Conditional Lipschitz shadowing for ordinary differential equations

TL;DR

This paper introduces conditional Lipschitz shadowing for certain ODEs, providing criteria for when solutions closely follow pseudo-orbits within specific sets, with applications to models like the logistic equation.

Contribution

It establishes new sufficient conditions for conditional shadowing in non-autonomous ODEs, expanding understanding of solution stability within prescribed sets.

Findings

Conditions are proven to be optimal through examples.

Applicable to important classes of models including the logistic equation.

Provides criteria based on hyperbolicity and logarithmic norm for shadowing.

Abstract

We 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 non\-auto\-nomous 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.