Exactly solvable Stuart-Landau models in arbitrary dimensions

Abstract

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.