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 & Continuous Dynamical Systems - A, 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. doi: 10.1142/S0218127410027246. Google Scholar

[2]

L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. Google Scholar

[3]

L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies,, J. Differential Equations, 246 (2009), 183. doi: 10.1016/j.jde.2008.06.009. Google Scholar

[4]

M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum,, Ergodic Theory Dynam. Systems, 20 (2000), 365. doi: 10.1017/S0143385700000171. Google Scholar

[5]

M. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum II: The theorem is sharp,, Discrete Contin. Dynam. Systems, 5 (1999), 599. doi: 10.3934/dcds.1999.5.599. Google Scholar

[6]

S. N. Elaydi, Nonautonomous difference equations: Open problems and conjectures,, Fields Inst. Commun., 42 (2004), 423. Google Scholar

[7]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Trans. Amer. Math. Soc., 352 (2000), 3305. doi: 10.1090/S0002-9947-00-02488-0. Google Scholar

[8]

M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets,, Proc. Symp. Pure Math., 14 (1970), 133. Google Scholar

[9]

J. Kennedy, S. Kocak and J. Yorke, A chaos lemma,, Amer. Math. Monthly, 108 (2001), 411. doi: 10.2307/2695795. Google Scholar

[10]

J. Kennedy and J. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513. doi: 10.1090/S0002-9947-01-02586-7. Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, 176 (2011). doi: 10.1090/surv/176. Google Scholar

[12]

H. Kokubu, D. Wilczak and P. Zgliczyński, Rigorous verification of the existence of cocoon bifurcation for the Michelson system,, Nonlinearity, 20 (2007), 2147. doi: 10.1088/0951-7715/20/9/008. Google Scholar

[13]

K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos,, Japan J. Indus. Appl. Math., 12 (1995), 205. doi: 10.1007/BF03167289. Google Scholar

[14]

J. Lewowicz, Lyapunov functions and topological stability,, J. Differential Equations, 38 (1980), 192. doi: 10.1016/0022-0396(80)90004-2. Google Scholar

[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. doi: 10.1016/j.jde.2010.06.019. Google Scholar

[16]

M.-C. Li and M.-J. Lyu, Covering relations and Lyapunov condition for topological conjugacy,, Dynamical Systems, 31 (2016), 60. doi: 10.1080/14689367.2015.1020286. Google Scholar

[17]

D. Richeson and J. Wiseman, Symbolic dynamics for nonhyperbolic systems,, Proc. Amer. Math. Soc., 138 (2010), 4373. doi: 10.1090/S0002-9939-2010-10434-3. Google Scholar

[18]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002). doi: 10.1007/978-3-642-14258-1. Google Scholar

[19]

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

[20]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, Second edition, (1999). Google Scholar

[21]

S. Smale, Diffeomorphisms with many periodic points,, Differential and Combinatorial Topology, (1965), 63. Google Scholar

[22]

P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos,, Topol. Methods Nonlinear Anal., 8 (1996), 169. Google Scholar

[23]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon function,, Nonlinearity, 10 (1997), 243. doi: 10.1088/0951-7715/10/1/016. Google Scholar

[24]

P. Zgliczyński, Covering relation, cone conditions and the stable manifold theorem,, J. Differential Equations, 246 (2009), 1774. doi: 10.1016/j.jde.2008.12.019. Google Scholar

[25]

P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32. doi: 10.1016/j.jde.2004.03.013. Google Scholar

show all references

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. doi: 10.1142/S0218127410027246. Google Scholar

[2]

L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. Google Scholar

[3]

L. Barreira and C. Valls, Lyapunov sequences for exponential dichotomies,, J. Differential Equations, 246 (2009), 183. doi: 10.1016/j.jde.2008.06.009. Google Scholar

[4]

M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum,, Ergodic Theory Dynam. Systems, 20 (2000), 365. doi: 10.1017/S0143385700000171. Google Scholar

[5]

M. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum II: The theorem is sharp,, Discrete Contin. Dynam. Systems, 5 (1999), 599. doi: 10.3934/dcds.1999.5.599. Google Scholar

[6]

S. N. Elaydi, Nonautonomous difference equations: Open problems and conjectures,, Fields Inst. Commun., 42 (2004), 423. Google Scholar

[7]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Trans. Amer. Math. Soc., 352 (2000), 3305. doi: 10.1090/S0002-9947-00-02488-0. Google Scholar

[8]

M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets,, Proc. Symp. Pure Math., 14 (1970), 133. Google Scholar

[9]

J. Kennedy, S. Kocak and J. Yorke, A chaos lemma,, Amer. Math. Monthly, 108 (2001), 411. doi: 10.2307/2695795. Google Scholar

[10]

J. Kennedy and J. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513. doi: 10.1090/S0002-9947-01-02586-7. Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, 176 (2011). doi: 10.1090/surv/176. Google Scholar

[12]

H. Kokubu, D. Wilczak and P. Zgliczyński, Rigorous verification of the existence of cocoon bifurcation for the Michelson system,, Nonlinearity, 20 (2007), 2147. doi: 10.1088/0951-7715/20/9/008. Google Scholar

[13]

K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos,, Japan J. Indus. Appl. Math., 12 (1995), 205. doi: 10.1007/BF03167289. Google Scholar

[14]

J. Lewowicz, Lyapunov functions and topological stability,, J. Differential Equations, 38 (1980), 192. doi: 10.1016/0022-0396(80)90004-2. Google Scholar

[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. doi: 10.1016/j.jde.2010.06.019. Google Scholar

[16]

M.-C. Li and M.-J. Lyu, Covering relations and Lyapunov condition for topological conjugacy,, Dynamical Systems, 31 (2016), 60. doi: 10.1080/14689367.2015.1020286. Google Scholar

[17]

D. Richeson and J. Wiseman, Symbolic dynamics for nonhyperbolic systems,, Proc. Amer. Math. Soc., 138 (2010), 4373. doi: 10.1090/S0002-9939-2010-10434-3. Google Scholar

[18]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lecture Notes in Mathematics, (2002). doi: 10.1007/978-3-642-14258-1. Google Scholar

[19]

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

[20]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, Second edition, (1999). Google Scholar

[21]

S. Smale, Diffeomorphisms with many periodic points,, Differential and Combinatorial Topology, (1965), 63. Google Scholar

[22]

P. Zgliczyński, Fixed point index for iterations, topological horseshoe and chaos,, Topol. Methods Nonlinear Anal., 8 (1996), 169. Google Scholar

[23]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon function,, Nonlinearity, 10 (1997), 243. doi: 10.1088/0951-7715/10/1/016. Google Scholar

[24]

P. Zgliczyński, Covering relation, cone conditions and the stable manifold theorem,, J. Differential Equations, 246 (2009), 1774. doi: 10.1016/j.jde.2008.12.019. Google Scholar

[25]

P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32. doi: 10.1016/j.jde.2004.03.013. Google Scholar

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