On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions

Journal of Differential Equations

Volumn 224, Issue 1, 2006, Pages 1-38

  Li, Desheng ^a   Kloeden, Peter E ^b  

^a TIANJIN UNIVERSITY   (China)

^b GOETHE UNIVERSITY   (Germany)

Author keywords

Differential inclusion; Existence; General dynamical system; Periodic; Robustness; Small time delay; Stability; Uniform forward attractor

Indexed keywords

EID: 33645957100     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2005.07.012     Document Type: Article

References (51)

1. Arino O., and Pituk M. More on linear differential systems with small delays. J. Differential Equations 170 (2001) 381-407
2. Aubin J.P. Dynamic Economic Theory (1997), Springer, Berlin
3. Aubin J.P., and Cellina A. Differential Inclusions (1984), Springer, Berlin
4. Cao J.D., and Chen T.P. Global exponentially robust stability and periodicity of delayed neural networks. Chaos Solitons Fractals 22 (2004) 957-963
5. Caraballo T., and Langa J.A. Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise. J. Math. Anal. Appl. 260 (2001) 602-622
6. Caraballo T., and Langa J.A. Attractors of differential equations with variable delays. J. Math. Anal. Appl. 260 (2001) 421-438
7. Caraballo T., Marin-Rubio P., and Robinson J.C. A comparison between two theories for multivalued semiflows and their asymptotic behaviour. Set-Valued Anal. 11 (2003) 297-322
8. Cheban D., Kloeden P.E., and Schmalfuß B. The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems. Nonlinear Dynamics Systems Theory 2 (2002) 9-28
9. Chepyzhov V.V., and Vishik M.I. Attractors of periodic processes and estimate of their dimensions. Math. Notes 57 (1995) 127-140
10. Clarke F.H., Ledyaev Yu.S., and Stern R.J. Asymptotic stability and smooth Lyapunov functions. J. Differential Equations 149 (1998) 69-114
11. R. Datko, Time delay perturbations and robust stability, in: K.D. Elworthy, W. Norrie Everitt, E. Bruce Lee (Eds.), Differential Equations, Dynamical Systems and Control Science, Lecture Notes in Pure and Applied Mathematics, vol. 152, Marcel Dekker, New York, 1994, pp. 457-468.
12. Deimling K. Multivalued Differential Equations (1992), De Gruyter, Berlin
13. Driver R.D. Linear differential systems with small delays. J. Differential Equations 21 (1976) 149-167
14. Filippov A.F. Differential Equations with Discontinuous Righthand Side (1998), Kluwer Academic Publishers, Dordrecht
15. Friesecke C.R. Convergence to equilibrium for delay-diffusion equations with small delay. J. Dynamical Differential Equations 5 (1993) 89-103
16. Ghidagalia J.M., and Temam R. Periodic dynamical systems with applications to Sine-Gordon equation: estimates on the fractal dimension of the universal attractor. Contemp. Math. 99 (1989) 223-249
17. Gopalsamy K., and Sariyasa S. Time delays and stimulus dependent pattern formation in periodic enviroments in isolated neurons-II. Dynamics Continuous Discrete Impulsive Systems 9 (2002) 39-58
18. Guo F.M., Yu W., and Huang F.L. Robustness with respect to small delays for exponential stability of non-autonomous systems. J. Math. Anal. Appl. 288 (2003) 671-679
19. Gyöni I., and Pituk M. Special solutions of neutral functions differential equations. J. Inequalities Appl. 6 (2001) 99-117
20. Hale J.K. Asymptotic Behavior of Dissipative Systems (1988), American Mathematical Society, Providence, RI
21. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional Differential Equations (1993), Springer, Berlin
22. Hale J.K., and Verduyn Lunel S.M. Effects of small delays on stability and control. Operator Theory and Analysis (Amsterdam, 1997), Operator Theory and Advanced Applications vol. 122 (2001), Birkhäuser, Basel 275-301
23. Haraux A. Attractors of asymptotically compact processesses and applications to nonlinear partial differential equations. Comm. Partial Differential Equations 13 (1988) 1383-1414
24. Huang F.L. On the stability with respect to small delays for linear differential equations in Banach spaces. Chinese J. Math. 6 (1986) 183-191 (in Chinese)
25. Huang H., Cao J.D., and Qu Y.Z. Global robust stability of delayed neural networks with a class of general activation functions. J. Comput. Systems Sci. 69 (2004) 688-700
26. Jiang H.J., and Teng Z.D. Boundedness and stability for nonautonomous bidirectional associative neural networks with delay. IEEE Trans. Circuits Systems-II: Express Briefs 51 (2004) 174-180
27. Kapustyan A.V., and Valero J. Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr. Appl. Anal. 5 (2000) 33-46
28. Kharitonov V.L. Robust stability analysis of time delay systems: a survey. Ann. Rev. Control 23 (1999) 185-196
29. Kloeden P.E. Asymptotic invariance and limit sets of general control systems. J. Differential Equations 19 (1975) 91-105
30. Kloeden P.E. General control systems. In: Coppel W.A. (Ed). Mathematical Control Theory, Lecture Notes in Mathematics vol. 680 (1978), Springer, Berlin 119-138
31. Kloeden P.E., and Kozyakin V.S. The inflation of attractors and discretization: the autonomous case. Nonlinear Anal. TMA 40 (2000) 333-343
32. P.E. Kloeden, J. Valero, Attractors of weakly asymptotically compact setvalued dynamical systems, Set-Valued Anal., 2005, to appear.
33. Kolmanovskii V.B., and Nosoc V.R. Stability of Functional Differential Equations (1986), Academic Press, New York
34. Li D.S. On dynamical stability in general dynamical systems. J. Math. Anal. Appl. 263 (2001) 455-478
35. D.S. Li, P.E. Kloeden, Robustness of asymptotic stability to small time delays, Discrete Continuous Dynamical Systems 13 (2005) 1007-1034.
36. Li X.J., and Liu K.S. The effects of small delayes in the feedbacks on boundary stabilization. Sci. China 36 (1993) 1443-1453
37. Li D.S., and Zhang X.X. On the stability in general dynamical systems and differential inclusions. J. Math. Anal. Appl. 274 (2002) 705-724
38. Logemann H., Rebarber R., and Weiss G. Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J. Control Optim. 37 (1996) 572-600
39. Mao X.R. Exponential stability of nonlinear differential delay equations. Systems Control Lett. 28 (1996) 159-165
40. Melnik V.S., and Valero J. On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 6 (1998) 83-111
41. Mizuno M., and Ikeda K. An unstable mode selection rule: frustrated optical instability due to competing boundary conditions. Physica D 36 (1989) 327-342
42. Oliva S.M. Reaction-diffusion equations with nonlinear boundary delay. J. Dynamical Differential Equations 11 (1999) 279-296
43. Pituk M. Convergence to equilibria in a differential equation with small delay. Math. Bohemica 127 (2002) 293-299
44. Pituk M. Convergence to equilibria in scalar nonquasimonotone functional differential equations. J. Differential Equations 193 (2003) 95-130
45. Singh V. Global robust stability of delayed neural networks: an LMI appraoch. IEEE Trans. Circuits Systems-II: Express Briefs 52 (2005) 33-36
46. Smith H.L. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (1995), American Mathematical Society, Providence, RI
47. Smith H.L., and Thieme H.R. Monotone semiflows in scalar non-quasi-monotone functional differential equations. Amer. Math. Anal. Appl. 150 (1990) 289-306
48. Ryabov Yu.A. Certain asymptotic properties of linear systems with small time lag. Trudy Sem. Teor. Diff. Uravnenii s Otklon. Argumentom Univ. Druzby Narodov Patrisa Lumumby 50 (1965) 153-164 (in Russian)
49. Wang Y.J., Li D.S., and Kloeden P.E. On the asymptotical behavior of nonautonomous dynamical systems. Nonlinear Anal. TMA 59 (2004) 35-53
50. Xu B. Stability robustness bounds for linear systems with multiple time-varying delayed perturbations. Internat. J. Systems Sci. 28 (1997) 1311-1317
51. Zhong C.K., Li D.S., and Kloeden P.E. Uniform attractors of periodic and almost periodic dynamical systems. Discrete Continuous Dynamical Systems 12 (2005) 213-232