Nonexistence of invariant manifolds in fractional order dynamical systems

TL;DR

This paper investigates the existence of invariant manifolds in fractional order dynamical systems, establishing conditions for invariance and proving the nonexistence of certain nonlinear invariant structures.

Contribution

It provides new theoretical results on the nonexistence of nonlinear invariant manifolds beyond linear subspaces in fractional order systems.

Findings

Invariant lines and parabolas in polynomial systems are characterized.

Conditions for invariance of linear subspaces in fractional systems are given.

Nonexistence of invariant manifolds other than linear subspaces is proven.

Abstract

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.