Combinatorial-topological framework for the analysis of global dynamics

Abstract

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conley's topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications.

Lattice structures for attractors I

“…Finally, these maps are robust with respect to either numerical or experimental perturbation. This strongly suggests that this approach provides a framework in which to develop a new computational theory of nonlinear dynamics, and the first steps in this direction have been taken [2,5,4,12]. For any given dynamical system ϕ, Att(ϕ) has at most countable elements while ANbhd(ϕ) typically contains uncountably many elements.…”

Databases for the Global Dynamics of Multiparameter Nonlinear Systems

“…Thus it allows us to vary the methods used to obtain an outer approximation. In particular we are no longer bound to use interval vectors [2]. We also began to study the feasibility of constructing outer approximations based on time series data and obtained probabilistic results concerning the ability to reconstruct the homotopy of a map from random data points [3].…”

“…To do this we developed and implemented clutching records which gives information about how the dynamics changes between the regions [2]. More explicitly, for each region of parameter space a Morse Graph is computed.…”

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].…”