Does nonuniform behavior destroy the structural stability?

Yuzhen Bai; Donal O’Regan; Yong-Hui Xia; Xiaoqing Yuan; Yuzhen Bai; Donal O’Regan; Yong-Hui Xia; Xiaoqing Yuan

doi:10.3934/math.2020359

AIMS Mathematics

2020, Volume 5, Issue 6: 5628-5638. doi: 10.3934/math.2020359

Previous Article Next Article

Research article Special Issues

Does nonuniform behavior destroy the structural stability?

1.
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P. R. China
2.
School of Mathematics, Statistics and Applied Mathematics, National university of Ireland, Galway, Ireland
3.
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, P. R. China

Received: 13 May 2020 Accepted: 23 June 2020 Published: 01 July 2020
MSC : 37F15, 34D09, 37B55, 37D25

This paper provides an answer if the nonuniform behavior can destroy the structural stability of nonlinear systems. We show that if the linear system $\dot{x}(t) = A(t)x(t)$ admits a nonuniform exponential dichotomy, then the perturbed nonautonomous system $\dot{x}(t) = A(t)x(t)+f(t, x)$ is structurally stable under suitable conditions.
- stability,
- exponential dichotomy,
- nonautonomous,
- nonlinear system
Citation: Yuzhen Bai, Donal O’Regan, Yong-Hui Xia, Xiaoqing Yuan. Does nonuniform behavior destroy the structural stability?[J]. AIMS Mathematics, 2020, 5(6): 5628-5638. doi: 10.3934/math.2020359

Related Papers:

Abstract

This paper provides an answer if the nonuniform behavior can destroy the structural stability of nonlinear systems. We show that if the linear system $\dot{x}(t) = A(t)x(t)$ admits a nonuniform exponential dichotomy, then the perturbed nonautonomous system $\dot{x}(t) = A(t)x(t)+f(t, x)$ is structurally stable under suitable conditions.

References

[1]	L. Barreira, Ya. Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press, 2007.
[2]	L. Barreira, C. Valls, Stability of Nonautonomous Differential Equations,Lecture Notes in Mathematics, vol. 1926, Springer, 2008.
[3]	J. Chu, Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math., 137 (2013), 1031-1047. doi: 10.1016/j.bulsci.2013.03.003
[4]	J. Chu, F. Liao, S. Siegmund, et al. Nonuniform dichotomy spectrum and reducibility for nonautonomous equations, Bull. Sci. Math., 139 (2015), 538-557. doi: 10.1016/j.bulsci.2014.11.002
[5]	J. Zhang, M. Fan, H. Zhu, Nonuniform (h,k,u,v)-dichotomy with applications to nonautonomous dynamical systems, J. Math. Anal. Appl., 452 (2017), 505-551. doi: 10.1016/j.jmaa.2017.02.064
[6]	J. Zhang, M. Fan, H. Zhu, Existence and roughness of exponential dichotomies of linear dynamic equations on time scales, Comput. Math. Appl., 59 (2010), 2658-2675. doi: 10.1016/j.camwa.2010.01.035
[7]	O. Perron, Die stabilitsfrage bei differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662
[8]	W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978.
[9]	Y. Gao, Y. Xia, X.Yuan, et al. Linearization of nonautonomous impulsive system with nonuniform exponential dichotomy, Abstr. Appl. Anal., Volume 2014, Article ID 860378, 7 pages.
[10]	J. Hong, R. Yuan, Z. J. Jing, Exponential dichotomies, almost periodic structurally stable differential systems, and an example, J. Math. Anal. Appl., 208 (1997), 71-84. doi: 10.1006/jmaa.1997.5290
[11]	L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315 (2006), 474-490. doi: 10.1016/j.jmaa.2005.05.042
[12]	S. G. Kryzhevich, V. A. Pliss, Structural stability of nonautonomous systems, Differential Equations, 39 (2003), 1395-1403. doi: 10.1023/B:DIEQ.0000017913.79915.b1
[13]	F. Lin, Exponential dichotomies, Anhui University Press, Hefei, China, 1999 (in Chinese).
[14]	F. Lin, Almost periodic structural stability of almost periodic differential equations, Ann. Diff. Equat., 5 (1989), 35-50.
[15]	K. Lu, Structural stability for scalar parabolic equations, J. Differential Equations, 114 (1994), 253-271. doi: 10.1006/jdeq.1994.1150
[16]	L. Markus, Structurally stable differential systems, Ann. Math., 73 (1961), 1-19. doi: 10.2307/1970280
[17]	K. J. Palmer, The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Differential Equations, 33 (1979), 16-25. doi: 10.1016/0022-0396(79)90076-7
[18]	J. Kurzweil, G. Papaschinopoulos, Structural stability of linear discrete systems via the exponential dichotomy, Czech. Math. J., 38 (1988), 280-284.
[19]	G. Papaschinopoulos, G. Schinas, Structural stability via the density of a class of linear discrete systems, J. Math. Anal. Appl., 127 (1987), 530-539. doi: 10.1016/0022-247X(87)90127-2
[20]	J. Wang, M. Fěckan, Y. Zhou, Center stable manifold for planar fractional damped equations, Appl. Math. Comput., 296 (2017), 257-269.
[21]	J. Wang, M. Li, M. Fěckan, Robustness for linear evolution equations with non-instantaneous impulsive effects, Bull. Sci. Math., 159 (2020), 102827.
[22]	L. Popescu, Topological classification and structural stability of strongly continuous groups, Integr. Equ. Oper. Theory, 79 (2014), 355-375. doi: 10.1007/s00020-014-2152-y
[23]	J. Shi, J. Zhang, The Principle of Classification for Differential Equations, Science Press, Beijing, 2003 (in Chinese).
[24]	J. C. Willems, Topological classification and structural stability of linear systems, J. Differential Equations, 35 (1980), 306-318. doi: 10.1016/0022-0396(80)90031-5
[25]	Y. Xia, X. Chen, V. G. Romanovski, On the linearization theorem of Fenner and Pinto, J. Math. Anal. Appl., 400 (2013), 439-451. doi: 10.1016/j.jmaa.2012.11.034
[26]	Y. Xia, R. Wang, K. Kou, et al. On the linearization theorem for nonautonomous differential equations, Bull. Sci. Math., 139 (2015), 820-846.
[27]	Y. Xia, Y. Bai, D. O'Regan, A new method to prove the nonuniform dichotomy spectrum theorem in Rn, Proc. Amer. Math. Soc., 149 (2019), 3905-3917.
[28]	C. Zou, Y. Xia, M. pinto, et al. Boundness and Linearisation of a class of differential equations with piecewise constant argument, Qual. Theor. Dyna. Syst., 18 (2019), 495-531. doi: 10.1007/s12346-018-0297-9
[29]	C. Zou, Y. Xia, M. Pinto, Hölder Continuity of Topological Equivalence Functions of DEPCAGs, Sci. China Math., 50 (2020), 847-872.
[30]	A. Algaba, N. Fuentes, E. Gamero, et al. Structural stability of planar quasi-homogeneous vector fields, J. Math. Anal. Appl., 468 (2018), 212-226. doi: 10.1016/j.jmaa.2018.08.005
[31]	X. Jarque, J. Llibre, Structural stability of planar Hamiltonian polynomial vector fields, Proc. Lond. Math. Soc., 68 (1994), 617-640.
[32]	J. Llibre, J. P. del Río, J. A. Rodríguez, Structural stability of planar homogeneous polynomial vector fields: Applications to critical points and to infinity, J. Differential Equations, 125 (1996), 490-520. doi: 10.1006/jdeq.1996.0038
[33]	J. Llibre, del Río, J. A. Rodríguez, Structural stability of planar semi-homogeneous polynomial vector fields: Applications to critical points and to infinity, Discrete Contin. Dyn. Syst., 6 (2000), 809-828. doi: 10.3934/dcds.2000.6.809
[34]	D. S. Shafer, Structure and stability of gradient polynomial vector fields, J. Lond. Math. Soc., 41 (1990), 109-121.
[35]	G. Wang, Y. L. Cao, Dynamical spectrum in random dynamical systems, J. Dynamic. Diff. Equat., 26 (2014), 1-20. doi: 10.1007/s10884-013-9340-3
[36]	R. Oliveira, Y. Zhao, Structural stability of planar quasi-homogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39-72. doi: 10.1007/s12346-013-0105-5

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)