Linearization of nonautonomous impulsive system with nonuniform exponential dichotomy. (English) Zbl 1474.34373

Gao, Yongfei; Xia, Yonghui; Yuan, Xiaoqing; Wong, P. J. Y.

Abstr. Appl. Anal. 2014, Article ID 860378, 7 p. (2014).

Summary: This paper gives a version of Hartman-Grobman theorem for the impulsive differential equations. We assume that the linear impulsive system has a nonuniform exponential dichotomy. Under some suitable conditions, we proved that the nonlinear impulsive system is topologically conjugated to its linear system. Indeed, we do construct the topologically equivalent function (the transformation). Moreover, the method to prove the topological conjugacy is quite different from those in previous works (e.g., see Barreira and Valls, 2006).

MSC:

34D09	Dichotomy, trichotomy of solutions to ordinary differential equations
34A37	Ordinary differential equations with impulses
34D10	Perturbations of ordinary differential equations

References:

[1]	Hartman, P., On the local linearization of differential equations, Proceedings of the American Mathematical Society, 14, 568-573 (1963) · Zbl 0115.29801 · doi:10.1090/S0002-9939-1963-0152718-3
[2]	Grobman, D., Homeomorphism of systems of differential equations, Doklady Akademii nauk SSSR, 128, 880-881 (1959) · Zbl 0100.29804
[3]	Palmer, K. J., A generalization of Hartman’s linearization theorem, Journal of Mathematical Analysis and Applications, 41, 3, 753-758 (1973) · Zbl 0272.34056 · doi:10.1016/0022-247X(73)90245-X
[4]	Fenner, J. L.; Pinto, M., On a Hartman linearization theorem for a class of ODE with impulse effect, Nonlinear Analysis: Theory, Methods and Applications, 38, 3, 307-325 (1999) · Zbl 0931.34007 · doi:10.1016/S0362-546X(98)00198-9
[5]	Xia, Y. H.; Chen, X.; Romanovski, V., On the linearization theorem of Fenner and Pinto, Journal of Mathematical Analysis and Applications, 400, 2, 439-451 (2013) · Zbl 1272.34048 · doi:10.1016/j.jmaa.2012.11.034
[6]	Xia, Y.; Cao, J.; Han, M., A new analytical method for the linearization of dynamic equation on measure chains, Journal of Differential Equations, 235, 2, 527-543 (2007) · Zbl 1126.34030 · doi:10.1016/j.jde.2007.01.004
[7]	Xia, Y. H.; Li, J.; Wong, P. J. Y., On the topological classication of dynamic equations on time scales, Nonlinear Analysis: Real World Applications, 14, 6, 2231-2248 (2013) · Zbl 1303.37007 · doi:10.1016/j.nonrwa.2013.05.001
[8]	Barreira, L.; Chu, J. F.; Valls, C., Lyapunov functions for general nonuniform dichotomies, Milan Journal of Mathematics, 81, 1, 153-169 (2013) · Zbl 1273.34057 · doi:10.1007/s00032-013-0198-y
[9]	Barreira, L.; Chu, J. F.; Valls, C., Robustness of nonuniform dichotomies with different growth rates, São Paulo Journal of Mathematical Sciences, 5, 2, 203-231 (2011) · Zbl 1272.34067
[10]	Chu, J. F., Robustness of nonuniform behavior for discrete dynamics, Bulletin des Sciences Mathématiques, 137, 37, 1031-1047 (2013) · Zbl 1291.37050 · doi:10.1016/j.bulsci.2013.03.003
[11]	Barreira, L.; Valls, C., Stability of nonautonomous differential equations in Hilbert spaces, Journal of Differential Equations, 217, 1, 204-248 (2005) · Zbl 1088.34053 · doi:10.1016/j.jde.2005.05.008
[12]	Barreira, L.; Valls, C., A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, Journal of Differential Equations, 228, 1, 285-310 (2006) · Zbl 1099.37022 · doi:10.1016/j.jde.2006.04.001