Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Abstract

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers.The proof of the above statement is conceptually simple, and combines several techniques from some of the authors' and Zgliczyński's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values.

Equilibrium validation in models for pattern formation based on Sobolev embeddings

“…In our previous work on the constructive implicit function theorem, our goal has been to give a systematic procedure for adapting these works to the context of parameter continuation. There are several papers that have already considered rigorous validation of parameter-dependent solutions for the Ohta-Kawasaki model [3,9,18,30,32,33,34,35,36,37]. Many of these papers also include methods of bounding the terms in a generalized Fourier series, and the estimates on the tail.…”

A functional analytic approach to validated numerics for eigenvalues of delay equations

“…Example 2 (eigenvalue validation for a system of DDEs): Consider now the delayed van der Pol equation (18). Recalling (19) and (20), this leads to the characteristic equation…”

“…In this section, we present applications of the Chebyshev series discretization approach to rigorously compute the number of eigenvalues outside circles of prescribed radii centered at 0 in the complex place. We apply our approach to Mackey-Glass (14), cubic , delayed van der Pol (18) and the predator-prey equation (21). Let us now present a rigorous computational procedure.…”

“…Some works of this kind include the validated numerics for Floquet theory developed in [16], the methods for validated Morse index computations (unstable eigenvalue counts) for infinite dimensional compact maps in [29,37], similar methods for equilibria of parabolic PDEs posed on compact domains in [41,1,44,54,40,55], the validated numerics for stability/instability of traveling waves in [3,2,7,5,6], stability analysis for periodic solutions of delay differential equations [28], the computerassisted proofs of instability for periodic orbits of parabolic partial differential equations found in [22], and the computer-assisted proofs for trapping regions of equilibrium solutions of parabolic PDEs in [18].…”

“…Consider the delayed van der Pol equation(18) with parameter values τ = 2, κ = −1 and ε = 0.15. Denote by u 0 ≡ (0, 0) T a steady state.…”