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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 to constructing outer approximations for systems in a class of integrodifference 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 [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 117-160], 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, semiconjugate symbolic dynamics, and a lower bound on topological entropy are computed for the Kot-Schaffer integrodifference operator from ecology with the exponential Ricker nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated. [Copyright &y& Elsevier]