Analysis and Visualization of High‐Dimensional Dynamical Systems’ Phase Space Using a Network‐Based Approach

Abstract

The concept of attractors is considered critical in the study of dynamical systems as they represent the set of states that a system gravitates toward. However, it is generally difficult to analyze attractors in complex systems due to multiple reasons including chaos, high-dimensionality, and stochasticity. This paper explores a novel approach to analyzing attractors in complex systems by utilizing networks to represent phase spaces. We accomplish this by discretizing phase space and defining node associations with attractors by finding sink strongly connected components (SSCCs) within these networks. Moreover, the network representation of phase space facilitates the use of well-established techniques of network analysis to study the phase space of a complex system. We show the latter by introducing a new node-based metric called attractivity which can be used in conjunction with the SSCC as they are highly correlated. We demonstrate the proposed method by applying it to several chaotic dynamical systems and a large-scale agent-based social simulation model.

Network representations of attractors for change point detection

“…Whilst the transformation from time series to reconstructed phase space is relatively straightforward (subject to appropriately selected delay lags), achieving a faithful and parsimonious representation of the attractor's structure and dynamics within phase space (reconstructed or otherwise) is challenging as these attractors are defined continuously in phase space. St. Luce & Sayama [32] proposed a method of achieving a transition network representation of an attractor by discretising phase space into voxels. This greatly simplifies the task of dealing with discrete observations of continuous data.…”

“…One of the drawbacks of the network-base method for identifying attractors presented by St. Luce et al [32] is the potentially poor computational scaling for increasing phase space dimension. This is due to the grid based voxel scheme used to discretise the entire phase space, where each voxel is represented by a single node [32].…”

“…One of the drawbacks of the network-base method for identifying attractors presented by St. Luce et al [32] is the potentially poor computational scaling for increasing phase space dimension. This is due to the grid based voxel scheme used to discretise the entire phase space, where each voxel is represented by a single node [32]. This results in a prohibitively large network for even modest phase space dimensions that requires extensive pruning before any further computation can be done.…”

“…Because A only contains points that are relevant to the attractor of the system, the resulting discretisation automatically excludes regions which are greater than a notional distance of δ from the attractor. This removes the need to trim off irrelevant regions of the discretised phase space, previously required in the the grid-based voxel approach by St. Luce et al [32].…”