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Pullback incremental attraction. (English) Zbl 1308.37009

Consider the system of ordinary differential equations with time-varying right hand side \[ {{dx}\over{dt}} = f(t,x)\;,\;x\in R^d \]
The paper studies the so-called pullback qualitative behavior, i.e., the limit for \(t_0\to-\infty\) of some solution. The existence of pullback attractors for pullback incrementally attracting processes on some Banach space is shown.

MSC:

37B25 Stability of topological dynamical systems
34D45 Attractors of solutions to ordinary differential equations
37B55 Topological dynamics of nonautonomous systems
93D30 Lyapunov and storage functions
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References:

[1] D. Angeli, A Lyapunov approach to the incremental stability properties, IEEE Trans. Automat. Control 47 (2002), 410-421. · Zbl 1364.93552
[2] T. Caraballo, M.J. Garrido Atienza and B. Schmalfuß, Existence of exponentially attracting stationary solutions for delay evolution equations. Discrete Contin. Dyn. Syst. Ser. A 18 (2007), 271-293. · Zbl 1125.60058
[3] T. Caraballo, P.E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50 (2004), 183-207. · Zbl 1066.60058
[4] C.M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971), 239-252.
[5] L. Grüne, P.E. Kloeden, S. Siegmund and F.R. Wirth, Lyapunov’s second method for nonautonomous differential equations, Discrete Contin. Dyn. Syst. Ser. A 18 (2007), 375-403. · Zbl 1128.37010
[6] P.E. Kloeden, Lyapunov functions for cocycle attractors in nonautonomous difference equations, Izvetsiya Akad Nauk Rep Moldovia Mathematika 26 (1998), 32-42.
[7] P.E. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Electron. J. Differ. Equ. Conf. 05 (2000), 91-102. · Zbl 0964.34041
[8] P.E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl. 28 (2010), 937-945. · Zbl 1205.60131
[9] P.E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations 253 (2012), 1422- 1438. · Zbl 1267.37018
[10] P.E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011.
[11] B.S. Rüffer, N. van de Wouw and M. Mueller, Convergent systems vs. incremental stability, Systems Control Lett. 62 (2013), 277-285. · Zbl 1261.93072
[12] E.D. Sontag, Comments on integral variants of ISS, Systems Control Lett. 34 (1998), 93-100. · Zbl 0902.93062
[13] A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996. · Zbl 0869.65043
[14] Fuke Wu and P.E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B 18, No.6, (2013), 1715-1734. · Zbl 1316.34083
[15] T. Yoshizawa, Stability Theory by Lyapunov’s Second Method. Math. Soc Japan, Tokyo, 1966. · Zbl 0144.10802
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