# Parameterized shadowing for nonautonomous dynamics

## TL;DR

This paper establishes conditions for shadowing and stability in nonautonomous, nonlinear differential and difference equations depending on parameters, including cases with non-hyperbolic linear parts.

## Contribution

It provides new sufficient conditions for $C^k$ shadowing and Hyers-Ulam stability in parameter-dependent nonautonomous systems, extending previous results to non-hyperbolic cases.

## Key findings

- Conditions for $C^k$ shadowing in nonhyperbolic systems
- Results on parameterized Hyers-Ulam stability for hyperbolic systems
- Applicability to a broad class of nonlinear, nonautonomous equations

## Abstract

For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit $C^k$, $k\in \N$ shadowing with respect to a parameter. Our results are applicable to situations when the linear part is not hyperbolic. In the case when the linear part is hyperbolic, we obtain results dealing with parameterized Hyers-Ulam stability.