Pragjyotish Bhuyan Gogoi

arXiv2025

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

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.

nlin.CD↗

arXiv2025

Exactly solvable Stuart-Landau models in arbitrary dimensions

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions $D >2$ and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has $N=\lfloor{D/2}\rfloor$ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd $D$ there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere $\mathbb{S}^{D-1}$ and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits each of which has the geometry of a torus $\mathbb{T}^N$ on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalisations.

nlin.CD↗