Author:
Backes Lucas,Dragičević Davor,Onitsuka Masakazu,Pituk Mihály
Abstract
AbstractWe introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain nonautonomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.
Funder
CNPq-Brazil PQ fellowship
Croatian Science Foundation
Japan Society for the Promotion of Science KAKENHI
Hungarian National Research, Development and Innovation Office
Publisher
Springer Science and Business Media LLC
Cited by
2 articles.
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1. Parameterized shadowing for nonautonomous dynamics;Journal of Mathematical Analysis and Applications;2024-01
2. Generalized Dichotomies and Hyers–Ulam Stability;Results in Mathematics;2023-12-04