Continuous-Time and Discrete-Time Quasilinear Systems with Asymptotically Unpredictable Solutions

TL;DR

This paper introduces the concept of asymptotically unpredictable solutions in quasilinear systems with delay, demonstrating their existence, uniqueness, and providing examples in continuous and discrete systems, advancing understanding of chaotic dynamics.

Contribution

It establishes the existence and uniqueness of asymptotically unpredictable solutions for delay quasilinear systems using contraction mapping, and introduces the notion of asymptotically unpredictable sequences.

Findings

Existence and uniqueness of asymptotically unpredictable solutions in quasilinear systems.

Introduction of asymptotically unpredictable sequences distinct from unpredictable ones.

Examples demonstrating asymptotically unpredictable solutions in continuous and discrete systems.

Abstract

A novel type of trajectory on semiflows, called asymptotically unpredictable, was proposed by Fen and Tokmak Fen [15]. The presence of sensitivity, which is an indispensable feature of chaotic dynamics, is a crucial property that arises from such trajectories. In the present paper, we show the existence and uniqueness of asymptotically unpredictable solutions for quasilinear systems with delay making benefit of the contraction mapping principle. Additionally, we introduce the notion of an asymptotically unpredictable sequence. It is verified that there exist asymptotically unpredictable sequences which are not unpredictable. Discrete-time equations possessing asymptotically unpredictable orbits are also under investigation. Examples of continuous-time and discrete-time systems with asymptotically unpredictable solutions are provided.