Phys. Rev. Lett. 119, 098301 (2017) - Topological Causality in Dynamical Systems

Learn about our response to COVID-19, including freely available research and expanded remote access support.

Go Mobile »
Access by University of Bath Library

Topological Causality in Dynamical Systems

Daniel Harnack, Erik Laminski, Maik Schünemann, and Klaus Richard Pawelzik

Phys. Rev. Lett. 119, 098301 – Published 1 September 2017

Abstract

Determination of causal relations among observables is of fundamental interest in many fields dealing with complex systems. Since nonlinear systems generically behave as wholes, classical notions of causality assuming separability of subsystems often turn out inadequate. Still lacking is a mathematically transparent measure of the magnitude of effective causal influences in cyclic systems. For deterministic systems we found that the expansions of mappings among time-delay state space reconstructions from different observables not only reflect the directed coupling strengths, but also the dependency of effective influences on the system’s temporally varying state. Estimation of the expansions from pairs of time series is straightforward and used to define novel causality indices. Mathematical and numerical analysis demonstrate that they reveal the asymmetry of causal influences including their time dependence, as well as provide measures for the effective strengths of causal links in complex systems.

Received 10 April 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.098301

© 2017 American Physical Society

Physics Subject Headings (PhySH)

Dynamics of networks

Nonlinear Dynamics

Authors & Affiliations

Daniel Harnack^*, Erik Laminski^†, Maik Schünemann^‡, and Klaus Richard Pawelzik^§

University of Bremen, 28359 Bremen, Germany and Center for Cognitive Science (ZKW), 28359 Bremen, Germany

^*daniel@neuro.uni-bremen.de
^†e.laminski@uni-bremen.de
^‡mschuene@uni-bremen.de
^§pawelzik@neuro.uni-bremen.de

Article Text

Click to Expand

Supplemental Material

Click to Expand

References

Click to Expand

Issue

Vol. 119, Iss. 9 — 1 September 2017

Reuse & Permissions

APS and the Physical Review Editorial Office Continue to Support Researchers

COVID-19 has impacted many institutions and organizations around the world, disrupting the progress of research. Through this difficult time APS and the Physical Review editorial office are fully equipped and actively working to support researchers by continuing to carry out all editorial and peer-review functions and publish research in the journals as well as minimizing disruption to journal access.

We appreciate your continued effort and commitment to helping advance science, and allowing us to publish the best physics journals in the world. And we hope you, and your loved ones, are staying safe and healthy.

Ways to Access APS Journal Articles Off-Campus

Many researchers now find themselves working away from their institutions and, thus, may have trouble accessing the Physical Review journals. To address this, we have been improving access via several different mechanisms. See Off-Campus Access to Physical Review for further instructions.

The relation of points and on multidimensional manifolds illustrated in one dimension. The joint manifold represented by (, ) can be interpreted as the function mediating the mappings between both spaces, and local linearizations of the mappings as the slope around a reference point. (a) When only , a one-to-one mapping from to exists, but not in the reverse direction: is not uniquely determined for all states . Locally, can be attributed a diverging expansion property, since close neighbors of a given point correspond to distant parts of the joint density (, ). The dashed lines visualize nonuniqueness. (b) Here, both couplings are nonzero, but . Larger independence of implies a stronger expansion by than by at most reference points, which is indicated by the higher slope of (, ) when seen from . (c) If no coupling exists, expansion diverges in both directions.
(a) The state-space dependent in a system given by Eqs. () is shown for , . (b) A segment of and the corresponding time courses of and for the same coupling weights. Different regimes of dominant causal direction give rise to different dynamical motifs. If is close to 0 for subsequent time points (light gray), and synchronize. If varies strongly around 0 (dark gray), and desynchronize. When the causal influence from one variable to the other is dominant, here from 1 to 2 (blue), the trajectory of shows higher amplitude excursions than the one of . (c) The mean asymmetry index for the same system with varying coupling strengths.
Two bidirectionally coupled Rössler systems [Eqs. ()] with , , and . The time series of data points were embedded with dimension . and are shifted to be aligned with the temporal mean of the corresponding reconstructions . (a) Local asymmetry of points shown on projections of the attractor to each system. (b) (black) for 200 consecutive time steps and the corresponding time series of (orange). The gray dashed line marks chance level. (c) Time series and used to estimate .
(a) Transitivity. A unidirectionally coupled chain is realized by a system of Eqs. () () with only and and . was averaged from randomly selected data points from time series of points with an embedding dimension . A bivariate increase of the excess of over chance level (CL) is observed, consistent with the theoretical prediction. (b) Common input investigated in a system of Eqs. () () with and only . is estimated with an embedding dimension . and (not shown) depend weakly on the common input and only become significant (marked by *) in the presence of high redundancy between 1 and 2, signified by a high Pearson correlation coefficient (PCC).

Sign up to receive regular email alerts from Physical Review Letters