Complexity measure of extreme events

Abstract

Complexity is an important metric for appropriate characterization of different classes of irregular signals, observed in the laboratory or in nature. The literature is already rich in the description of such measures using a variety of entropy and disequilibrium measures, separately or in combination. Chaotic signal was given prime importance in such studies while no such measure was proposed so far, how complex were the extreme events when compared to non-extreme chaos. We address here this question of complexity in extreme events and investigate if we can distinguish them from non-extreme chaotic signal. The normalized Shannon entropy in combination with disequlibrium is used for our study and it is able to distinguish between extreme chaos and non-extreme chaos and moreover, it depicts the transition points from periodic to extremes via Pomeau-Manneville intermittency and, from small amplitude to large amplitude chaos and its transition to extremes via interior crisis. We report a general trend of complexity against a system parameter that increases during a transition to extreme events, reaches a maximum, and then starts decreasing. We employ three models, a nonautonomous Lienard system, 2-dimensional Ikeda map and a 6-dimensional coupled Hindmarh-Rose system to validate our proposition.

Figures (6)

• Figure 1: Forced Liénard system: Bifurcation diagrams of $y_{\max}$ against forcing frequency $\omega$. (a) PM intermittency: Periodic oscillation transition to large amplitude events at a critical $\omega \approx 0.642212$ and it continues for larger $\omega$. (b) Interior-crisis-induced extreme events: Small amplitude chaos transition to large amplitude chaos that continues to appear for decreasing $\omega$. Normalized Shannon entropy $S$ (solid black line) and disequilibrium distance $D$ (dashed black line) in the PM intermittency regime (c) and interior-crisis region (d). Behavior of complexity $C$ against $\omega$ are shown for the PM intermittency region (e) and interior crisis route (f). Number of bins $m=50$, for both cases, other parameters are $\alpha=0.45$, $\beta=0.50$, and $\gamma=-0.50$. A significant height threshold $T=\mu +6\sigma$ is used as a referral marker of extreme events.
• Figure 2: Forced Liénard system. PM intermittency: Temporal dynamics of extremes for (a) $\omega$= $0.642236$, non-extreme events for (c) $\omega$= $0.643$. PDF in semi-log scale for (b) $\omega$=0.642236 and (d) $\omega=0.643$ during PM intermittency. Interior crisis: Temporal dynamics of extreme events for (e) $\omega=0.731693$ and non-extremes for (g) $\omega$= $0.73007$. PDF in semi-log scale for (f) $\omega=0.731693$ and (h) $\omega=0.73007$. Extreme events are larger than the significant height (horizontal dashed red lines) in (a) and (e) when their PDF of events (b) and (f), respectively, show rare occurrence of large events beyond the vertical dashed marker line. Non-extreme chaotic events are very frequent in (c) and (g) when all the events are of lower height than the significant height and show no events beyond the marker $T$ (vertical dashed line) in (d) and (h), respectively.
• Figure 3: Ikeda map. (a) Bifurcation diagram of $y_{min}$ and significant height $T$ (solid red line) against $p$. Sudden expansion of the attractor at a critical point $p \approx 7.26884894$ via interior crisis. (b) Complexity measure $C$ (black solid line) against $p$. $C$ remains low during non-extreme chaotic oscillation until it transitions to a high value at the critical point $p \approx 7.26884894$ with the onset of extremes. Parameters $A = 0.85, B = 0.9$ and $k = 0.4$.
• Figure 4: Ikeda map. Temporal dynamics of $y$ in the pre-crisis non-extreme bounded chaos regime for $p = 7.265$ (a), post-crisis extremes for $p=7.275$ (c), and post-crisis frequent non-extreme large events for $p = 7.295$(e). PDF for pre-crisis region (b), extreme events (d) and post-crisis frequent large events (f).
• Figure 5: Coupled Hindmarsh-Rose neuron model: (a) Bifurcation diagram of $x_{||}$ (blue dots) and extreme event marker $T$ (red line) against $\epsilon$. (b) Complexity measure $C$ (black line) of the coupled HR model against $\epsilon$. (c, e) Temporal dynamics of $x_{||}$ for $\epsilon = - 0.07127$ (A) and, $- 0.1431$ (B) when their corresponding distribution (PDF) of events are in (d) and (f), respectively.