# American Institute of Mathematical Sciences

January  2016, 21(1): 227-243. doi: 10.3934/dcdsb.2016.21.227

## Realization of arbitrary hysteresis by a low-dimensional gradient flow

 1 Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, United States

Received  May 2015 Revised  July 2015 Published  November 2015

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.
Citation: Dmitrii Rachinskii. Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 227-243. doi: 10.3934/dcdsb.2016.21.227
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