Abstract:
In this survey, we have made an attempt to present the contemporary ideas and methods of investigation of qualitative dynamics of infinite dimensional dissipative systems. Essential concepts such as dissipativity and asymptotic smoothness of dynamical systems, global and fractal attractors, determining functionals, regularity of asymptotic dynamics are presented. We place the emphasis on the quasi-stability method developed by I. Chueshov and I. Lasiecka. The method is based on an appropriate decomposition of the difference of the trajectories into a stable and a compact parts. The existence of this decomposition has a lot of important consequences: asymptotic smoothness, existence and finite dimensionality of attractors, existence of a finite set of determining functionals, and (under some additional conditions) existence of a fractal exponential attractor. The rest of the paper shows the application of the abstract theory to specific problems. The main attention is paid to the demonstration of the scope of the quasi-stability method.
Key words and phrases:
infinite dimensional dynamical systems, asymptotic behavior, global attractors, fractal exponential attractors, determining functionals, finite fractal dimension, quasi-stability, stability, PDEs.
The last two authors were partially supported by the Volkswagen Foundation grant within the frameworks of the international project “Modeling, Analysis, and Approximation Theory toward Applications in Tomography and Inverse Problems”.
Citation:
Igor Chueshov, Tamara Fastovska, Iryna Ryzhkova, “Quasi-stability method in study of asymptotic behavior of dynamical systems”, Zh. Mat. Fiz. Anal. Geom., 15:4 (2019), 448–501
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\by Igor~Chueshov, Tamara~Fastovska, Iryna~Ryzhkova
\paper Quasi-stability method in study of asymptotic behavior of dynamical systems
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2019
\vol 15
\issue 4
\pages 448--501
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\crossref{https://doi.org/10.15407/mag15.04.448}
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Linking options:
https://www.mathnet.ru/eng/jmag739
https://www.mathnet.ru/eng/jmag/v15/i4/p448
This publication is cited in the following 1 articles:
Tamara Fastovska, Dirk Langemann, Iryna Ryzhkova, “Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam”, Front. Appl. Math. Stat., 10 (2024)
Citation in format AMSBIB
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