Abstract. A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decompositions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the existence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional, density-dependent, Leslie population model. An interactive visualization of the results of computations discussed in the paper can be accessed at the website http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available.
A classical multidimensional scaling (CMDS) method is employed to visualize an intrinsic reaction coordinate (IRC) and a global reaction route map consisting of the equilibrium minima and transition state structures connected by the IRC network. As demonstrations, the method was applied to the IRCs of the intramolecular proton transfer in malonaldehyde and the S2 reaction of OH + CHF → CHOH + F, which are both well described by two principal coordinates. Next, the method was applied to the global reaction route map of the Au cluster; the resulting map shows appropriate positions of five minima and 14 transition states in a reduced 2- or 3-dimensional coordinate space successfully.
We propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real Hénon family, we prove the existence of many regions of hyperbolic parameters in the parameter plane of the family.
Following our recent work to reduce a dimension of a set of reference structures along the intrinsic reaction coordinate (IRC) by a classical multidimensional scaling (CMDS) approach (J. Chem. Theory Comput. 2018, 14, 4263−4270), we propose the method to project on-the-fly trajectories into a reduced-dimension subspace determined by the IRC network, using the out-of-sample extension of CMDS. The method was applied to the SN2 reaction, OH -+ CH3F, in which trajectories show a bifurcating nature around the highly-curved region of the IRC path, and to the structural transformation of Au5 cluster in which the global reaction path network consists of five equilibrium structures and 14 IRCs. It was demonstrated that the present analysis can visualize the dynamics effect by showing a dynamic reaction route on the basis of the static reaction paths.
Abstract. We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real Hénon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex Hénon map.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.