A catastrophic approach to designing interacting hysterons

TL;DR

The paper addresses how to predict and design collective transitions in networks of interacting hysterons by merging catastrophe theory with gradient-dynamics, linking fold bifurcations to state-switch sequences under a global drive $\\gamma$ and exploring higher-codimension bifurcations that alter transition graphs. It introduces a concrete procedure to construct transition graphs from fold points, uses Sotomayor-type nondegeneracy conditions to locate folds, and leverages unstable-manifold dynamics to determine escape routes, with explicit analysis of a two-hysteron example. The work extends the framework to cusp and crossing bifurcations (codimension-2) and discusses higher-codimension scenarios, showing how parameter variations carve the two-parameter space into regions each associated with a distinct graph topology and phenomena such as avalanches and Garden of Eden states. The findings illuminate both the design potential of metamaterials encoding memory and computation and the practical challenges of scaling to larger systems, including the need for advanced numerical methods to locate higher-codimension bifurcations in large networks.

Abstract

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.

Figures (7)

• Figure 1: (a) Schematic of a single hysteron showing its two stable configurations, 0 and 1. Bottom: A hysteron with continuous nonlinear relationship between the two conjugate variables $\gamma$ and $\theta,$ which we call a continuous hysteron. Top: A binary hysteron. (b) Mechanical rotors connected to a driving field and to other hysterons through linear springs paulsen2024mechanical. Below the schematic, the equilibrium equations for the system are shown, where $F_i$ represents applied torque on rotor $i$ from the displacement field, while $F_{ij}$ represents torque applied on hysteron $i$ from hysteron $j.$ (c) A system of connected balloons inflated under volume control muhaxheri2024bifurcations, and the equilibrium equations for that system.
• Figure 2: Obtaining the transition graph for an example of two hysterons. (a) Each fold bifurcation point with varying $\gamma$ is shown. Creation and annihilation label on the points assumes increasing $\gamma,$ while that labeling is inverted when $\gamma$ is decreasing. The black (orange) arrows represent transitions with increasing (decreasing) $\gamma.$ The figure shows that the system stays in a state until it is annihilated through a fold bifurcation point, in which case it transitions to a different state - that is shown through the diagonal arrows. (b) The transition graph for this system is shown. The dashed lines represent transitions out of a 'Garden of Eden' state, which is a state that cannot be accessed by varying global field $\gamma.$ The 'Garden of Eden' state is denoted by an orange vertex. (c) The dynamics of the system are shown for the $\gamma$ domain highlighted in part (a), explicitly showing the formation of the escape route, denoted by the red arrow, during the fold bifurcation involving the state $(0,0).$ The escape route directs the system to transition from $(0,0)$ to $(1,0).$
• Figure 3: The expanded bifurcation diagram for the example in Sec. \ref{['sec: example']}, with varying parameter $c_1$ showing a dual cusp bifurcation involving the state $(0,0)$ which separates the diagram into two domains of the parameter $c_1,$ with each domain showing a different transition graph topology. In the transition graphs, an orange vertex denotes a 'Garden of Eden' state.
• Figure 4: The expanded bifurcation diagram for the example in Sec. \ref{['sec: example']}, with varying interaction parameter, $k_{12},$ showing a crossing of fold bifurcation curves involving the states $(0,0)$ and $(0,1)$ which separates the diagram into two domains of the parameter $k_{12}.$ In the lower domain, when the state $(0,0)$ annihilates, it can transition to state $(0,1),$ but that is not possible in the upper domain after the crossing of the curves. The state $(0,0)$ can only transition to state $(1,1),$ thus causing an avalanche, and turning state $(0,1)$ into a Garden of Eden state.
• Figure 5: The reduced two-dimensional parameter space that includes the varying parameters $c_1$ and $k_{12}$ is shown, together with curves that represent codimension $2$ bifurcations. Each point in the dashed curves represents a dual cusp bifurcation point while each point in the undashed curves represents a crossing of fold bifurcation curves. Essentially, each point in any of these curves represents a codimension $2$ bifurcation point, and the points at which these curves meet represent bifurcations of codimension $3,$ represented by the black dots in the figure. Dashed arrows in the transition graphs are used for Garden of Eden states, and thicker arrows indicate an avalanche.