Symmetry-driven layered dynamics in the Kuramoto-Sivashinsky equation

Abstract

In this work, we uncover a layered organization of the state space in the Kuramoto-Sivashinsky equation with periodic boundary conditions, in which multiple invariant sets coexist at fixed system parameters and are selected by the initial condition. Within this framework, both chaotic attractors and periodic orbits (traveling waves) can be systematically generated by amplifying a single initial condition and parameterized by the initial energy. As the energy increases, the period of the periodic orbits decreases according to an inverse scaling law. In transitional parameter regions, periodic dynamics at low initial energy is found to coexist with strange attractors at higher energy levels, revealing a unique layered landscape governed by the viscosity and the initial condition. We conjecture that this behavior is linked to continuous spatial translational symmetry, which is reflected in the degeneracy of the neutral part of the Lyapunov spectrum.

Figures (4)

• Figure 1: First two Lyapunov exponents, $\lambda_1$ and $\lambda_2$, as a function of the viscosity parameter $\nu$.
• Figure 2: Four chaotic attractors chosen out of 20 generated by the amplification of a single initial condition. On the top left the magnitude of the initial energy for each different attractor.
• Figure 3: Periodic orbits generated by $20$ different initial conditions (see text for details). Colors are proportional to the initial energy $\varepsilon_0^{(k)} = ||u_0^{(k)}||_2$ injected into the system.
• Figure 4: Scaling of (a) the mean final energy of the system computed on a window of $\Delta t = 5 \times 10^3$ time steps at the end of the simulation with the initial energy and (b) the period of the periodic orbits with the initial energies.