Nonlinear Dynamics

, Volume 92, Issue 4, pp 1975–1983 | Cite as

Autonomous models of self-crossing pinched hystereses for mem-elements

  • Wieslaw Marszalek
Original Paper


This paper discusses autonomous implicit models yielding self-crossing trajectories (hystereses with odd symmetry), typical for mem-elements. In particular, the lemniscates of Gerono and Devil result in two systems of differential implicit equations having both a folded saddle at the origin and a pair of equilibria (centers), each located on the opposite sides of singularity (fold curve). The folded saddle at the origin allows a self-crossing pinched hysteretic dynamics. Also, two dual circuits for the Gerono case are presented with the pinched hystereses for inductive (flux versus current) and capacitive (charge versus voltage) mem-elements. When parameters of the circuits change to yield an increased frequency, then the area enclosed by the hysteresis decreases. The time zero-crossing property of the mem-circuits’ signals is also discussed. The Devil’s lemniscate case is linked to Duffing’s equation. The autonomous models are differentiable, and the use of the sign and abs terms, typical in modeling of hystereses, is avoided.


Mem-elements Autonomous implicit equations Pinched hystereses Folded saddles 



The author would like to thank two anonymous reviewers for their constructive and helpful comments.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

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