Shadowing in the Neighborhood of a Hyperbolic Equilibrium Point for Fractional Equations
DOI:
https://doi.org/10.52575/2687-0959-2025-57-1-41-51Keywords:
Fractional Equations, Semilinear Cauchy Problems in Banach Space, Hyperbolic Equilibrium Point, Compact Convergence of Resolvents, General Approximation Scheme, ShadowingAbstract
In this paper, we study the behavior of trajectories of abstract parabolic problems with fractional time derivative in the neighborhood of a hyperbolic equilibrium point, where the fractional derivative is understood in the Caputo – Djerbashyan sense. It is well known that for dynamical systems with integer derivative, the phase space in the neighborhood of a hyperbolic equilibrium point splits in such a way that this initial value problem reduces to initial value problems with exponentially decreasing solutions in opposite directions. In the case of a fractional derivative, the situation changes dramatically. First, there is no exponential decay. Second, the spectrum of the linearized operator admits an expansion different from the classical picture. Nevertheless, we manage to prove analogs of the results on shadowing. The main conditions of our results are satisfied, in particular, for operators with a compact resolvent and can be verified for the finite elements method and difference methods.
Acknowledgements
The paper was carried out at the Research Computing Center of Moscow State University named after M. V. Lomonosov as part of the research work on the topic "Research and development of methods, algorithms and software in the field of computational mathematics"and with the support of the Russian Science Foundation (grant No. 23-21-00005).
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Vu Quoc Phong. The spectral radius, hyperbolic operators and Lyapunov’s theorem. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 187–194, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.
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Larsson S. Numerical analysis of semilinear parabolic problems. (English) Ainsworth, Mark (ed.) et al., The graduate student’s guide to numerical analysis ’98. Lecture notes from the 8th EPSRC summer school in numerical analysis. Leicester, GB, July 5-17, 1998. Berlin: Springer. Springer Ser. Comput. Math. 26, 83–117 (1999).
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Kokurin MM. The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space. Russian Mathematics. 2013;57:16–30.
Popov AYu, Sedletskiy AM. Distribution of the roots of the Mittag – Leffler functions Sovremennaja mathematica. Fundamental’nye Napravlenija. 2011;40:3–171.
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Liu R., Li M., Piskarev SI.: Approximation of semilinear fractional Cauchy problem. Applied Mathematics and Computation. 2015;15:203–212.
Siegmund S, Piskarev S. Approximations of stable manifolds in the vicinity of hyperbolic equilibrium points for fractional differential equations. Nonlinear Dynamics (NODY), 2019;95(1):685–697.
Piskarev S, Siegmund S. Unstable manifolds for fractional differential equations. Eurasian Journal of Mathematical and Computer Applications. 2022;10(3):58 – 72.
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Tuan Hoang The, Siegmund Stefan, Son Doan Thai, Cong Nguyen. An instability theorem for nonlinear fractional differential systems. Discrete and Continuous Dynamical Systems Series B. 2017;22(8):3079–3090.
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