Figure 1

(a) Two-parameter phase diagram of a two-oscillator system showing the dependence on $\alpha$ and $\beta$. The asymptotic states can be classified into three types: symmetric, asymmetric and chaotic. In the bistable region, both the symmetric and asymmetric states are stable. The bottom graphs elucidate that the nature of $\Lambda (\Delta \varphi )$ changes with the parameter $\beta$. (b)–(e) Phase portraits of the system in the ($k_{12}$, $k_{21}$) plane obtained using an adiabatic approximation for each state. The blue and green lines represent the nullclines for $k_{12}$ and $k_{21}$, respectively. The black circles indicate stable fixed points.Reuse & Permissions

Figure 2

(a) Phase diagram of an $N$ oscillator system ($N=200$). The asymptotic states appearing in the regions depicted are a two-cluster state, a coherent state with a fixed phase relation, and a chaotic state with frustration. (b)–(d) Three typical simulation results. (left) Time evolution of the order parameters and the normalized rate of change averaged over all weights. (middle) Distribution of the phase ${\varphi }_i$ at $t=1000$. The inset displays the autocorrelation function of the phase pattern. (right) The weight matrix $k_{ij}$ in the final state ($t=1000$). The indices $i$ and $j$ of oscillators are arranged in order of increasing phase. The parameter value is $ϵ=0.005$, and the initial conditions for $k_{ij}$ were chosen randomly from a uniform distribution on $[-1,1]$.Reuse & Permissions

Figure 3

(a) A schematic illustration of co-evolution of both phase oscillators and network connections. 1. The phase pattern causes the structure of the weighted network to change. 2. The change undergone by the weights causes a new phase pattern to appear. 3. The change of the phase pattern results in further modulation of the weights of the network. 4. This process repeats. (b) A trajectory for the chaotic state of the two-oscillator system, with $\alpha =0.1\pi$, $\beta =0.55\pi$, and $ϵ=0.005$. (c) Lyapunov exponents as functions of $ϵ$. The inset displays a log-log plot, with a fitting curve satisfying $\lambda \propto {ϵ}^{1/2}$. (d) Mutual information between the initial and final phase patterns and the entropy of the final phase pattern for the three asymptotic states. The parameter values here are the same as in (b), (c), and (d) of Fig. 2. The horizontal dashed line represents the maximum attainable entropy for the phase pattern.Reuse & Permissions