Route to extreme events in excitable systems

Systems of FitzHugh–Nagumo units with different coupling topologies are capable of self-generating and -terminating strong deviations from their regular dynamics that can be regarded as extreme events due to their rareness and recurrent occurrence. Here we demonstrate the crucial role of an interior crisis in the emergence of extreme events. In parameter space we identify this interior crisis as the organizing center of the dynamics by employing concepts of mixed-mode oscillations and of leaking chaotic systems. We find that extreme events occur in certain regions in parameter space, and we show the robustness of this phenomenon with respect to the system size.

Figure 1

Exemplary temporal evolutions of both model systems for different coupling strengths. For consistency with the continuation calculations (cf. Fig. 4), we use $x_1$ instead of $\overline{x}$ as the main observable for system A.

Figure 2

Top row: Interevent intervals $t_{\mathrm{IEI}}$ for system A observed in $100000$ time units in dependence of the coupling strength $k$. For each $k$, the vertical slice of the diagram corresponds to a histogram with logarithmic bins, where the number of occurrences for each bin is color coded. Middle row: Same as top but for local maxima $x_1^{\max }$ of $x_1$, with a linear ordinate, and for 20 observations of 5000 time units each (different initial conditions). We consider a value $x_1\left(t\right)$ a local maximum if $x_1(t-h)<x_1\left(t\right)>x_1(t+h)$, where $h$ is the sampling time. The H marks the Hopf bifurcation. Bottom row: Event rate $R$.

Figure 3

Top: Detail of the second row of Fig. 2. Bottom: Attractor projection on the $(x_1,y_1)$ plane before ($k=0.127$, left) and after the interior crisis ($k=0.128$, right).

Figure 4

Continuation of certain periodic orbits (primary mixed-mode oscillations) and the origin superimposed over the original bifurcation diagram. The H marks a Hopf bifurcation, PD a period doubling, and LPC corresponds to the limit point (saddle-node) bifurcation of cycles. $L^S$ marks a mixed-mode oscillation with $L$ consecutive high-amplitude and $S$ consecutive low-amplitude oscillations.

Figure 5

Same as Fig. 2 but with ${10}^8$ time units of observation of system B, with the local maxima of $\overline{x}$ being used in the second row, and with a zoom-in of the second row in the bottom row. The two arrows in the second row mark $k=0.2078$ and $k=0.2151$ (see Fig. 6).

Figure 6

Left: Third maximum map for $k=0.2078$ in the vicinity of the largest amplitude peak, showing an approaching tangent bifurcation which stabilizes the $2^1$ MMO (see also Fig. 5). Right: Fourth maximum map showing the approaching tangent bifurcation before the stabilization of the primary $3^1$ MMO at $k=0.2151$ (left). The black solid lines show the identity function for reference.

Figure 7

Top: Event rate $R$ (color-coded) depending on the number of oscillators $n$ and on the normalized coupling strength $k$ for generalizations of system B, using the same parameters. Middle: Estimated fraction $q\left(\tilde{R}\right)$ of $k$ values with $0<R<\tilde{R}$ for several thresholds $\tilde{R}$, i.e., $q\left(\tilde{R}\right):={\int }_{0.125}^{0.200}\Theta \left(R\left(k\right)\right)\Theta [\tilde{R}-R\left(k\right)]dk$, where $\Theta$ denotes the Heaviside function and $\Theta \left(0\right):=0$. For $n>101$, the scaling of the abscissa is centupled, and the resolution of the ordinate and thus the accuracy of $q\left(\tilde{R}\right)$ is reduced. Bottom: $R$ in dependence of $n$ for selected $k$: black dashed line: $k=0.129$; red solid line: $k=0.139$.

Figure 8

Left: Same as the top row of Fig. 2 but with $k$ fixed to $0.128$ and $a$ varied. Right: Same as the top row of Fig. 5 but with $k$ fixed to $0.128$ and $a$ varied.

Figure 9

Top: Zoom into the bifurcation diagram shown in Fig. 4. The H marks a Hopf bifurcation. Bottom: Dependence of the two largest Lyapunov exponents ${\Lambda }_1$ (black) and ${\Lambda }_2$ (green) on the coupling strength $k$.