One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone.
Abstract. The aim of this paper is to introduce a method for computing rigorous lower bounds for topological entropy. The topological entropy of a dynamical system measures the number of trajectories that separate in finite time and quantifies the complexity of the system. Our method relies on extending existing computational Conley index techniques for constructing semi-conjugate symbolic dynamical systems. Besides offering a description of the dynamics, the constructed symbol system allows for the computation of a lower bound for the topological entropy of the original system. Our overall goal is to construct symbolic dynamics that yield a high lower bound for entropy. The method described in this paper is algorithmic and, although it is computational, yields mathematically rigorous results. For illustration, we apply the method to the Hénon map, where we compute a rigorous lower bound for topological entropy of 0.4320.
Abstract. This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semi-conjugacy between an attractor for the Swift-Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits.
Abstract. We present a numerical method to prove certain statements about the global dynamics of infinitedimensional maps. The method combines set-oriented numerical tools for the computation of invariant sets and isolating neighborhoods, the Conley index theory, and analytic considerations. It not only allows for the detection of a certain dynamical behavior, but also for a precise computation of the corresponding invariant sets in phase space. As an example computation we show the existence of period points, connecting orbits, and chaotic dynamics in the Kot-Schaffer growth-dispersal model for plants.
Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting homology can be computed. In this paper we present an algorithm for correctly computing the homology of one-and two-dimensional nodal domains. The approach relies on constructing an appropriate cubical approximation for the nodal domain based on the behavior of the defining function at the vertices of a fixed grid. Homology groups for these cubical sets are readily computable using [22,23]. Here, we present a technique to verify that the cubical representation is homeomorphic to the nodal domain, and therefore preserves homology. To illustrate this approach we consider examples from three classes of nodal domains, including the time-dependent patterns generated by the Cahn-Hilliard model for spinodal decomposition. We use these results to examine the probability of correct homology computations given specific grid sizes as related to the analytic estimates presented in [12,31,32,36].
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