Chaotic Switching In The Minimal Pendula Network

Abstract

We report the chaotic switching phenomenon in the minimal $N = 3$ pendula network with global coupling. Analyzing the stability conditions of the chimera states and their dependence on the parameters, three scenarios of chaotic switchings are identified: 1) a riddling bifurcation scenario, where an unstable periodic orbit inside the chimera manifold becomes transversally unstable, 2) a blowout bifurcation scenario, where the switching is caused by the transverse destabilization of the chaotic chimera with respect to its manifold, and 3) switchings between "laminar" saddle chimeras within a global "turbulent" attractor. The results are obtained based on the detailed examination of the existing regimes including chimera states, limit cycles and fixed points, their multistability and switching regime. In the parameter regions where the chaotic chimeras coexist with stable non-chaotic solutions, the switching trajectory can eventually escape to a stable solution, causing an additional unpredictability in the system behavior, as it is difficult to predict the escaping moment.

Figures (9)

• Figure 1: Parameter regions and sample solutions of the system (\ref{['PN']}). (a) Stability regions in the $(\alpha, \mu)$-parameter plane including the standard chimera state $(1:2:2)$ in the green region, the mixed-mode chimera no.1 $(0:1:1)$ in purple region, the in-phase mixed-mode chimera no.2 (0:0:1) in the blue region bounded by the bifurcation curves $T_{c2}$ and $T_{c1}$, the out-of-phase mixed-mode chimera no.2 (0:0:1) in the hatched blue region, the chaotic inverted chimera $(1:1:a)$ in the light green region, the oscillatory state where all the pendula have zero mean frequency in the red area, the asynchronous fixed point in the dashed-dotted region beyond curve $T_{Afp}$ and $\mu>1$, the synchronous fixed point in the dotted region for $\mu<1$ and any $\alpha$, and the chaotic switching states in the orange dotted region. The sample solutions include (b) chaotic standard chimera at $(\alpha, \mu) = (1.58, 2.5)$, (c) chaotic inverted chimera at $(\alpha, \mu) = (1.8, 2.5)$, (d) chaotic mixed-mode chimera no.1 at $(\alpha, \mu) = (1.6, 2.0)$, (e) chaotic out-of-phase mixed-mode chimera no.2 at $(\alpha, \mu) = (2.6, 1.55)$ and (f) frequency-locked oscillatory state at $(\alpha, \mu) = (2.6, 1.2)$.
• Figure 2: Riddling bifurcation scenario of the chaotic switching. (a) Lyapunov exponents of the inverted chimera state are calculated for fixed phase-shift $\alpha = 1.8$ and varying coupling strength: the inverted chimera is hyperchaotic with two positive Lyapunov exponents. After the riddling bifurcation at $\mu_r \approx 2.41$, decreasing the coupling strength $\mu$ the switching periods become shorter (b) $\mu_1 = 2.32$, (c) $\mu_2 = 2.25$, and (d) $\mu_3 = 1.9$. (e) the average switching period of the inverted chimeras plotted against the distance from the riddling bifurcation point. (f) phase-velocity portrait of the three inverted chimeras.
• Figure 3: Basins of attraction of the inverted chimera at fixed phase-shift $\alpha = 1.8$. Each color denotes one of the replicas, see the legend. Before the riddling bifurcation at $\mu_r \approx 2.41$, each chimera replica has large basins of attraction with fractal basin boundary $\mu = 2.45$. After the riddling bifurcation for the coupling strength $\mu = 2.2$, the basin of each chimera replica is riddled with the points belonging to the other replicas. Further decrease of coupling strength $\mu = 1.9$, the basins of attraction are completely riddled.
• Figure 4: Blowout bifurcation scenario of chaotic switching. (a) Lyapunov exponents of the standard chimera state are calculated for fixed coupling strength $\mu = 2.3$ and varying phase-shift: the standard chimera is chaotic with one positive Lyapunov exponent. The switching dynamics of the standard chimera begins with riddling followed by consecutive transversal destabilization of infinite number of unstable periodic orbits. Switching examples for (b) $\alpha_1 = 1.589067$ and (b) $\alpha_2 = 1.589070$. The standard chimera then undergoes a blowout bifurcation at $\alpha_b = 1.589091$, where the transversal Lyapunov exponent of the chimera manifold becomes positive. (d) switching example after the blowout bifurcation $\alpha_3 = 1.5891$. Due to coexistence of the standard and inverted chimeras for these parameters, system trajectory can include switching between these two chimera types. (e) the average switching period of between the standard chimeras plotted against phase-shift $\alpha$. (f) phase-velocity portrait of the three standard chimeras.
• Figure 5: Laminar-Turbulent scenario of chaotic switching. (a) Lyapunov exponents of the out-of-phase mixed-mode chimera no.2 are calculated for fixed phase-shift $\alpha = 2.6$ and varying coupling strength: the chimera is chaotic with one positive Lyapunov exponent of the order $\lambda_{1, max} \approx 0.01$. In contrast to the previous two scenarios, the switching dynamics occur with the turbulence such that with increase of the coupling strength, the average period of the turbulence increases. Examples are shown for (b) $\mu_1 = 1.5067$, (c) $\mu_2 = 1.79$ and (d) $\mu_3 = 1.85$. We note that, the out-of -phase mixed-mode chimera no.2 undergoes a blowout bifurcation at $\mu_b = 1.82$ where the transversal Lyapunov exponent of the chimera manifold becomes positive.