Abstract
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are a priori unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from the observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from electroencephalogram signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions.
Highlights
The reason is simple, nature and man-made systems are filled with such examples, where many units interact dynamically and are able to collectively self-organise —as our brains, composed of billions of neurons interconnected in complex synaptic networks, or our powergrids, composed of steady power-plants, fluctuating renewable power-sources, and randomly demanding consumers, all inter-connected by a complex network of transmission lines
An important way to characterise these complex systems and their emerging behaviours is by using measures from Information Theory, which require the calculation of invariant probabilities from observed data, a process that is never trivial
In this work we present a procedure that uses an Information theoretical perspective to approximate a Generating Markov Partition (GMP) for a complex system from finite resolution and finite time interval trajectories
Summary
An important way to characterise these complex systems and their emerging behaviours is by using measures from Information Theory, which require the calculation of invariant probabilities from observed data, a process that is never trivial. The invariant probability measure (IPM)[1,2,3,4], μ(Γ), of a complex system or a dynamical system, is the probability measure, μ, that is preserved under the system’s equations of motion, and gives the probability density of finding the system at a given point in state space, Γ. From a practical point-of-view, it is advantageous to derive a coarse-grained discrete IPM by making some finiteresolution observations, only during a finite time-interval, and on a projected lower-dimensional space, but from where all relevant statistical quantities can be well estimated. Instead of dealing with a continuous IPM, we need to transform the system’s trajectories into finite symbolic-sequences conforming a finite alphabet[1,2,3,4] and find a discrete IPM for the symbols’ probabilities
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