Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.
- Nonlinear waves
- Parametric excitation
Bakri, T., Meijer, H. G. E., & Verhulst, F. (2009). Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations. Journal of nonlinear science, 19(5), 571-596. [10.1007/s00332-009-9045-2]. https://doi.org/10.1007/s00332-009-9045-2