TOPOLOGICAL LINEARIZATION OF DEPCAGS WITH UNBOUNDED NONLINEAR TERMS

Changwu Zou; Jinlin Shi

doi:10.11948/2017021

2017 Volume 7 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(1): 309-333

Next Article Previous Article

Changwu Zou, Jinlin Shi. TOPOLOGICAL LINEARIZATION OF DEPCAGS WITH UNBOUNDED NONLINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 309-333. doi: 10.11948/2017021

Citation:

Changwu Zou, Jinlin Shi. TOPOLOGICAL LINEARIZATION OF DEPCAGS WITH UNBOUNDED NONLINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 309-333. doi: 10.11948/2017021

TOPOLOGICAL LINEARIZATION OF DEPCAGS WITH UNBOUNDED NONLINEAR TERMS

Changwu Zou^1,2,
Jinlin Shi²

1 School of Mathematical Sciences, Peking University, Beijing, 100871, China;
2 College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

Fund Project:

Abstract

Abstract

In this paper, we study the global topological linearization of a differential equation with piecewise constant argument of generalized type (DEPCAG) when the nonlinear term is unbounded. Some sufficient conditions are established for the topological conjugacy between a nonlinear system and its linear system. Our work generalizes the main result of Pinto and Robledo in[25].
- Piecewise constant argument /
- global topological linearization /
- unbounded nonlinear term /
- exponential dichotomy
MSC: 34A30;34D09;34K34

FullText(HTML)

References

[1]	A. R. Aftabizadeh, J. Wiener and J. M. Xu, Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc., 99(1987), 673-679. Google Scholar
[2]	M. U. Akhmet, Nonlinear Hybrid Continuous/Discrete Time Models, Atlantis Press, Paris, 2011. Google Scholar
[3]	M. U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument, Discontinuity, Nonlinearity and Complexisty, 1(2012), 337-352. Google Scholar
[4]	M. U. Akhmet and E. Yilmaz, Neural Networks with Discontinuous/Impact Activations, Nonlinear Systems and Complexisty, Springer, New York, 2014. Google Scholar
[5]	L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics, J. Diff. Eqs., 228(2006), 285-310. Google Scholar
[6]	L. Barreira and C. Valls, A simple proof of the Grobman-Hartman theorem for the nonuniformly hyperbolic flows, Nonlinear Anal., 74(2011), 7210-7225. Google Scholar
[7]	A. Cabada, J. B. Ferreiro and J. J. Nieto, Green's function and comparison principles for first order differential equations with piecewise constant arguments, J. Math. Anal. Appl., 291(2004), 690-697. Google Scholar
[8]	K. L. Cooke and J. Wiener, Oscillations in systems of differential equations with piecewise constant delays, J. Math. Anal. Appl., 137(1989), 221-239. Google Scholar
[9]	L. Dai, Nonlinear Dynamics of Piecewise Constants Systems and Implementation of Piecewise Constants Arguments, World Scientific, Singapore 2008. Google Scholar
[10]	D. R. Grobman, The topological classification of vicinity of a singular point in n-dimensional space, Math. Ussr-sb, 56(1962), 77-94. Google Scholar
[11]	P. Hartman, On the local linearization of differential equation, Proc. Amer. Math. Soe., 14(1963), 568-573. Google Scholar
[12]	Z. Huang, Y. Xia and X. Wang, The existence and exponential attractivity of k-almost periodic sequence solution of discrete time neural networks, Nonlinear Dyn., 50(2007), 13-26. Google Scholar
[13]	L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315(2006), 474-490. Google Scholar
[14]	L. Jiang, Strongly topological linearization with generalized exponential dichotomy, Nonlinear Anal., 67(2007), 1102-1110. Google Scholar
[15]	J. Kurzweil and G. Papaschinopoulos, Topological equivalence and structural stability for linear difference equations, J. Diff. Eqs., 89(1991), 89-94. Google Scholar
[16]	J. López-Fenner and M. Pinto, On a Hartman linearization theorem for a class of ODE with impulse effect, Nonlinear Anal., 38(1999), 307-325. Google Scholar
[17]	Y. Nakata, Global asymptotic stability beyond 3/2 type stability for a logistic equation with piecewise constants arguments, Nonlinear Anal., 73(2010), 3179-3194. Google Scholar
[18]	K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41(1973), 752-758. Google Scholar
[19]	K. J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl., 51(1975), 243-255. Google Scholar
[20]	G. Papaschinopoulos, Exponential dichotomy, topological equivalence and structural stability for differential equations with piecewise constant argument, Analysis, 145(1994), 239-247. Google Scholar
[21]	G. Papaschinopoulos, A linearization result for a differential equation with piecewise constant argument, Analysis, 16(1996), 161-170. Google Scholar
[22]	M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Diff. Equ. Appl., 17(2011), 721-735. Google Scholar
[23]	C. Pötzche, Topological decoupling, linearization and perturbation on inhomogeneous time scales, J. Diff. Eqs., 245(2008), 1210-1242. Google Scholar
[24]	A. Seuret, A novel stability analysis of linear systems under asynchronous samplings, Automatica, 48(2012), 177-182. Google Scholar
[25]	J. Shi and K. Xiong, On Hartman's linearization theorem and Palmer's linearization theorem, J. Math. Anal. Appl., 92(1995), 813-832. Google Scholar
[26]	J. Shi and J. Zhang, Classification of the Differential Equations, Science Press, Beijing, 2003. Google Scholar
[27]	J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. Google Scholar
[28]	L. Wang, Y.Xia and N. Zhao, A characterization of generalized exponential dichotomy, J. Appl. Anal. Comput., 5(2015), 662-687. Google Scholar
[29]	Y. Xia, J. Cao and M. Han, A new analytical method for the linearization of dynamic equation on measure chains, J. Diff. Eqs, 235(2007), 527-543. Google Scholar
[30]	Y. Xia, X. Chen and V. G. Romanovski, On the linearization theorem of Fenner and Pinto, J. Math. Anal. Appl., 400(2013), 439-451. Google Scholar
[31]	Y. Xia, J. Li and P. J. Y. Wong, On the topological classification of dynamic equations on time scales, Nonlinear Analysis:RWA, 14(2013), 2231-2248. Google Scholar
[32]	R. Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise constant argument, Nonlinear Anal., 48(2002), 1013-1032. Google Scholar