Equations governing the nonlinear dynamics of complex systems are usually unknown, and indirect methods are used to reconstruct their manifolds. In turn, they depend on embedding parameters requiring other methods and long temporal sequences to be accurate. In this paper, we show that an optimal reconstruction can be achieved by lossless compression of system’s time course, providing a self-consistent analysis of its dynamics and a measure of its complexity, even for short sequences. Our measure of complexity detects system’s state changes such as weak synchronization phenomena, characterizing many systems, in one step, integrating results from Lyapunov and fractal analysis.

Dynamics of natural systems is often described by nonlinear equations. When those equations are unknown, we can reproduce the system dynamics through the reconstruction of the manifold from the time course of one of its variables [

According to the Whitney theorem, the diffeomorphism on

To estimate the embedding dimension, methods that involve the fact that entropies are diffeomorphism invariants have been proposed and include, for example, differential entropy [

In this paper, we show that optimal embedding dimension can be estimated through a measure of the Kolmogorov complexity, which is here evaluated using the compression algorithm introduced by Lempel and Ziv [

To estimate the optimal embedding dimension

This means that the optimum embedding dimension is the one at which entropy rate has at least a component that behaves as a nonlinear function of

To estimate the optimal dimension for the embedding, avoiding the evaluation of the optimal time delay

Estimated embedding dimensions for low-dimensional chaotic system. (a) First derivative of

In coupled chaotic systems with a drive-response configuration, generalized synchronization (GS) may occur if the state of response system

For instance, it has been proven that synchronization occurs when all of the conditional Lyapunov exponents are negative [

Here, we show that the analysis of the dimension of the response system through lossless complexity measures can easily detect the emergence of GS. To this aim, we studied synchronization phenomena between two unidirectional chaotic systems, where GS takes place as a function of coupling factor

As a first example, we considered a unidirectionally coupled system in which the autonomous driver

This type of system was investigated in previous works [

Lorenz driven by Rossler system. (a) Estimated

For each coupling value, we estimated the optimal embedding dimension as the average across 50 realizations of the system dynamics. Time series with 1000 time points were used for the estimation. When approaching the synchronization threshold

A second example we considered is the unidirectionally coupled system formed by two identical Hénon maps [

We computed Lyapunov exponents using the pull-back method with 5000 time points and we found that the conditional exponent takes negative values in two different intervals of couplings: in a window

Coupled Hénon maps. (a) Estimated embedding

In conclusion, we have shown that complexity measures used to reconstruct the geometry of the manifold of a dynamical system can be used to gain many insights about the system itself, even when the underlying governing equations are not known. We observed how the irregularity of the dynamics, expressed by entropy rate estimates, reaches a plateau and remains constant by increasing the dimension of the manifold, providing a robust and parameter-free estimate of the intrinsic optimal dimension. Our measure is quite stable for different values of time delay

We choose to relate complexity of the system to the way at which entropy rate measures depart from extensive functions and become nonlinear functions of the number of system dimensions. How to properly evaluate complexity has been a debated topic in last years. One of the most debated issue is the fact that information theoretic estimates like Shannon entropy measure the degree of randomness of the system and do not take into account system’s dynamical organization, whereas ideal complexity measures should treat both random and lower distributions as minimally complex [

To detect synchronization, usually quantities related to the randomness of the dynamics [

The data that support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

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