A catastrophic approach to designing interacting hysterons

We present a framework for analyzing collections of interacting hysterons through the lens of catastrophe theory. By modeling hysteron dynamics as a gradient system, we show how to construct hysteron transition graphs by characterizing the fold bifurcations of the dynamical system. Transition graphs represent the sequence of hysterons switching states, providing critical insights into the collective behavior of driven disordered media. Extending this analysis to higher codimension bifurcations, such as cusp bifurcations and crossings of fold curves, allows us to map out how the topology of transition graphs changes with variations in system parameters. This approach can suggest strategies for designing metamaterials capable of encoding targeted memory and computational functionalities, but it also highlights the rapid increase of design complexity with system size, further underscoring the computational challenges of controlling large hysteretic systems.

💡 Research Summary

This paper introduces a novel framework for analyzing and designing networks of interacting hysterons (bistable elements) through the application of catastrophe theory. The core idea is to model the system of N hysterons as a gradient dynamical system, where the state of each hysteron is described by a continuous variable θ_i, and its time evolution follows the gradient of a potential energy function V(θ, γ, Ω). This potential combines a local quartic potential for each hysteron (providing two stable states) and pairwise harmonic interaction potentials between hysterons.

The primary methodological contribution is a three-step procedure to construct a “transition graph,” which maps the sequence of stable state switches as a global driving field γ is varied adiabatically. First, the authors identify all “fold bifurcation” points where stable states are created or annihilated. This is achieved by solving the combined system of equilibrium equations (F(θ,γ,Ω)=0) and the singularity condition of the Jacobian (det J=0), followed by applying Sotomayor’s theorem to classify each bifurcation. Plotting these bifurcation points against γ creates an “expanded bifurcation diagram,” revealing all metastable states available at different driving levels, including “Garden of Eden” states inaccessible via global driving.

Second, to determine the destination state after an annihilation event, the concept of an “escape route” is introduced. At a fold bifurcation, a stable node collides with a saddle point. The unstable manifold of this saddle defines the path the system takes immediately after the stable state vanishes. By integrating the gradient dynamics from near the bifurcation point, the destination stable state is unambiguously identified.

The framework is demonstrated using a concrete example of two interacting hysterons. The authors specify parameters, compute the fold bifurcations, characterize them, and use escape route analysis to build the complete directed transition graph, visually illustrating the process.

The paper further explores the implications of extending the analysis to higher-codimension bifurcations, such as cusps or crossings of fold curves, which occur when multiple system parameters (Ω) are varied. These points represent boundaries in parameter space where the topology of the transition graph itself changes, offering potential knobs for designing systems with specific memory or computational pathways. However, the paper concludes with a significant caveat: the complexity of the bifurcation landscape grows explosively with the number of hysterons N. This underscores a fundamental computational challenge in inversely designing large hysteretic systems with targeted functionalities, even with a powerful forward-analysis framework like the one presented. Thus, the work simultaneously opens a path for principled design and highlights the inherent difficulty in scaling such control.