Identifying dynamical structures in the physical space of stochastic processes

¹ Departamento de Física, Universidade Federal do Paraná - 81531-980, Curitiba-PR, Brazil
² Departamento de Física and National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte - 59078-970, Natal- RN, Brazil
³ Departamento de Física, Universidade do Estado de Santa Catarina - 89219-710, Joinville-SC, Brazil
⁴ Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco 50670-901, Recife-PE, Brazil
⁵ Physics Department, Clarkson University - Potsdam, NY 13699-5820, USA

Received: 19 October 2018
Accepted: 11 January 2019

Abstract

Characterizing dynamical patterns in the (physical) state space of stochastic processes can be a challenging task. From two visualization techniques, the observable-representation and k-means clustering, a unified framework to identify such structures is developed. The only information required is the system transition matrix R (a quantity that can be directly accessed from experimental data). The approach is illustrated through the analysis of random searches for targets distributed in patchy environments. The protocol —for R constructed from a typical tracked long trajectory— is able to reveal the shape and locations of all the landscape patches. The method constitutes a valuable new tool to study the underlying geometry of general stochastic processes.