Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Inferring symbolic dynamics of chaotic flows from persistence

“…The only discrete symmetry-reduction method for highdimensional systems in the literature known to us is the invariant polynomials for reflection-type symmetries. 43 We are going to present the symmetry reduction of the Kuramoto-Sivashinsky system in appendix A.…”

“…Generically, this set of periodic orbits can be found via recurrence-based searches 4,5,18,19,44 or following bifurcations 17,45 and unstable manifolds of known solutions. 43,46 While there exists variational, 46 Levenberg-Marquardt search-based, 44 and possibly various other optimization methods for numerically locating unstable periodic orbits, the current community standard for very-high-dimensional flows is the Newton-Krylov-hookstep method of Viswanath. 19…”

“…49 Owing to its computational simplicity, nowadays the Kuramoto-Sivashinsky system is frequently chosen as the testing ground for methods to study high-dimensional chaos and turbulence. 43,44,[50][51][52][53] In (1 + 1) dimensions, the Kuramoto-Sivashinsky equation reads…”

“…u(x, t) = u(x + L, t). The domain length L is the sole control parameter of the Kuramoto-Sivashinsky system, whose dynamics become chaotic when L is large enough 43,44 . The Kuramoto-Sivashinsky equation (17) is equivariant under continuous translations…”

“…This problem was addressed in Ref. 43, which combined the first Fourier mode slice method of Ref. 36 with an invariantpolynomial method to obtain a fully symmetry-reduced representation of the Kuramoto-Sivashinsky state space.…”