Abstract:
We have recently applied Clifford’s geometric algebra to generalise
the well-known Stuart Landau system to arbitrary dimensions D>2. This
approach recognises the special symmetries of the two-dimensional case
and retains this feature in all dimensions, revealing a rich geometric
and dynamical structure of the general model. We obtain exact
solutions and describe the dynamical attractors and their basins of
attraction in all dimensions. Since the model has D dimensional
rotational symmetry, we further study the dynamics of an ensemble of
oscillators in higher dimensions with coupling chosen to either
preserve or break this symmetry. The emergent collective states can
have unusual dynamical behaviour such as partial synchronization (when
only a subset of the variables coincide and oscillate) and partial
oscillation death (when a subset of variables asymptote to different
stationary values). Similar Clifford extensions of other limit cycle
oscillator systems can be made, offering a consistent procedure for
generalising dynamical systems to arbitrary dimensions.