A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y'(t) = - \alpha y(t-1) [ 1 + y(t) ]$ for all $\alpha \in (0,\tfrac{\pi}{2}]$. This has been proven to be true for a subset of parameter values $\alpha$. We extend the result to the full parameter range $\alpha \in (0,\tfrac{\pi}{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\tfrac{\pi}{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha\in(\tfrac{\pi}{2} , \tfrac{\pi}{2} + 6.830 \times 10^{-3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha=\tfrac{\pi}{2}$ is globally parametrized by $\alpha > \tfrac{\pi}{2}$.Comment: 45 page

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system

Validated computations for connecting orbits in polynomial vector fields

In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simultaneously solves for the local (un)stable manifolds and the orbit which connects these. We obtain explicit rigorous control on the distance between the numerical approximation and the heteroclinic orbit. Transversality of the stable and unstable manifolds is also proven.Comment: 60 pages, 11 figure

Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions

In this paper we develop a symmetry preserving method for the rigorous computation of stationary states of the Ohta-Kawasaki partial differential equation in three space dimensions. By preserving the relevant symmetries we achieve an enormous reduction in computational cost. This makes it feasible to construct computer-assisted proofs of complex three-dimensional structures. In particular, we provide the first existence proofs for both the double gyroid and body centered cubic packed sphere solutions to this problem

Continuation of homoclinic orbits in the suspension bridge equation: a computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation $u""+\beta u" + e^u-1=0$ for all parameter values $\beta \in [0.5,1.9]$. For each $\beta$, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.Comment: 37 pages, 6 figure

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system