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We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilibrium. An important component of our method is that we obtain mathematically rigorous lower bounds on the size of this neighborhood, as well as validated a posteriori error bounds for the polynomial approximation. In the second step we use a rigorous numerical integrating scheme to propagate the boundary of the local stable/unstable manifold as long as possible, i.e., as long as the integrator yields validated error bounds below some desired tolerance. The procedure exploits adaptive remeshing strategies which track the growth/decay of the Taylor coefficients of the advected curve. In order to highlight the utility of the procedure, we study the embedding of some two dimensional manifolds in the Lorenz system. [ABSTRACT FROM AUTHOR]

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]