Exactly solvable Stuart-Landau models in arbitrary dimensions
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| Format: | Preprint |
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2025
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| author | Gogoi, Pragjyotish Bhuyan Ghosh, Rahul Ghoshal, Debashis Prasad, Awadhesh Ramaswamy, Ram |
| author_facet | Gogoi, Pragjyotish Bhuyan Ghosh, Rahul Ghoshal, Debashis Prasad, Awadhesh Ramaswamy, Ram |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exactly solvable Stuart-Landau models in arbitrary dimensions Gogoi, Pragjyotish Bhuyan Ghosh, Rahul Ghoshal, Debashis Prasad, Awadhesh Ramaswamy, Ram Chaotic Dynamics Exactly Solvable and Integrable Systems 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. |
| title | Exactly solvable Stuart-Landau models in arbitrary dimensions |
| topic | Chaotic Dynamics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2511.05160 |