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Topological conjugacy for Lipschitz perturbations of non-autonomous systems
1. | Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300 |
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. |
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), 2591-2636.
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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