Phase chimera states: frozen patterns of disorder

Abstract

Coupled oscillators can serve as a testbed for larger questions of pattern formation across many areas of science and engineering. Much effort has been dedicated to the Kuramoto model and phase oscillators, but less has focused on oscillators with variable amplitudes. Here we examine the simplest such oscillators -- Stuart-Landau oscillators -- and attempt to elucidate some puzzling dynamics observed in simulation by us and others. We demonstrate the existence and stability of a previously unreported state which we call a ``phase chimera state.'' Remarkably, in this state, the amplitudes of all oscillators are identical, but one subset of oscillators phase-locks while another subset remains incoherent in phase. We also show that this state can take the form of a ``multitailed phase chimera state'' where a single phase-synchronous cluster of oscillators coexists with multiple groups of phase-incoherent oscillators.

Figures (5)

• Figure 1: Phase chimera. Top: long-term average of the frequencies $\dot{\phi}$; middle: snapshot of the amplitude profile $|W|$; bottom: snapshot of profile $\phi$. $N=1200$ oscillators. Parameter values are $K = 0.25$, $x = 0.4$, $C_1 = 1.4$ and $C_2=-C_1$. The initial conditions consist of a two-cluster state with (nearly) uniform amplitude of $\sqrt{1-K}$ and phases centered on either $0$ and $\pi$ (each oscillator is perturbed by a random value from $\mathcal{N}(0, 0.01)$).
• Figure 2: Zoology of states in the Stuart-Landau system. Amplitude (top row), phase (middle row), and distributions in the complex plane (bottom row) for various states observed in the Stuart-Landau system of coupled oscillators. Here, $N=600$ and $C_2=-C_1$. For all but the chaotic state, $C_1=1.4$. For the chaotic state, $C_1=3$ and $K=0.6$ with initial condition (IC) randomly distributed on the unit disk. For the amplitude mediated chimera (AMC), $K=0.55$ and IC is a two-cluster state ($x=0.4$ with amplitude $\sqrt{1-K}$ and phases at $0$ and $\pi$). For the phase chimera, multitailed phase chimera, and incoherent states $K=0.25$. ICs for these three states are, respectively: a two-cluster state ($x=0.4$ with amplitude $\sqrt{1-K}$ and phases at $0$ and $\pi$); a three cluster state ($1/3$ of oscillators in each cluster) with amplitudes for two of the clusters $\sqrt{1-K}$ and phases $\pi$ and $0$, and the third cluster also at phase $0$ at amplitude $0.97\sqrt{1-K}$ (though many choices work similarly); uniform randomly distributed oscillators on the unit disk. Note that supplemental videos accompanying this paper also illustrate several of these states.
• Figure 3: Phase chimera and $x$-$\alpha$ relationship. Location on the unit disk of $N$ oscillators with fraction $x$ in the locked group (larger blue dot) and $1-x$ in the incoherent group (set of smaller red dots). All oscillators are located at amplitude $|W|=\sqrt{1-K}$. The oscillators in the incoherent group are uniformly distributed from $-\alpha$ to $\alpha$.
• Figure 4: Stability boundaries. Blue dotted, yellow solid, and red dot-dashed curves show stability boundaries for the phase chimera with varying fractions of locked oscillators $x = 0$, $x=2/5$, and $x=1/2$, respectively, and $C_2=-C_1$. The phase chimera is stable in region I above the curves and unstable elsewhere. $x=0$ case is equivalent to the stability curve for the fully incoherent state. Inset shows a zoomed version of the stability curve for $x=2/5$ and demonstrates agreement between analytical results (blue solid curve) and numerical computation of eigenvalues for system with $N=5$.
• Figure 5: Minimal chimera. Location on the unit disk of the oscillators for small $N$ values ($N=2,3,4,5,6$). Parameter values are chosen in the range where the phase chimera and splay states are stable with values $K = 0.25$, $C_1 = 1.4$ and $C_2=-C_1$. In each panel a fraction $x$ of oscillators is locked at $\theta =\pi$ (blue dots) and a fraction $1-x$ of oscillators are distributed uniformly about $\theta=0$ (red dots); all amplitudes are $|W|=\sqrt{1-K}$. Oscillators in blue (located at $\theta=\pi$). Dot sizes are proportional to the number of oscillators in the group.