In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., 53, 617-641 (2006)] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comp., 79, 1565-1584], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition. Remark 1.2. We fix the domain of our approximate parameterization to be the unit ball in C m (where m is the number of (un)stable eigenvalues, i.e. the dimension of the manifold) and vary the scalings of the eigenvectors in order to optimize with respect to the constraints. Another (theoretically equivalent approach) would be to fix the scalings of the eigenvectors and vary the size of the domain. However the scalings of the eigenvectors determine the decay rates of the power series coefficients, and working with analytic functions of fast decay seems to stabilize the problem numerically. Remark 1.3. In many previous applications of the parameterization method the free constants were selected by some "numerical experimentation." See for example the introduction and discussion in Section 5 of [8], Remark 3.6 of [9], Remark 2.18 and 2.20 of [10], the discussion of Example 5.2 in [11], Remark 2.4 of [12], and the discussion in Sections 4.2 and 6 of [12]. This motivates the need for systematic procedures developed here.

In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.

In this paper, we establish smoothness of moments of the solutions of discrete coagulation-diffusion systems. As key assumptions, we suppose that the coagulation coefficients grow at most sub-linearly and that the diffusion coefficients converge towards a strictly positive limit (those conditions also imply the existence of global weak solutions and the absence of gelation).

Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical Müller-Brown test potential.

Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modelling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte-Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds and heteroclinic orbits.

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u + βu + e u − 1 = 0 for all parameter values β ∈ [0.5, 1.9]. For each β, 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.for some Y (j) > 0 and Z (j) : R + → R + : r → Z (j) (r). The goal of the radii polynomial approach is to provide an efficient way to prove that an operator is a uniform contraction over a subset of X. This subset consists of small balls around the line of centers, provided by the linear interpolation between two numerical approximations of solutions at different parameter values.