Exactly solvable Stuart-Landau models in arbitrary dimensions

Area of Science:

• Mathematics
• Physics
• Dynamical Systems

Background:

• The Stuart-Landau system is a fundamental model for oscillatory dynamics near a Hopf bifurcation.
• Extending this system to higher dimensions (D>2) presents significant mathematical challenges.

Purpose of the Study:

• To generalize the Stuart-Landau system to arbitrary dimensions D>2 using Clifford's geometric algebra.
• To find an exact analytical solution for the extended oscillator equations.
• To investigate the complex dynamics and multistability arising in higher dimensions.

Main Methods:

• Application of Clifford's geometric algebra to formulate the extended Stuart-Landau system.
• Analysis of Jacobian matrix eigenvalues at the fixed point to identify bifurcation conditions.
• Characterization of asymptotic dynamics and limiting orbits in higher-dimensional spaces.

Main Results:

• An exact solution for the generalized Stuart-Landau system in D dimensions is derived.
• The system exhibits a supercritical Hopf bifurcation with N=⌊D/2⌋ pairs of complex conjugate eigenvalues crossing the imaginary axis.
• For odd D, an additional real eigenvalue also crosses the imaginary axis.
• Asymptotic dynamics are confined to a hypersphere S^{D-1}.
• Extreme multistability is observed, with infinite coexisting limiting orbits on tori T^{N} exhibiting periodic or quasiperiodic motion.

Conclusions:

• Clifford's geometric algebra provides a powerful framework for extending oscillator models to higher dimensions.
• The generalized system reveals rich dynamics, including extreme multistability and complex limit cycle behavior.
• The findings offer insights into generalized limit cycle oscillator systems and their potential applications.