Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling

Cyclops states are intriguing cluster patterns observed in oscillator networks, including neuronal ensembles. The concept of cyclops states formed by two distinct, coherent clusters and a solitary oscillator was introduced by Munyaev et al. [Phys. Rev. Lett. 130, 107201 (2023)], where we explored the surprising prevalence of such states in repulsive Kuramoto networks of rotators with higher-mode harmonics in the coupling. This paper extends our analysis to understand the mechanisms responsible for destroying the cyclops' states and inducing dynamical patterns called breathing and switching cyclops states. We first analytically study the existence and stability of cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional oscillators with inertia as a function of the second coupling harmonic. We then describe two bifurcation scenarios that give birth to breathing and switching cyclops states. We demonstrate that these states and their hybrids are prevalent across a wide coupling range and are robust against a relatively large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and switching cyclops states promise to strongly manifest in other physical and biological networks, including coupled theta neurons.

Figure 1

Snapshot of a breathing cyclops state in network (1) of 11 oscillators. Periodically oscillating $x\left(t\right)$ and $y\left(t\right)$ govern the phase difference between the synchronous clusters (blue and pink circles) and the solitary oscillator (gray circle). The solitary oscillator's phase is chosen at ${\theta }_M=0$ as a reference. Parameters are ${\alpha }_1=1.7, {\varepsilon }_2=0.08,{\alpha }_2=-0.3$.

Figure 2

(a, b) The color shows the number of distinct stationary cyclops states in the network (1) as a function of the second-harmonic coupling and phase lag parameters $({\alpha }_2,{\varepsilon }_2).$ Other parameters are (a) $N=5, {\alpha }_1=2.0$ and (b) $N=11, {\alpha }_1=1.7$. The number of cyclops is calculated by numerically finding solutions of system (6). The arrow points to the dashed area corresponding to the stability diagram of Fig. 3. (c) Snapshots of three distinct stationary cyclops states (up to permutation of clusters $x⟷y$) for the parameter set $N=11, {\alpha }_1=1.7, {\alpha }_2=0.0, {\varepsilon }_2=6.0$ corresponding to the open diamond in the green area in panel (b). The oscillator coloring corresponds to the intercluster differences $x$ and $y$ according to positive and negative values depicted from the horizontal color bar.

Figure 3

(a) The stability diagram for cyclops states. Regions of stable stationary cyclops states (CS) are shown in blue, switching cyclops states (SCS) in yellow, breathing cyclops states (BCS) in green, and two-cluster regimes ($5\text{:}6$) in white. Analytical boundaries: the blue solid line corresponds to the stability boundary (15), the red dashed line to $\text{Re}{\lambda }_1^{\mathrm{tran}}=0$, and the green dash-dotted curve to $\text{Re}{\lambda }_2^{\mathrm{tran}}=0$. Two numerical curves marked by the solid circles separate the stability regions of the switching and breathing stationary cyclops states. The black dotted line $\mathrm{\Gamma }$ corresponds to $H^{\prime }\left(0\right)=0.$ Values above the curve make the coupling attractive and full synchronization locally stable. Stationary cyclops states are found as a solution of system (6) and used as initial conditions. The round, diamond, and circled times correspond to the parameters used in Figs. 4, 5, 6. (b) The real part of the eigenvalues, associated with the stationary cyclops state, that determine the (internal) stability of the intercluster phase differences (blue solid line) and transversal (external) stability of the first (red dashed line) and second cluster (green dash-dotted line) for fixed ${\varepsilon }_2={\varepsilon }_2^{*}$ and varying ${\alpha }_2$ [along the white dashed horizontal line in panel (a)]. The background color indicates the type of the emerged cyclops states as in panel (a). (c) The diagram is similar to panel (b), but for fixed ${\alpha }_2={\alpha }_2^{*}$ and varying ${\varepsilon }_2$ [along the black-white dashed vertical line panel (a)]. The shaded area indicates the bistability of switching and breathing cyclops states. Parameters: $N=11, {\varepsilon }_1=1.0, {\alpha }_1=1.7, {\varepsilon }_2^{*}=0.08, {\alpha }_2^{*}=0.78$.

Figure 4

Breathing cyclops state. (a) The colors depict the phase differences ${\theta }_k\left(t\right)-{\theta }_6\left(t\right).$ The gray strip indicates the reference solitary oscillator. (b) The corresponding values of $r_1$ and $r_2$. (c) The eigenvalues associated with the destabilized stationary cyclops state. Some eigenvalues are repeated. The round (triangular) labels correspond to the internal (transversal) stability. Note a pair of complex eigenvalues with a positive real part (red) that emerged due to an Andronov-Hopf bifurcation and yielded periodic oscillations of intercluster differences. (d) Phase distributions ${\theta }_k$ at several time instants. The arrows indicate the direction of periodic phase clusters' oscillations (see Supplemental Movie 1 [68] demonstrating this breathing cyclops state). The oscillators' coloring represents their relative phase difference with the solitary oscillator as in Fig. 2. Parameters $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7, {\varepsilon }_2=0.08, {\alpha }_2=-0.1$ correspond to the open circle label in Fig. 3.

Figure 5

Switching cyclops state. (a) The colors depict the phase differences ${\theta }_k\left(t\right)-{\theta }_6\left(t\right).$ The strips with solid black borders indicate the reference solitary oscillator during the lifetime of a cyclops state configuration (the first stage). Note that clusters disintegrate to form a new cyclops state with a different solitary oscillator (the second stage). (b) The corresponding values of $r_1$ and $r_2$. The gray fragments correspond to the zoomed-in insets (right panels). (c) The eigenvalues associated with the destabilized stationary cyclops state. Some eigenvalues are repeated. The round (triangular) labels correspond to the internal (transversal) stability. Note a positive real eigenvalue (red) corresponding to the loss of the transversal stability of the stationary cyclops state due to Statement 2. (d) Phase distributions ${\theta }_k$ corresponding to a death-birth process in which a cyclops state existing at $t=t_1$ disintegrates to form a new cyclops state at $t=t_4$ (see Supplemental Movie 2 [68] for the details of this dynamical evolution). Parameters $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7, {\varepsilon }_2=0.08, {\alpha }_2=0.78$ correspond to the diamond label in Fig. 3.

Figure 6

Switching-breathing cyclops state. The notations are as in Fig. 5. One cluster of the breathing cyclops state [depicted in orange in panel (a)] eventually disintegrates, forming a reshuffled synchronous cluster and a new solitary oscillator. Note the weak internal and transversal instability of the destabilized stationary cyclops state due to the three eigenvalues with small positive real parts [red circles and nabla in panel (c)]. Supplemental Movie 3 [68] animates the sequence given in panel (d). Parameters $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7, {\varepsilon }_2=0.0578, {\alpha }_2=0.78$ correspond to the circled times label in Fig. 3.

Figure 7

Rotobreathing cyclops state. The notations are as in Fig. 5. From left to right: the relative phase between the first synchronous cluster and the sixth reference oscillator oscillates, whereas the phase of the second synchronous cluster passes zero and rotates until the clusters exchange their roles. Supplemental Movie 4 [68] details this process. Parameters are $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7, {\varepsilon }_2=0.08, {\alpha }_2=-2.0$.

Figure 8

Stability and prevalence of cyclops states. (a) Stability diagram extending Fig. 3 to the full range of the phase lag parameter ${\alpha }_2.$ The notations are similar to Fig. 3, with the addition of rotobreathers (pink). The shaded vertical strip corresponds to the parameter region of Fig. 3. Stationary cyclops states in the region $CS$ are chosen as initial conditions and further continued by changing the parameter ${\alpha }_2$ right and left from each point on the line ${\alpha }_2=0.0$ for each value of ${\varepsilon }_2.$ The initial conditions for the subsequent calculation are carried over from the final state of the preceding computation. The double-shaded areas (inclined stripes) indicate overlapping stability regions and correspond to the bistability of different cyclops state types. The two dash-dotted horizontal lines indicate the values of ${\varepsilon }_2$ used in panels (b) and (c). (b, c) Probability of cyclops states' emergence (all types). The number of trials is 1000. The initial phases are uniformly distributed in the segment $[-\pi ,\pi ]$, and the initial velocities are uniformly distributed in the segment $[-1.0,1.0]$. The black dashed vertical lines in panel (b) indicate the stability boundary of full synchronization. In panel (c) full synchronization is unstable. Parameters are $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7$. (b) ${\varepsilon }_2=0.08$, (c) ${\varepsilon }_2=0.05$.

Figure 9

The probability of occurrence of two-cluster ($N=10$) and cyclops states ($N=11$) under attractive and repulsive interactions. The histograms indicate the percentage occurrence of dynamical states in 1000 simulations with random initial conditions. The initial phases ${\mathrm{\Theta }}_k\left(0\right)$ are uniformly distributed on $[-\pi ,\pi ]$; the initial velocities ${\dot{\mathrm{\Theta }}}_k\left(0\right)$ are uniformly distributed on $[-5.0,5.0]$. The snapshots correspond to the established dynamical states with the relative phases ${\theta }_k$ in the rotating frame as in (4). (a, b) Attractive coupling: complete synchronization is prevalent with a $72\%$ ($N=10$) and $62\%$ ($N=11$) probability. It coexists for $N=10$ with a stationary two-cluster 5:5 state with an even partition of five oscillators in each cluster and a constant intercluster phase difference (a) and for $N=11$ with a stationary cyclops state 5:1:5 (b). (c, d) Repulsive coupling. (c) $N=10$: stationary two-cluster 5:5 states coexist with four-cluster states 4:4:1:1 with nonstationary intercluster phase differences. (d) $N=11$: breathing cyclops states 5:1:5 with oscillating intercluster phase differences are global attractors emerging with a $100\%$ probability. The black arrows indicate the directions of evolving intercluster differences. Parameters are $\mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.7$; (a, b) ${\varepsilon }_2=0.12, {\alpha }_2=0.0$; (c, d) ${\varepsilon }_2=0.04, {\alpha }_2=0.6$.

Figure 10

Odd vs even-sized large networks: the transition between cyclops and two-cluster states as $N$ changes. The colors depict the relative phases ${\theta }_k,k=1,...,N.$ (a) A cyclops state with the three-cluster partition 50:1:50 in the 101-oscillator network. The cyclops state with small phase offsets, randomly chosen from a uniform distribution $[-0.01;0.01]$ was selected as the initial condition. The horizontal black stripe indicates the solitary oscillator with ${\theta }_{51}=0.$ (b) The emergence of a two-cluster state 50:50 in the 100-oscillator network. The cyclops state from (a) with one oscillator removed from the second cluster was selected as the initial condition. (c) Removing one oscillator from the two-cluster state in (b) induces a cyclops state 49:1:49 in the 99-oscillator network. (d) The corresponding snapshots of phases ${\theta }_k$ at several time instants: $t_1=1250, t_2=3750, t_3=5200, t_4=6250, t_5=11250$, and $t_6=13750$. The black arrow indicates the direction of the solitary oscillator's phase evolution. Parameters are $\mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=3.1, {\varepsilon }_2=0.002, {\alpha }_2=0.2$.

Figure 11

Persistence of cyclops states in system (1) with mismatched frequencies ${\omega }_k$ distributed evenly over the interval $[\omega -\delta ,\omega +\delta ],$ where $\omega =1.7$ and $\delta$ is a frequency detuning. Global maxima (circles) and minima (crosses) of order parameter $r_2$ for three cyclops states (red, orange, and cyan). The global maximum was determined in the time interval $5\times {10}^3\le t\le {10}^4$ following the transition process. A stationary cyclops state is indicated by overlapping circles and crosses of the same color, while separated circles and crosses denote a switching cyclops state. The first cyclops state (red) originates from the original intrinsic frequency distribution. Subsequently, modifying the distribution by exchanging the intrinsic frequency of the $M\mathrm{th}$ solitary oscillator, ${\omega }_M$, with ${\omega }_1$ (or ${\omega }_{M+1}$) induces the emergence of the second (or third) cyclops state, depicted by orange and cyan, respectively. As the parameter mismatch $\delta$ increases, the distribution of mismatched frequencies widens. To induce each of the three cyclops states as $\delta$ increases, a stationary cyclops state from the identical frequency case was used as the initial condition for $\delta =0$. The final phase distribution at a given $\delta$ is then used as initial conditions for subsequent simulations at a higher value of $\delta$. The values of $\delta <{\delta }_1$ preserve all three stable stationary cyclops states. Increasing $\delta >{\delta }_1$ destabilizes the first stationary cyclops state (red) and turns it into a switching cyclops state. Further increasing $\delta >{\delta }_2$ leads to disintegrating the second cyclops state (orange) at $\delta ={\delta }_2$. The third stationary cyclops state (cyan) persists to $\delta ={\delta }_3$. The cyclops states are found from direct numerical simulations of system (1) for three sets of natural frequency distributions ${\omega }_k$ with a continuous increase in $\delta$ from zero. The inset shows instantaneous phase distributions ${\theta }_k$ for the third cyclops state with nonidentical frequencies. Parameters are $N=11, \mu =1.0, {\varepsilon }_1=1.0, {\alpha }_1=1.8, {\varepsilon }_2=0.12.$ The bifurcation parameter values are ${\alpha }_2=0.2, {\delta }_1=0.034, {\delta }_2=0.105, {\delta }_3=0.183$.

Figure 12

The supports $S_1, S_2$ (black dots) and the corresponding Newton polytopes $P_1, P_2$ (shaded regions) of (a) the first and (b) second polynomials of system (A1). (c) The Minkowski sum $P_1\oplus P_2$.