The discontinuous dynamics and non-autonomous chaos

TL;DR

This paper demonstrates that impulsive differential equations can generate complex, chaotic behavior with unique properties like intermittency, highlighting their significance in understanding non-autonomous dynamical systems.

Contribution

It introduces a new class of chaos in impulsive differential equations and shows their potential to exhibit complex dynamics distinct from continuous systems.

Findings

Existence of a chaotic attractor in impulsive differential equations

Presence of intermittency phenomena in solutions

Impulsive systems can exhibit complex behaviors different from continuous dynamics

Abstract

A multidimensional chaos is generated by a special initial value problem for the non-autonomous impulsive differential equation. The existence of a chaotic attractor is shown, where density of periodic solutions, sensitivity of solutions and existence of a trajectory dense in the set of all orbits are observed. The chaotic properties of all solutions are discussed. An appropriate example is constructed, where the intermittency phenomenon is indicated. The results of the paper are illustrating that impulsive differential equations may play a special role in the investigation of the complex behavior of dynamical systems, different from that played by continuous dynamics.