-autonomous systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5011-5024. doi: 10.3934/dcds.2016017
##### References:
 [1] F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Int. J. Bifurc. Chaos, 20 (2010), 2591-2636. doi: 10.1142/S0218127410027246. [2] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential equations, 228 (2006), 285-310. doi: 10.1016/j.jde.2006.04.001. [3] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies, J. Differential Equations, 246 (2009), 183-215. doi: 10.1016/j.jde.2008.06.009. [4] M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum, Ergodic Theory Dynam. Systems, 20 (2000), 365-377. doi: 10.1017/S0143385700000171. [5] M. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum II: The theorem is sharp, Discrete Contin. Dynam. Systems, 5 (1999), 599-616. doi: 10.3934/dcds.1999.5.599. [6] S. N. Elaydi, Nonautonomous difference equations: Open problems and conjectures, Fields Inst. Commun., 42 (2004), 423-428. [7] J. Franks and D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc., 352 (2000), 3305-3322. doi: 10.1090/S0002-9947-00-02488-0. [8] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Symp. Pure Math., 14 (1970), 133-163. [9] J. Kennedy, S. Kocak and J. Yorke, A chaos lemma, Amer. Math. Monthly, 108 (2001), 411-423. doi: 10.2307/2695795. [10] J. Kennedy and J. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [12] H. Kokubu, D. Wilczak and P. Zgliczyński, Rigorous verification of the existence of cocoon bifurcation for the Michelson system, Nonlinearity, 20 (2007), 2147-2174. doi: 10.1088/0951-7715/20/9/008. [13] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indus. Appl. Math., 12 (1995), 205-236. doi: 10.1007/BF03167289. [14] J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2. [15] M.-C. Li and M.-J. Lyu, Topological dynamics for multidimensional perturbations of maps with covering relations and strong Lyapunov condition, J. Differential Equations, 250 (2011), 799-812. doi: 10.1016/j.jde.2010.06.019. [16] M.-C. Li and M.-J. Lyu, Covering relations and Lyapunov condition for topological conjugacy, Dynamical Systems, 31 (2016), 60-78. doi: 10.1080/14689367.2015.1020286. [17] D. Richeson and J. Wiseman, Symbolic dynamics for nonhyperbolic systems, Proc. Amer. Math. Soc., 138 (2010), 4373-4385. doi: 10.1090/S0002-9939-2010-10434-3. [18] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 2002, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. [19] C. Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8. [20] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second edition, CRC Press, Boca Raton, FL, 1999. [21] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton University, Princeton, NJ, 1965, 63-80. [22] P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos, Topol. Methods Nonlinear Anal., 8 (1996), 169-177. [23] P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon function, Nonlinearity, 10 (1997), 243-252. doi: 10.1088/0951-7715/10/1/016. [24] P. Zgliczyński, Covering relation, cone conditions and the stable manifold theorem, J. Differential Equations, 246 (2009), 1774-1819. doi: 10.1016/j.jde.2008.12.019. [25] P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.

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##### References:
 [1] F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Int. J. Bifurc. Chaos, 20 (2010), 2591-2636. doi: 10.1142/S0218127410027246. [2] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Differential equations, 228 (2006), 285-310. doi: 10.1016/j.jde.2006.04.001. [3] L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies, J. Differential Equations, 246 (2009), 183-215. doi: 10.1016/j.jde.2008.06.009. [4] M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum, Ergodic Theory Dynam. Systems, 20 (2000), 365-377. doi: 10.1017/S0143385700000171. [5] M. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum II: The theorem is sharp, Discrete Contin. Dynam. Systems, 5 (1999), 599-616. doi: 10.3934/dcds.1999.5.599. [6] S. N. Elaydi, Nonautonomous difference equations: Open problems and conjectures, Fields Inst. Commun., 42 (2004), 423-428. [7] J. Franks and D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc., 352 (2000), 3305-3322. doi: 10.1090/S0002-9947-00-02488-0. [8] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Symp. Pure Math., 14 (1970), 133-163. [9] J. Kennedy, S. Kocak and J. Yorke, A chaos lemma, Amer. Math. Monthly, 108 (2001), 411-423. doi: 10.2307/2695795. [10] J. Kennedy and J. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [12] H. Kokubu, D. Wilczak and P. Zgliczyński, Rigorous verification of the existence of cocoon bifurcation for the Michelson system, Nonlinearity, 20 (2007), 2147-2174. doi: 10.1088/0951-7715/20/9/008. [13] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indus. Appl. Math., 12 (1995), 205-236. doi: 10.1007/BF03167289. [14] J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2. [15] M.-C. Li and M.-J. Lyu, Topological dynamics for multidimensional perturbations of maps with covering relations and strong Lyapunov condition, J. Differential Equations, 250 (2011), 799-812. doi: 10.1016/j.jde.2010.06.019. [16] M.-C. Li and M.-J. Lyu, Covering relations and Lyapunov condition for topological conjugacy, Dynamical Systems, 31 (2016), 60-78. doi: 10.1080/14689367.2015.1020286. [17] D. Richeson and J. Wiseman, Symbolic dynamics for nonhyperbolic systems, Proc. Amer. Math. Soc., 138 (2010), 4373-4385. doi: 10.1090/S0002-9939-2010-10434-3. [18] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 2002, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. [19] C. Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations, 22 (1976), 28-73. doi: 10.1016/0022-0396(76)90004-8. [20] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second edition, CRC Press, Boca Raton, FL, 1999. [21] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton University, Princeton, NJ, 1965, 63-80. [22] P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos, Topol. Methods Nonlinear Anal., 8 (1996), 169-177. [23] P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon function, Nonlinearity, 10 (1997), 243-252. doi: 10.1088/0951-7715/10/1/016. [24] P. Zgliczyński, Covering relation, cone conditions and the stable manifold theorem, J. Differential Equations, 246 (2009), 1774-1819. doi: 10.1016/j.jde.2008.12.019. [25] P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.
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