Symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line II

TL;DR

This work advances computer-assisted proofs for infinite-dimensional dynamics by introducing a robust $\mathcal{C}^1$ algorithm to integrate variational equations of dissipative PDEs and applying it to the KS equation on the line with odd periodic boundary conditions. The authors establish countably many homoclinic and heteroclinic connections between two periodic orbits through a carefully constructed chain of covering relations and cone conditions on Poincaré maps, supported by rigorous infinite-dimensional bounds. A key contribution is the rigorous handling of variational equations in infinite dimensions via convergence and isolation properties, enabling finite-dimensional topological methods to extend to PDEs. They also prove an attracting periodic orbit for $\nu=0.127$ using symmetry and a Schauder-based fixed-point argument, illustrating practical stability results alongside symbolic dynamics. Overall, the framework provides a general, verifiable approach to proving complex dynamical phenomena in dissipative PDEs and broadens the scope of computer-assisted analysis in infinite-dimensional settings.

Abstract

We prove the existence of infinite number of homoclinic and heteroclinic orbits to two periodic orbits for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and for some fixed parameter value of the system. The proof is computer assisted and it is based on a new algorithm for rigorous integration of the variational equation for a class of dissipative PDEs on the torus.

Figures (6)

• Figure 1: Two approximate time-periodic orbits $u^1$ and $u^2$.
• Figure 2: Numerically observed chaotic attractor for (\ref{['eq:KS']})--(\ref{['eq:KSbc']}) obtained by simulation of a finite-dimensional projection of the corresponding infinite-dimensional ODE for the Fourier coefficients in $u(t,x)=\sum_{k=1}^\infty u_k(t)\sin (kx)$. Projection onto $(u_2,u_3)$ plane of the intersection of the observed attractor with the Poincaré section $u_1=0, u_1'>0$ is shown along with an approximate location of the two periodic points $u^1, u^2$ appearing in Theorems \ref{['thm:symdynKS']} and \ref{['thm:connectingOrbits']}. The point $u^1$ is a fixed point for the Poincaré map and $u^2$ is of period two.
• Figure 3: An example of an h-set in three dimensions with $u(N)=1$ and $T_N=D_2$ -- a two-dimensional closed disc. Here $N_c=\overline{B_1}\oplus D_2$.
• Figure 4: An example of an $f-$covering relation: $N\stackrel{f}{\Longrightarrow}M$. In this case, the homotopy joining $f_c(x,y)$ with a linear map $(L(x),0)$ and satisfying [CR1]--[CR4] is simply given by $H(t,x,y) = t(L(x),0) + (1-t)f_c(x,y)$.
• Figure 5: Numerically observed heteroclinic connections $a^{i}_{1\to 2}$, $a^{i}_{2\to 1}$$i=0,\ldots,10$ between approximate periodic points $a^1$ and $a^2$. All these points are located in the section $\Theta=\{a\in l_2 : a_1=0 \wedge a_1'=f_1(a)>0\}$ .

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Theorem 4
• Definition 3
• Definition 4
• Theorem 5
• Lemma 6