Rigorous Enclosures of Solutions of Neumann Boundary Value Problems

This paper is dedicated to the problem of isolating and validating zeros of non-linear two point boundary value problems. We present a method for such purpose based on the Newton-Kantorovich Theorem to rigorously enclose isolated zeros of two point boundary value problem with Neumann boundary conditions.Comment: 21 pages and 2 figure

Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation

Rigorous numerics for piecewise-smooth systems : a functional analytic approach based on Chebyshev series

In this paper, a rigorous computational method to compute solutions of piecewise-smooth systems using a functional analytic approach based on Chebyshev series is introduced. A general theory, based on the radii polynomial approach, is proposed to compute crossing periodic orbits for continuous and discontinuous (Filippov) piecewise-smooth systems. Explicit analytic estimates to carry the computer-assisted proofs are presented. The method is applied to prove existence of crossing periodic orbits in a model nonlinear Filippov system and in the Chua’s circuit system. A general formulation to compute rigorously crossing connecting orbits for piecewise-smooth systems is also introduced

Dynamics of a class of ODEs via wavelets

The objective of this paper is to study a perturbed linear hyperbolic differential equation. The first part of this work is dedicated to study perturbation of the equilibrium (special solution) of a perturbed hyperbolic system. On the second part we analyze the stable and the unstable manifolds of a perturbed semilinear differential equation. We assume that the perturbed forcing function belongs to an L2 class and that it is developed in a series of wavelets. Then we analyze the effect of this development on the special solution of the perturbed equation. Similar study is provided for the stable and unstable manifolds of this special solutions.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaFundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil

Homology and symmetry breaking in Rayleigh-Benard convection: Experiments and simulations

Algebraic topology (homology) is used to analyze the weakly turbulent state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Benard convection.The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present.Comment: 21 pages with 6 figure

Inferring Long-term Dynamics of Ecological Communities Using Combinatorics

In an increasingly changing world, predicting the fate of species across the globe has become a major concern. Understanding how the population dynamics of various species and communities will unfold requires predictive tools that experimental data alone can not capture. Here, we introduce our combinatorial framework, Widespread Ecological Networks and their Dynamical Signatures (WENDyS) which, using data on the relative strengths of interactions and growth rates within a community of species predicts all possible long-term outcomes of the community. To this end, WENDyS partitions the multidimensional parameter space (formed by the strengths of interactions and growth rates) into a finite number of regions, each corresponding to a unique set of coarse population dynamics. Thus, WENDyS ultimately creates a library of all possible outcomes for the community. On the one hand, our framework avoids the typical ``parameter sweeps'' that have become ubiquitous across other forms of mathematical modeling, which can be computationally expensive for ecologically realistic models and examples. On the other hand, WENDyS opens the opportunity for interdisciplinary teams to use standard experimental data (i.e., strengths of interactions and growth rates) to filter down the possible end states of a community. To demonstrate the latter, here we present a case study from the Indonesian Coral Reef. We analyze how different interactions between anemone and anemonefish species lead to alternative stable states for the coral reef community, and how competition can increase the chance of exclusion for one or more species. WENDyS, thus, can be used to anticipate ecological outcomes and test the effectiveness of management (e.g., conservation) strategies.Comment: 25 pages, 9 figure