Global temporal dynamics on the centre manifold for spatially periodic solutions

Develop a rigorous, global-in-time description of the temporal dynamics on the centre manifold for the coupled Swift–Hohenberg system (1) near a 1:1 resonant Turing and Turing–Hopf instability. In particular, prove persistence and classify the global dynamics (e.g., periodic, quasiperiodic, or more complex behavior) of the reduced normal-form flow under the presence of highly oscillatory higher-order terms induced by the differing phase velocities, thereby extending beyond local or bounded-in-time existence results for spatially periodic solutions.

Background

Using center manifold reduction and normal form theory, the paper constructs globally bounded, spatially periodic solutions close to the resonant instability. However, the reduced dynamics on the centre manifold contains fast oscillatory higher-order terms coming from interactions of modes with different phase velocities, complicating the temporal classification of solutions.

While the cut-off normal form without oscillatory perturbations admits a clear characterization of time-periodic states, establishing their persistence and the full global temporal dynamics in the presence of oscillatory terms (which can lead to complicated behaviors such as quasiperiodicity) remains unresolved.