Mixed higher-order coupling stabilizes new states

Abstract

Understanding how higher-order interactions affect collective behavior is a central problem in nonlinear dynamics and complex systems. Most works have focused on a single higher-order coupling function, neglecting other viable choices. Here we study coupled oscillators with dyadic and three different types of higher-order couplings. By analyzing the stability of different twisted states on rings, we show that many states are stable only for certain combinations of higher-order couplings, and thus the full range of system dynamics cannot be observed unless all types of higher-order couplings are simultaneously considered.

Figures (4)

• Figure 1: Stability of twisted states under isolated and mixed higher-order coupling. (a--b) For a network of size $N=100$ with interaction radius $r=3$, the dominant non-trivial eigenvalue $\lambda_D$ of the Jacobian for all possible twisted states $\bm{\theta}(w)$, where $w$ is the normalized winding number $q/N$. Results for individual Jacobians $J_1$, $J_{2a}$, $J_{2b}$, $J_3$, and the mixed coupling case $J=K_1J_1+K_{2a}J_{2a}+K_{2b}J_{2b}+K_3J_3$ are plotted in orange diamonds, blue crosses, red squares, green triangles, and black circles, respectively. Note that (b) shows a zoomed-in view of the eigenvalue near zero to more easily identify stability, i.e., when $\lambda_D<0$. We summarize the respective regions of stability for twisted states in panel (c). (d--g) The evolution of a perturbation, $\|\bm{\theta}(t)-\bm{\theta}^*\|$ towards or away from the twisted states for $w=0.04$, $0.21$, $0.27$, and $0.42$. Solid orange, dashed blue, dot-dashed red, dotted green, and thick black curves represent systems with couplings given by only dyadic coupling, type a triadic coupling, type b triadic coupling, tetradic coupling, and mixed coupling, respectively. These winding numbers are also marked by vertical black lines in panels (a)--(c).
• Figure 2: Stability regions of twisted states for different coupling combinations. For a network of size $N=200$ with interaction radius $r=3$, regions where the twisted state $\bm{\theta}(w)$ are not stable (white), stable under some combination of dyadic and only one type of higher-order coupling (light blue), and stable only under some combination that includes two or more types of higher-order coupling (dark blue) as a function of the winding number $w=q/N$ and dyadic coupling $K_1$.
• Figure 3: Balancing of the eigenvalue sprectra explains how mixed higher-order interactions can stabilize states that are unstable under any individual interaction. For a network of size $N=100$ with interaction radius $r=3$, the eigenvalues $\lambda_p$ of the Jacobians for the twisted state with $w=0.42$ as a function of the index $p$ defining the eigenvector $\bm{v}^p$ with indices $v_j^p=\cos[2\pi(p-1)j/N]$. Eigenvalues for Jacobians $J_1$, $J_{2a}$, $J_{2b}$, $J_3$, and the mixed coupling case $J=K_1J_1+K_{2a}J_{2a}+K_{2b}J_{2b}+K_3J_3$ are plotted in orange diamonds, blue crosses, red squares, green triangles, and black circles, respectively.
• Figure 4: Heat map for the continuum limit eigenvalue function $\lambda^\infty(w,y)$ for the case $r=3$ with coupling $K_1=1/10$, $K_{2a}=1/2$, $K_{2b}=1/10$, and $K_3=3/10$. Regions where $\lambda^\infty(w,y)>0$ are colored white, i.e., the $y^\text{th}$ mode of the $w^\text{th}$ twisted state is unstable. The $w^\text{th}$ twisted state is stable when $\lambda^\infty(w,\cdot)<0$. The region $\mathcal{W}/\left(\mathcal{W}_1\cup\mathcal{W}_{2a}\cup\mathcal{W}_{2b}\cup\mathcal{W}_3\right)$, where only a mixture of higher-order coupling yields stability, is indicated with the red vertical bars.