Extreme events in the forced Liénard system

We observe extremely large amplitude intermittent spikings in a dynamical variable of a periodically forced Liénard-type oscillator and characterize them as extreme events, which are rare, but recurrent and larger in amplitude than a threshold. The extreme events occur via two processes, an interior crisis and intermittency. The probability of occurrence of the events shows a long-tail distribution in both the cases. We provide evidence of the extreme events in an experiment using an electronic analog circuit of the Liénard oscillator that shows good agreement with our numerical results.

Figure 1

(a) Autonomous Liénard system: large periodic orbits in blue (black) lines coexist with a stable focus in black spiral at $(1,0)$, a saddle focus in red (dashed) spiral at $(-1,0)$, and a saddle origin for $\alpha =0.45$, $\beta =0.50$, and $\gamma =-0.50$. An innermost periodic orbit in blue (black) line close to a HO at the saddle origin is shown. (b) Basins of attraction of the autonomous system (white) and the forced (light gray) Liénard system for $A=0.2$, $\omega =0.7315$. The red (black) squares indicate a Poincaré section of the quasiperiodic attractor (gray lines) shown in (c), while the blue (black) circles represent the Poincaré section of the dense chaotic attractor in the middle with occasional large excursions shown in scattered blue (black) circles. A more clear picture of the chaotic attractor is given in Fig. 6. Trajectories of the forced Liénard system for $A=0.2$ and $\omega =0.7315$; bounded chaos with occasional large excursion of the trajectory in blue (black) lines that coexists with quasiperiodic attractor (gray line).

Figure 2

Numerical bifurcation diagrams are drawn with varying $\omega$ for two different initial conditions showing signatures of multistability. For initial conditions inside the HO, the top panel (a) shows a periodic regime at the left-hand side with a sudden expansion of attractor at $\omega =0.6423$ (arrow L) for increasing $\omega$. A similar expansion of the attractor is seen at $\omega =0.7315$ (arrow R) from a bounded chaos at a larger $\omega =0.7316$ (arrow C), when decreasing the $\omega$ value. Bottom panel (b) of bifurcation diagram is plotted for a choice of initial conditions outside the HO, which manifest a quasiperiodic motion with no significant change in amplitude with forcing frequency.

Figure 3

Temporal dynamics in (a)–(d) and three-dimensional (3D) trajectories in (e)–(h) of the Liénard system. $\alpha =0.45$, $\beta =0.50$, $\gamma =-0.50$, $A=0.20$, and varying $\omega$. (a) and (e) bounded chaos for $\omega =0.7316$, (b) and (f) extreme event for $\omega =0.7315$, (c) and (g) multiple large amplitude events for $\omega =0.68$, (d) and (h) extreme events at $\omega =0.6423$. The horizontal red (black) lines in (a)–(d) indicate the extreme event qualifier threshold $H_S$. The 3D trajectories of chaos and extreme events are shown in red (black) lines and the coexisting quasiperiodic orbits in green (gray) lines.

Figure 4

Numerical PDF of the peaks for different $\omega$ values. Long-tail distribution exceeding the significant height $H_S$ in vertical red (black) dashed lines for (a) $\omega =0.7315$, and (b) $\omega =0.6423$.

Figure 5

Phase diagram of extreme events in the ($\omega \text{−}A$) plane of a periodically forced Liénard system. Regions of extreme events (EE, black region) and nonextreme events (NEE, white region) are delineated in the parameter space.

Figure 6

Poincaré surface of sections of chaotic trajectories in blue (black) circles and quasiperiodicity in red (black) squares. A open black circle denotes the saddle point. For $\omega =0.7316$, chaotic trajectory comes close to the saddle point in (a). Inset shows the chaotic trajectory in blue (black) circles is still far away from the saddle point. At $\omega =0.7315$, the chaotic trajectory in (b) almost reaches the saddle point and thus collides with the saddle orbit and thereby made recurrent large excursions as extreme events denoted by the faraway scattered blue (black) circles. Inset shows a largest event in blue (black) circle near the quasiperiodic orbit in red (black) squares. For $\omega =0.6423$, extreme events are seen in (c), in the intermittency regime, as scattered blue (black) circles when the trajectory collides occasionally with the saddle orbit. A largest event is again shown in the inset. All the extreme events remain bounded within the quasiperiodic boundary.

Figure 7

Phase slips during extreme events. Top panels show time series of extreme events and bottom panels manifest the phase difference ($\mathrm{\Delta }\varphi$) for $\omega =0.7315$ (right panel), and $\omega =0.6423$ (left panel).

Figure 8

Analog circuit realization of a forced Liénard type oscillator.

Figure 9

Oscilloscope pictures of time series $V_{C_2}\left(t\right)$ of the forced Liénard type oscillator. (a) Bounded chaotic oscillation for $\omega =2.85\mathrm{KHz}$, (b) extreme events at $\omega =2.8\mathrm{KHz}$, (c) more frequent large amplitude oscillation for $\omega =2.75\mathrm{KHz}$, and (d) extreme events at $\omega =2.7\mathrm{KHz}$.

Figure 10

Experimental PDF of the peaks for different $\omega$ values. Long-tail distribution exceeding the significant height $H_S$ in vertical red (black) dashed line for (a) $\omega =2.8\mathrm{KHz}$ and (b) $\omega =2.7\mathrm{KHz}$.