Rigorous Computation of the Global Dynamics of Integrodifference Equations with Smooth Nonlinearities

Abstract. Topological tools, such as Conley index theory, have inspired rigorous computational methods for studying dynamics. These methods rely on the construction of an outer approximation, a combinatorial representation of the system that incorporates small, bounded error. In this work, we present an automated approach for constructing outer approximations for systems in a class of integrodi↵erence operators with smooth nonlinearities. Chebyshev interpolants and Galerkin projections form the basis for the construction, while analysis and interval arithmetic are used to incorporate explicit error bounds. This represents a significant advance to the approach given by Day, Junge, and Mischaikow (A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems SIAM J. Appl. Dyn. Syst., 3(2):117160, 2004), extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set. As a demonstration of the techniques, a Morse decomposition of the global dynamics, a list of validated periodic orbits, semi-conjugate symbolic dynamics, and a lower bound on topological entropy are computed for the Kot-Scha↵er integrodi↵erence operator from ecology with the exponential Ricker nonlinearity.