Figure 1

(a) An example of the AKP for our model [Eqs. (1, 2)] with $N=10$ and $\beta =10$. The AKP is computed by fixing the $y$ value at the abscissa and numerically integrating the equations of the fast variables for ${10}^3$ time units from 10 randomly selected initial conditions. The ordinate $⟨h⟩$ is then computed from the temporal average of the last $8\times {10}^2$ time units of each initial condition. The $y$ value is incremented by 0.005 to obtain the plot. (b) The time series of $y$ for $ϵ=0.0001$. There are two coexisting attractors: a fixed point (green) and a limit cycle (red).Reuse & Permissions

Figure 2

(a) The AKP with $N=30$ for a given $J$ that produces complex oscillation. The AKP is computed in the same manner as Fig. 1 but from $3\times {10}^5$ to $3.5\times {10}^5$ time units over 20 randomly chosen initial conditions. The branches are labeled with lower case letters. Because of the symmetry in the model, branches for $y>0$ and $y<0$ are indicated by $+$ and $-$, respectively. (The branches $i$ and $j$ happen to take almost the same $⟨h⟩$ value but belong to different $\{x_j\}$ attractors). The numbers correspond to the time course plotted in part (b). (b) Time series of the slow variable $y$ for $ϵ=0.0001$. The numbers ($1,2,\dots ,12$) correspond to the branches visited there, as displayed in part (a). (c) The time series of the fast variables $x_i$ ($i=1,2,\dots ,9$) for the branches $b$, $c$, and $h$.Reuse & Permissions

Figure 3

Time series of the slow variables $y$ corresponding to Fig. 2, except for $ϵ=2\times {10}^{-6}$ (a), $ϵ=1\times {10}^{-5}$ (b), and $ϵ=1\times {10}^{-4}$ (c). The time series of (c) corresponds to Fig. 2b but is plotted for a longer time span.Reuse & Permissions

Figure 4

(a) The AKP computed for a different matrix $J$ with $N=20$ and the time average between ${10}^2$ to $2\times {10}^3$ time units from 15 initial conditions. The maximum Lyapunov exponent of the dynamics of $\{x_i\}$ is also computed over the interval of $3\times {10}^3$ to $4\times {10}^3$ time units for each branch of a given $y$. The segment of the branch with positive exponent (whose value is about $\sim .02$) is colored as blue. (See Supplemental Material [28], Fig. 2 for the Lyapunov exponents at each branch). (b) The time series of $y$ for $ϵ=0.001$.Reuse & Permissions

Figure 5

Frequency of the switching in the branches corresponding to the dynamics in Fig. 4 as a function of $ϵ$. For each $ϵ$ value, the number of $a+$ to $b+$ switching events divided by those from $a+$ to $a-$ are computed as the fraction of the number of times that $y$ goes beyond 0.2 divided by the number of times that it goes below 0 within 500 time units. Since at the branch $b+$, $y$ goes beyond 0.2, while at the branch $a+$ it cannot go beyond 0.11 as shown in Fig. 4, the threshold 0.2 is adopted to select the branch $b+$.Reuse & Permissions