Realization of Arbitrary Hysteresis by a Low-dimensional Gradient Flow

TL;DR

This paper demonstrates that any graph representing state transitions in a gradient system with a scalar parameter can be realized by a two-dimensional gradient flow, providing insights into the genesis of hysteresis phenomena.

Contribution

It proves that all admissible transition graphs can be realized by 2D gradient flows, linking bifurcation dynamics to hysteresis modeling.

Findings

Any admissible graph can be realized in 2D gradient flows.

Transitions between stable equilibria are modeled by saddle-node bifurcations.

The results relate to the genesis of hysteresis phenomena, exemplified by the Preisach model.

Abstract

We consider gradient systems with an increasing potential that depends on a scalar parameter. As the parameter is varied, critical points of the potential can be eliminated or created through saddle-node bifurcations causing the system to transit from one stable equilibrium located at a (local) minimum point of the potential to another minimum along the heteroclinic connections. These transitions can be represented by a graph. We show that any admissible graph has a realization in the class of two dimensional gradient flows. The relevance of this result is discussed in the context of genesis of hysteresis phenomena. The Preisach hysteresis model is considered as an example.