Complexity

Volume 2018, Article ID 8259496, 16 pages

https://doi.org/10.1155/2018/8259496

## Synchronization Measure Based on a Geometric Approach to Attractor Embedding Using Finite Observation Windows

^{1}Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania^{2}Department of Engineering Mechanics, Hohai University, Nanjing 210098, China

Correspondence should be addressed to Minvydas Ragulskis; tl.utk@sikslugar.sadyvnim

Received 24 February 2018; Revised 9 May 2018; Accepted 7 June 2018; Published 8 August 2018

Academic Editor: Daniel Novák

Copyright © 2018 Inga Timofejeva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple and effective algorithm for the identification of optimal time delays based on the geometrical properties of the embedded attractor is presented in this paper. A time series synchronization measure based on optimal time delays is derived. The approach is based on the comparison of optimal time delay sequences that are computed for segments of the considered time series. The proposed technique is validated using coupled chaotic Rössler systems.

#### 1. Introduction

The reconstruction of geometry from a time series is one of the paradigmatic algorithms used for the computational analysis of nonlinear dynamical systems. There is a number of algorithms for the analysis of time series that consider phase spaces, that is, the set of all possible system states, of dynamical systems. For a deterministic dynamical system, the phase space is known from the mathematical model (equations of motion). Thus, with the knowledge of the system state at a particular time, it is always possible to determine the future states. However, real-world or experimental dynamical systems are usually too complex, and phase spaces of such systems are unknown due to the chaotic nature of the analyzed data. Therefore, phase space reconstruction methods had to be developed. A branch of applied dynamical system theory, analyzing such algorithms, that is, extracting the information about geometrical and topological properties of the phase space of time series obtained by performing measurements on an evolving system, is called embedology [1]. One of the first methods for reconstructing the phase space system from a time series by using a time delay variable was proposed several decades ago [2]. A complex time series has an inherent geometry, and Packard et al. [2] were first to show that a representative geometry of a dynamical system can be obtained by using a time series of one of its observables.

The much celebrated Takens embedding theorem proved that it is possible to reconstruct the attractor of a dynamical system using only a single time sequence of scalar measurements [3]. The vectors accomplishing this reconstruction have the form , where are the observations on the system, the integer is the embedding dimension, and is the time difference between consecutive components (also called the time lag or the delay time) and is a multiple of the sampling time step. Takens assumed an infinite sequence of noise-free measurements and proved the existence of a diffeomorphism between the original and the reconstructed attractors for almost any choice of positive delay times and a sufficiently big dimension .

Although the application of the Takens time delay technique is straightforward for almost any time series, the selection of optimal reconstruction parameters and is a nontrivial problem due to their dependence on the nature of the analyzed data. Thus, numerous studies have been made in this area. The selection of optimal time delay value is usually based on the optimization of a particular target function that corresponds to some measures of the quality of phase space reconstruction. Classical statistical approaches for the construction of the target function include the autocorrelation function method and the mutual information method [4–6]. Such techniques are based on maximizing the independence between the coordinates of the delay vector. This approach is heavily reliant on the geometry of the phase space reconstruction, which is appropriate for the study of parameters such as fractal dimension and Lyapunov exponents. However, the selection of such target function does not always yield best results for nonlinear time series prediction problems [7].

Another class of methods for the selection of an optimal is based on the concept of phase space expansion. The “fill factor” as a spatial measure of the phase space was first introduced by Buzug and Pfister in [8, 9]. The fill factor quantifies an attractor’s utilization of the embedding space as a function of . While this method does not possess the drawbacks of statistical techniques with respect to nonlinear data, it is computationally intensive (for ) and is prone to overfolding of the attractor [10]. A more computationally effective algorithm based on the expansion of the area for all pairwise planar projections of the embedded attractors is presented in [11]. Similar approaches based on the spatial properties of phase spaces include the average displacement method [10], the SVF method [12], and the wavering product [13] method.

A different viewpoint considers both phase space reconstruction parameters simultaneously since some experiments indicate that an irrelevant relationship between and may impact the congruence between the original system and the reconstructed phase space. This class of methods includes the small-window solution [14] and the C-C method [15].

Novel approaches to the selection of embedding parameters have been developed in recent years. It is shown in [16] that an improved phase space reconstruction method based on manifold embedding and Laplacian eigenmaps can reveal the structure of the chaotic attractor. A new algorithm for the estimation of the dimension of chaotic dynamical systems using neural networks and robust location estimate is proposed in [17]. Statistical analysis of the nearest neighbors is exploited for the reliable estimation of minimum embedding dimension for noisy time series in [18].

It has been demonstrated in [11, 19–21] that nonuniform embedding (when all time delays are not equal) performs better than uniform embedding in a variety of applications—typical examples are causality and coupling detection and time series prediction. However, the procedure for the selection of time delays is usually related to a particular application and is implemented by introducing a specific target function which determines the utility generated by a concrete nonuniform embedding.

This paper has two main objectives. The first objective is to introduce a simple and effective algorithm for the identification of optimal time delays which is based on the geometrical properties of the embedded attractor and is applicable for short time series.

The second objective is to introduce a time series synchronization measure based on optimal time delays. A short review of commonly applied methods for the detection of synchronization is given below.

The first linear synchronization measures based on correlation analysis, namely, cross correlation and coherence functions, are widely applied due to their computational effectiveness [22]. However, such measures can only detect the most straightforward regimes of synchronization.

Nonlinear synchronization measures were introduced in order to quantify more complex synchronization effects. The mutual information measure is based on Shannon entropy and takes into account not only linear but also nonlinear dependencies [23]. Phase synchronization measure is used to quantify similarity between cyclic signals and time series. Two approaches—based on the Hilbert or wavelet transform—can be used to implement the phase synchronization detection algorithm [22].

A new form of synchronization between coupled chaotic oscillators called amplitude envelope synchronization has been discovered in [24]. Generalized synchronization of dynamical systems occurs if dynamical variables from one subsystem are a function of the variable of another subsystem [25]. A nonlinear interdependence of dynamical systems based on state space reconstruction is a similar approach to generalized synchronization but does not require a functional relationship between the dynamics of the underlying systems [26]. The function projective synchronization in relay coupled systems is studied in [27]. A novel sort of synchronization called complex antilag synchronization is introduced in [28]. A new approach for the investigation of hybrid chaos synchronization in discrete-time hyperchaotic dynamical systems based on stability theory of linear discrete-time systems and Lyapunov stability theory was proposed in [29].

This paper presents a novel time series synchronization measure based on the geometrical approach towards optimal time delays. This approach is based on the determination of time lags that maximize the volume of the state space occupied by the embedded segments of the time series. The sequences of obtained time lags reveal the level of synchronization between 2 time series. The presented algorithm for the identification of synchronization between two time series is validated using coupled chaotic Rössler systems.

#### 2. Preliminaries

##### 2.1. Two-Dimensional Delay Coordinate Space

Let us consider a harmonic function , where is the angular frequency and is the phase of harmonic oscillations, and the amplitude is set to 1. A pair of function values and , where is the time lag, is mapped into a point in the phase plane , where and are the coordinates of the delay coordinate plane. Two-dimensional time delay embedding maps the harmonic function into an ellipse; the equation of that ellipse reads

The geometrical shape of the ellipse can be exploited for the quantification of the quality of the embedding. Such geometric approach (for a two-dimensional phase plane) was firstly proposed by Buzug and Pfister in 1992 [9]. The radiuses of the embedded ellipse and can be directly expressed from (1):

Thus, the area of the ellipse reads

The maximum area of the ellipse is (when the ellipse becomes a circle). Now, the function representing the quality of embedding into a two-dimensional delay coordinate space can be defined as a ratio between the area of the ellipse and the area of the circle:

##### 2.2. -Dimensional Delay Coordinate Space

###### 2.2.1. The Embedding Quality Function

The geometrical approach for embedding a harmonic function into a two-dimensional phase plane is generalized for the -dimensional delay coordinate space in [11]. The coordinates of the reconstructed point in the -dimensional delay coordinate space read where is the time delay vector; is the total embedding window. It can be observed that there exist planar projections of the embedded ellipse. The function representing the quality of embedding into the -dimensional delay coordinate space is constructed as an arithmetic average of all quality functions is all possible planar projections [30]:

In the case of the uniform embedding (when ), (6) reads

The equality implies that the harmonic function is compressed into line segments in all possible planar projections. The maximization of in respect of time lags results in a maximum average area of ellipses in all possible planar projections.

###### 2.2.2. A Nonharmonic Function

Let us consider a function defined in the time interval , where and are the limits of the observation window. Every harmonic component of is affected by the quality function when the appropriate harmonic signal is embedded into the -dimensional delay coordinate space. Harmonic components with frequencies where is small will be suppressed (in average) in all possible planar projections more than those harmonic components where is large. Such motivation did help to construct the optimization problem for the identification of an optimal set of time delays which do result in the richest representation of the attractor in the delay coordinate space: where

is the optimal set of time delays and is the Fourier amplitude spectrum of the signal in the observation window [11]. The integral in the denominator is used in order to normalize the target function in respect of the signal (this integral can be computed once at the beginning of the optimization process). The multiplier is used to normalize the target function in respect of the white noise (the value of the target function for the white noise signal now becomes equal to 1 for any time delays not equal to 0) [11].

#### 3. The Proposed Algorithm

The technique proposed in [11] enables a fast and effective determination of the optimal set of time delays by assessing the geometrical properties of the embedded attractor. However, the optimization problem in [11] does not asses the phase spectrum of the Fourier transform. In other words, [11] is an approximate algorithm for the determination of optimal time delays when every discrete harmonic component of the Fourier amplitude spectrum is treated as a separate individual harmonic component. Nevertheless, such an approximating approach appears to be useful in practical applications—especially in time series forecasting algorithms based on fuzzy neural networks [31–33]. However, it is clear that the Fourier phase spectrum does have an impact on the geometrical representation of the reconstructed attractor in the phase space. In other words, a more accurate assessment of the geometrical properties of the reconstructed attractor is required. Let us consider a continuous function in the observation window and the -dimensional delay coordinate space with the set of time lags . Then, instead of computing any projections of the embedded attractor into a 2-dimensional phase planes, we compute the average distance of the attractor points to the origin of the embedding frame:

##### 3.1. Properties of : Embedding a Harmonic Function

Let us consider that the embedding is uniform and the embedded function is a harmonic function , where , , and are the amplitude, the circular frequency, and the phase, accordingly. Then,

The change of variables yields

Now,

Let us denote . Then,

Note that when . Thus, is the maximal value for the harmonic function. On the opposite, the pole is a removable singularity at and is the minimal value for the harmonic function.

##### 3.2. The Comparison between and

We continue with the harmonic function and uniform embedding. Then, and (9) yields , where is the Dirac delta function. Without the loss of generality, we assume that and . Then, ; . Graphs of and are illustrated in Figure 1.