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.

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Cited by in corpus (1)

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