# Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems

## TL;DR

This paper explores chaotic behavior in time-varying discrete dynamical systems using Furstenberg families and scrambled pairs.

## Contribution

The paper introduces and studies G-scrambled pairs and defines new types of chaos for time-varying systems.

## Key findings

- G-scrambled pairs are defined and their properties are analyzed.
- Conditions for a system to be generically strongly G-chaotic are established.
- New definitions of chaos in time-varying discrete dynamical systems are proposed.

## Abstract

Assume that (Y,ρ) is a nontrivial complete metric space, and that (Y,g1,∞) is a time-varying discrete dynamical system (T-VDDS), which is given by sequences (gl)l=1∞ of continuous selfmaps gl:Y→Y. In this paper, for a given Furstenberg family G and a given T-VDDS (Y,g1,∞), G-scrambled pairs of points of the system (Y,g1,∞) (which contains the well-known scrambled pairs) are provided. Some properties of the set of G-scrambled pairs of a given T-VDDS (Y,g1,∞) are studied. Moreover, the generically G-chaotic T-VDDS and the generically strongly G-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly G-chaotic is also presented.