Regular Attractor by Strict Lyapunov Function for Random Dynamical Systems
DOI:
https://doi.org/10.31185/wjps.638Keywords:
Lyapunov's second method, strict Lyapunov functionAbstract
The main objective of this paper is to study some types of random attractors in random dynamical systems based on the random strict Lyapunov function. Where we first defined the random strict Lyapunov function and then the definition of the gradient random dynamical system and studied the finding of global attractors based on the existence of the random strict Lyapunov function. Last but not least, we introduced a new type of random attractor, which we called the random regular attractor, and proved the relationship between the existence of a regular attractor and the existence of a random strict Lyapunov function.
References
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin/ Heidelb-erg/New York, (1998).
Ludwig Arnold and Björn Schmalfuss, Lyapunov’s Second Method for Random Dynamical Systems, Journal of Differential Equations 177, 235–265, (2001).
A. Bacciotti and L. Rosier, Lyapunov and Lagrange stability inverse theorem for discontinuous system, Math. Control Signals Systems, 11, 101–128, (1998).
N. Bhatia and G. Szegö, Stability Theory of Dynamical Systems, Springer-Verlag, Berlin/Heidelberg/New York, (1970).
D. C. Biles and E. Schechter, Solvability of a finite or infinite system of discontinuous quasmonotone differential equations, Proc. Amer. Math. Soc., 128( 11), 3349– 3370, (2000).
Caraballo, J. A. Lange, and Z. Liu, Gradient Infinite-Dimensional Random Dynamical Systems, Siam J. Applied Dynamical Systems, 11(4), 1817–1847, (2012).
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probable. Theory Related Fields 100, 375–393, (1994).
H. Crauel, L.H. Duc and S. Siegmund, Towrds a Morse theory for random dynamical systems, Stochastics and Dynamics, 4 ,277-296, (2004).
V. S. Chellaboina, A. Leonessa and W. M. Haddad, Generalized Lyapunov and invariant set theorem for nonlinear dynamic systems, System & Control Letters, 38, 289– 295, (1999).
I. Chueshov, Monotone Random Dynamical Systems, Springer-Verlag Berlin Heidelberg, (2002).
C. Conley(1978), Isolated Invariant Sets and the Morse Index, (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society).
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise, Stochastics Stochastics Reports 59, 21–45, (1996).
D. Gulick, Encounters with Chaos,2nd edition. McGraw-Hill, Inc, (2012).
X. Ju, A. Qi and J. Wang, Strong Morse–Lyapunov functions for Morse decompositions of attractors of random dynamical systems, Stochastics and Dynamics 18(1), 1850012, (2018).
I. J. Kadhim and M. S. Imran, Stability and Asymptotic Stability of Closed Random Sets, Journal of Physics: Conference Series 1664 , 012049, (2020).
P. Kloeden, Lyapunov functions for cocycle attractors in nonautonomous difference equations, Izv. Akad. Nauk Respub. Moldova Mat. 26, 32–42, (1998).
P. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Electron. J. Diff. Eqns., Conf. 05, 91–102, (2000).
M. A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci.Toulouse 9. [Translation of the Russian edition, Kharkov 1892, reprinted by Princeton University Press, Princeton, NJ, 1949 and 1952.], (1907).
Z. Liu, The random case of Conley’s theorem: II. The complete Lyapunov function", Nonlinearity 20, 1017-1030, (2007).
Z. Liu, Conley index for random dynamical systems, Journal of Differential Equations, 244(7), 1603-1628, (2008).
Z. Liu , S. Ji and M. Su, Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems, Stochastic and Dynamics 08 (04), 625-641, (2008).
V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197–231, (1968).
G.Q. Xu and S.P.Yung, Lyapunov stability of abstract nonlinear dynamic system in Banach space , Ima Journal Of Mathematical Control And Information, 20 (1), 105-127, (2003).
J. P. Parker, The Lorenz system as a gradient-like system, Nonlinearity, IOP Publishing Ltd & London Mathematical Society 37 (9), 095022 (17pp), (2024).
S. V. Zelik, Attractors. Then and now", Achievements of Mathematical Sciences 78, 4 (472), 53-198, (2023).
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