A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems

Abstract

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.

Recent development in rigorous computational methods in dynamical systems

“…A partial order ≺ between the computed combinatorial Morse sets can be determined by the analysis of paths in G connecting those sets: i ≺ j if i = j and there exists a path in G from any vertex in M j to any vertex in M i . With the use of the graph G, a combinatorial Morse decomposition can be computed by algorithms introduced in [3,5] based on the standard DFS ("Depth-First Search") strategy. The complexity of these algorithms is linear (i.e., they are very fast).…”

“…The practicality of the approach based on the analysis of combinatorial Morse decompositions is illustrated in [3] with the application of this method to the 2-dimensional overcompensatory Leslie population model and f : R 2 × R 4 → R 2 in (6) given by (7) f (x 1 , x 2 ; θ 1 , θ 2 , κ, p) = θ 1 x 1 + θ 2 x 2 e −κ(x1+x2) , px 1…”

“…In [3], a bounded rectangular region R ⊂ R 2 is fixed which is proven in [39] to contain the dynamics of interest, and bounded ranges of parameters θ 1 , θ 2 and p are considered. The parameter κ is a rescaling factor, so it is arbitrarily set to 0.1, like in [39].…”

“…The automatization of this process allows one to scan large sets of parameters of a given family of dynamical systems to determine changes in dynamics automatically and to search for "interesting" regions of parameters worth further attention. In particular, [3] is a large project which benefits from these features, and we are going to describe it in this section.…”

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…[37]), and also in an automatized method for the computation of the Conley index from index pairs constructed as cubical sets in R n [41,48], used e.g. in a method for editing vector fields and extraction of periodic orbits [8], and also very helpful in an automatic method for classification of dynamics in multi-parameter systems [4,7], which found applications e.g. in population biology [38] and in physics of plasmas [49].…”