Conditional Lipschitz shadowing for ordinary differential equations

L. Backes, D. Dragicevic, M. Onitsuka and M. Pituk

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 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.

Journal of Dynamics and Differential Equations

36

3535

3552

https://doi.org/10.1007/s10884-023-10246-6