Dynamics of coupled $D$-dimensional Stuart-Landau oscillators

TL;DR

The paper extends the Stuart-Landau oscillator to $D>2$ to explore how SO($D$) symmetry and its breaking shape collective dynamics in two coupled oscillators for $D=3$ and $D=4$. It employs Clifford algebra-based bivector parametrization and normal-form decoupling to analyze symmetry-preserving and symmetry-breaking couplings, revealing regimes of phase locking, quasiperiodicity, amplitude death, and novel partial quenching phenomena. Key contributions include identifying partial amplitude death and partial oscillation death, establishing phase diagrams in parameters such as the coupling strength and bivector alignment, and showing how symmetry-breaking couplings enrich dynamical states beyond the $D=2$ case. The findings have implications for controlling synchronization and quenching in high-dimensional nonlinear systems and offer a framework for studying larger networks and diverse coupling forms in high dimensions.

Abstract

The Stuart-Landau oscillator generalized to $D > 2$ dimensions has SO($D$) rotational symmetry. We study the collective dynamics of a system of $K$ such oscillators of dimensions $D =$ 3 and 4, with coupling chosen to either preserve or break rotational symmetry. This leads to emergent dynamical phenomena that do not have analogs in the well-studied case of $D=2$. Further, the larger number of internal parameters allows for the exploration of different forms of heterogeneity among the individual oscillators. When rotational symmetry is preserved there can be various forms of synchronization as well as multistability and $partial$ amplitude death, namely, the quenching of oscillations within a subset of variables that asymptote to the same constant value. The oscillatory dynamics in these cases are characterized by phase-locking and phase-drift. When the coupling breaks rotational symmetry we observe $partial$ synchronization (when a subset of the variables coincide and oscillate) and $partial$ oscillation death (when a subset of variables asymptote to different stationary values), as well as the coexistence of these different partial quenching phenomena.

Figures (13)

• Figure 1: Schematic phase diagram of two coupled oscillators in $D=2$. For different values of $\varepsilon$ and ${\gamma}$, periodic and quasiperiodic motion, and amplitude death can be observed. The boundaries of the different regions are constructed based on the largest two Lyapunov exponents.
• Figure 2: (Color online) Trajectories (solid black and dashed blue lines) of the two coupled three-dimensional oscillators outlining the different forms of multistability present in Eq. (\ref{['eq:3D2a']}). For $\gamma = 0.05$, there are three attractors: (a) for $\varepsilon=0.5$ one is quasiperiodic and two are fixed points, (b) for $\varepsilon=1.5$, all three are fixed points, and (c) for $\varepsilon=3$ there are two fixed points and one periodic orbit. (The plots correspond to $\Omega=0.2$ and $\varrho=1$.)
• Figure 3: (Color online) Trajectories (solid black and dashed blue lines) of the two coupled three-dimensional oscillators in Eq. (\ref{['eq:3D2a']}) with identical frequency $\boldsymbol{\mu}^{(1)} = \boldsymbol{\mu}^{(2)}$ and parameters $\varrho_1=1$, $\varrho_2=1.5$, $\varepsilon_1=0.5$ and $\varepsilon_2=0.2$. They approach two circular limit cycles located on the spheres of radii $r_{1*}\approx 1.04$ and $r_{2*}\approx 1.21$.
• Figure 4: (Color online) (a) Bifurcation diagram showing the extrema (red dash-dotted lines) of the variable $x_1^{1}$ [in Eq. (\ref{['eq:3DC3']})] as a function of $\varepsilon$ for the values $\varrho=1$, $\mu^{(1)}_{13}=0.3$ and $\mu^{(1)}_{12}=-0.2$. One of the two fixed points of the system, calculated using Eq. (\ref{['eq:Ch3DC4']}), is shown in black and marked FP. (b) The (equal) real parts of the complex conjugate pair of eigenvalues associated with the fixed points which become positive at $\varepsilon\approx 0.106$ marking a Hopf bifurcation (HB).
• Figure 5: (Color online) The evolution of ${\mathbf x}_1$, Eq. (\ref{['eq:3DC3']}), corresponding to $\gamma=1$, $\mu^{(1)}_{13} = 0.2$, and $\mu^{(1)}_{12} = -0.3$. (a) For $\varepsilon = 0.01$, it gradually approaches the fixed point FP, tracing out a portion of the sphere $r_1\approx\sqrt{\varrho}$ [see Eq. (\ref{['eq:Ch3radial']})]. (b) For $\varepsilon = 0.18$, the trajectory approaches a limit cycle.