Jan Bouwe van den Berg

Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray–Scott equation.

Formal asymptotics of bubbling in the harmonic map heat flow

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behavior of singularities arising in this flow for a special class of solutions which generalizes a known (radially symmetric) reduction. Specifically, the rate at which blowup occurs is investigated in settings with certain symmetries, using the method of matched asymptotic expansions. We identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.

Global smooth solution curves using rigorous branch following

In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f: ℝ l 1 × B 1 → ℝ l 2 ×B 2 , where B 1 and B 2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation

We prove that the stationary Swift–Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semiconjugacy to a subshift of finite type shows that the dynamics is chaotic.

Vanishing beyond all orders: Stokes lines in a water-wave model equation

We study the existence of solutions homoclinic to a saddle centre in a family of singularly perturbed fourth order differential equations, originating from a water-wave model. Due to a reversibility symmetry, the occurrence of such embedded solitons is a codimension-1 phenomenon. By varying a parameter a countable family of solitary waves is found. We examine the asymptotic frequency at which this phenomenon of persistence in the singular limit occurs, by performing a refined Stokes line analysis. In the limit where the parameter tends to infinity, each Stokes line splits into a pair, and the contributions of these two Stokes lines cancel each other for a countable set of parameter values. More generally, we derive the full leading order asymptotics for the Stokes constant, which governs the (exponentially small) amplitude of the (minimal) oscillations in the tails of nearly homoclinic solutions. True homoclinic trajectories are characterized by the Stokes constant vanishing. This formal asymptotic analysis is supplemented with numerical calculations.

Ultra fast electromagnetic field computations for RF multi-transmit techniques in high field MRI

A new, very fast, approach for calculations of the electromagnetic excitation field for MRI is presented. The calculation domain is divided in different homogeneous regions, where for each region a general solution is obtained by a summation of suitable basis functions. A unique solution for the electromagnetic field is found by enforcing the appropriate boundary conditions between the different regions. The method combines the speed of an analytical method with the versatility of full wave simulation methods and is validated in the pelvic region against FDTD simulations at 3 and 7 T and measurements at 3 T. The high speed and accurate reproduction of measurements and FDTD calculations are believed to offer large possibilities for multi-transmit applications, where it can be used for on-line control of the global and local electric field and specific absorption rate (SAR) in the patient. As an example the method was evaluated for RF shimming with the use of 7 T simulation results, where it was demonstrated that the magnetic excitation field could be homogenized, while both the local and average SAR were reduced by 38% or more.

Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

In this work we introduce a rigorous computational method for finding heteroclinic solutions of a system of two second order differential equations. These solutions correspond to standing waves between rolls and hexagonal patterns of a two-dimensional pattern formation PDE model. After reformulating the problem as a projected boundary value problem (BVP) with boundaries in the stable/unstable manifolds, we compute the local manifolds using the parameterization method and solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85--110] about the coexistence of hexagons and rolls.

TRAVELLING WAVES FOR FOURTH ORDER PARABOLIC EQUATIONS

We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x, t )= U (x + ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

Parametric dependence of exponents and eigenvalues in focusing porous media flows

We study the hole-filling problem for the porous medium equation ut = 1 m ∆u with m > 1 in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution u= t2β−1F(|x|t−β), and we establish that the self-similarity exponent β is a monotone function of the parameter m. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any m > 1 against perturbations with 2-fold symmetry. In addition, we prove that as m is varied there are bifurcations from the radial solution to self-similar solutions with k-fold symmetry for each k= 3, 4, 5, . . . . These bifurcations are simple and occur at values m3 > m4 > m5 > · · · → 1.

Closed characteristics of fourth-order twist systems via braids

Abstract For a large class of second order Lagrangian dynamics, one may reformulate the problem of finding periodic solutions as a problem in solving second-order recurrence relations satisfying a twist condition. We project periodic solutions of such discretized Lagrangian systems onto the space of closed braids and apply topological techniques. Under this reformulation, one obtains a gradient flow on the space of braided piecewise linear immersions of circles. We derive existence results for closed braided solutions using Morse–Conley theory on the space of singular braid diagrams.