Dynamic self-loops in networks of passive and active binary elements

Abstract

Models of coupled binary elements capture memory effects in complex dissipative materials, such as transient responses or sequential computing, when their interactions are chosen appropriately. However, for random interactions, self-loops - cyclic transition sequences incompatible with dissipative dynamics - dominate the response and undermine statistical approaches. Here we reveal that self-loops originate from energy injection and limit cycles in the underlying physical system. We furthermore introduce interaction ensembles that strongly suppress or completely eliminate self-loops, allowing statistical studies of memory in large dissipative systems. Our work opens a route towards a unified description of passive and active multistable materials using hysteron models.

Figures (31)

• Figure 1: (a) Crumpled sheet with bistable elements (ridges $i$ and $j$) shohat2022memory. (b) Abstract hysteron models employ interacting binary elements. (c) Variations in strength of physical elements map to asymmetric hysteron interactions ($|c_{ij}|>|c_{ji}|$). (d) Self-loop in a (partial) transition graph. When the switching thresholds satisfy $H^{+}(S^{1})\!<\!H^{+}(S^{0})$, $H^{-}(S^{2})\!>\!H^{+}(S^{0})$ and $H^{-}(S^{3})\!>\!H^{+}(S^{0})$ and the system is in state $S^{0}$, increasing the driving $H$ above its threshold $H^{+}(S^{0})$ triggers a self-loop as all states are unstable (Supplemental Material).
• Figure 2: Statistical measures for gaps and self-loops scale when plotted as function of $N J_0$ ($10^5$ samples; color from light to dark as $N$ increases from $2$ to $10$). (a) Probability $P_g^0$ of finding a gap at $H = 0$ (dashed line indicates slope $4$). (b) Averaged fraction of gaps $f_G$, where $f_G$ is defined as the ratio of the size of intervals where no stable states exist divided by $H^{-}(++\dots) - H^{+}(--\dots)$ (dashed line indicates slope $4$). (c) Probability $P_{sl}^0$ of finding at least one self-loops at $H = 0$ (dashed line indicates slope $4$). (d) Probability $P_{sl}$ of finding at least one self-loop for any value of $H$ (dashed line indicates slope $3$). Inset: The probability to be self-loop free, $1-P_{sl}$, decays to zero exponentially with $N$ for large couplings ($NJ_0 = 10^2$).
• Figure 3: Self-loops in networks containing active elements. (a-b) Force-displacement curves for a dissipative element with $d_i>0$ and $\Delta E_i<0$ (a, blue), and for an active element with $d_i < 0$ and $\Delta E_i>0$ (b, red). (c) Two serially coupled elements, with total displacement $U = x_1 + x_2$. (d) Four classes of behaviors for two serially-coupled elements as function of $d_1$ and $d_2$; here $(x_1^-, x_1^+) = (-1,1)$, $(x_2^-, x_2^+) = (0.2,0.4)$. (e) Serially coupling one active element and a linear spring (representing one of the elastic branches of the second element) can produce a gap without stable states ($U^+(\triangle) < U^-(\triangledown)$) if $d_1$ is strongly negative (Supplemental Material). (f) Regions where self-loops emerge for two coupled hysterons; for fixed $\sigma_1 = \sigma_2 = 0$ (green areas), and for $(\sigma_1,\sigma_2)$ determined for the physical system represented by the red dot in (d) (red boundaries). The blue hatched regions in (d,f) represent purely dissipative systems. Orange lines in (d,f) indicate active elements that lead to symmetric hysteron interactions. (g) Two $L=4$ self-loops observed in (d,f).
• Figure 4: Simulations of large systems of coupled hysterons in the constant-columns (left) and symmetric (right) ensembles ($N\!=\!16, 32, \dots, 512$ for increasingly dark colors). (a-b) Ensemble averaged avalanche size $\langle A \rangle$. To determine these, we initialize the system at a stable state $S^0$ at $H=0$, increase $H$, and measure the number of flips before the system settles on a stable state. (c-d) Ensemble averaged transient $\langle \tau \rangle$, where $\tau$ is the number of cyclic drive cycles after which the system reaches a periodic orbit (Supplemental Material).
• Figure 5: Relations between active and passive multistable physical systems, hysteron models, self-loops and symmetry of $c_{ij}$. Both active and passive systems can map to hysteron model with symmetric couplings ($c_{ij} = c_{ji}$, blue region); in this case, self-loops are forbidden. The region where self-loops occur forms a complex cloud of polytopes in the space of hysteron models (SL, green regions).