Naslov
A general approach to nonautonomous shadowing for nonlinear dynamics

Autori
Backes, Lucas ; Dragičević, Davor

Izvornik
Bulletin des sciences mathématiques (0007-4497) 170 (2021); 102996, 30

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
shadowing ; nonautonomous systems ; nonlinear dynamics ; dichotomies

Sažetak
Given a nonautonomous and nonlinear differential equation \begin{; ; equation}; ; \label{; ; DE}; ; x'=A(t)x+f(t, x) \quad t\geq 0, \end{; ; equation}; ; on an arbitrary Banach space $X$, we formulate very general conditions for the associated linear equation $x'=A(t)x$ and for the nonlinear term $f:[0, +\infty)\times X\to X$ under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that $x'=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behaviour as a very particular case. In addition, we require that $f$ is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting a discrete time versions of our results.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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