Nonlinear Filtering Preserves Chaotic Synchronization via Master-Slave System

Abstract

We present a study on a class of interconnected nonlinear systems and give some criteria for them to behave like a filter. Some chaotic systems present this kind of interconnected nonlinear structure, which enables the synchronization of a master-slave system. Interconnected nonlinear filters have been defined in terms of interconnected nonlinear systems. Furthermore, their behaviors have been studied numerically and theoretically on different input signals.

1. Introduction

In the study of nonlinear dynamics, much research was devoted to synchronization of chaotic systems, which have been unidirectional coupling [1], and one of the main applications is in secure communications systems [2]. Under unidirectional coupling, there is a master system that forces a slave system. Let us begin by considering a master system whose temporal evolution is ruled by the following equation: 𝐲̇=𝑓(𝐲), (1) where 𝐲∈ℝ^𝑛 is the state vector, with 𝑓 defining a vector field 𝑓∶ℝ^𝑛→ℝ^𝑛. The slave system is given by 𝐱̇=𝑔(𝐱,𝐲), (2) where 𝐱∈ℝ^𝑚 is the state vector, and the function 𝑔∶ℝ^𝑚×ℝ^𝑛→ℝ^𝑚 describes the dynamics of the slave system and coupling. The master system behaves as an autonomous system and the slave system is a forced system that under certain conditions could behave as a nonautonomous system, where its dynamic is completely determined by the master system; that is, it means that the slave system acts in function of the master system. Hence, the slave system can be seen as a driven system by an external force, and then it can be studied as a filter when the input signal comes from a master system or any external signal unidirectional coupled.

There are several asymptotical behaviors reported based on master-slave configuration. Some cases are summarized as follows: Identical Synchronization (IS) implies coincidence of the corresponding states of the interacting systems [1]. Lag Synchronization (LS) occurs when the trajectory of one oscillator is delayed by a specific time and is identical to the trajectory of the other oscillator [3]. Phase Synchronization (PS) means the lock of chaotic oscillator phases, regardless of their amplitudes [4]. Frequency Entrainment Synchronization (FES) occurs when two systems are oscillating with the same frequency [5]. Generalized Synchronization (GS) is defined as the presence of some functional relationship between the states of the slave and master systems [6]. Multimodal Generalized Synchronization (MGS) is presented when there are several basins of attraction for the slave system, and generalized synchronization is also presented [7, 8]. The aforementioned chaotic synchronization phenomena tell us that some types of synchronization are stronger than others. Another characteristic of these asymptotic behaviors is that they always can be described in terms of a functional relationship. For example, let Φ be the GS function which relates the master system with the slave system; that is, 𝐲=Φ(𝐱). In general, the way how synchronization phenomena can be detected is by means of the auxiliary method [6]; thus the way one can know that unidirectional coupled systems present GS is when they present asymptotic behavior: lim_𝑡→∞𝐱¹(𝑡)−𝐱²(𝑡)=0, (3) where 𝐱¹(𝑡) and 𝐱²(𝑡) are solutions of the slave and auxiliary systems, respectively, initialized with different initial condition 𝐱¹(0)≠𝐱²(0). If this limit is satisfied, it indicates that the slave system has lost its sensitivity to initial conditions, thus losing their autonomy. Thence, the trajectory of the response system depends on the input signal from master system. If we consider the slave system as a forced system, then we will see that its behavior depends on an external forcing, like a filter does.

Forced Synchronization (FS) is defined as a phenomenon that occurs when oscillations of several chaotic systems 𝐱⁽¹⁾(𝑡), 𝐱⁽²⁾(𝑡),…,𝐱^(𝑘)(𝑡) show correlated behavior because of an external signal applied to these oscillators [9]. Meanwhile, the conditions in order that a chaotic system behaves as a filter for an external signal under a specific coupling are given in [10] where this phenomenon was called nonlinear filtering (NF). So, these three phenomena (GS, FS, and NF) can be recognized when the limit (3) is satisfied and can be studied from the viewpoint of forced chaotic systems. GS contains NF and MGS because these two phenomena present asymptotic behavior. Nevertheless, there exists a difference between NF and MGS. For MGS, the initial conditions determine the basin of attraction where the trajectory of the slave system converges, the fact despite that there exists a functional relationship between the slave and the master systems, the slave system does not behave like a filter. For NF, the slave system must act in function of the external force and never in function of its initial conditions.

The dynamic of at least one state of several chaotic systems has similar structure to low pass filter, for example, the first equation of the Lorenz system 𝑥̇=−𝜎𝑥+𝜎𝑦, where 𝑥 and 𝑦 can be seen as the output and input signals of a low pass linear filter, respectively. The second equation of the Lorenz system can be seen as a low pass nonlinear filter, 𝑦̇=−𝑦+𝑥(𝜌−𝑧), where 𝑦 is the output and 𝑥 and 𝑧 are the input signals of a low pass nonlinear filter. Without loss of generality, from the viewpoint of continuous dynamical systems, a filter can be seen as a forced dynamical system in which its response depends on its structure, given by their equations, and can be linear or nonlinear. Then a natural question would be the following: what is the effect on the response of forced nonlinear system given an input signal? In order to answer this question, we make a study of a chaotic system forced with different kinds of signals.

We are interested in explaining synchronization phenomenon of chaotic systems based on nonlinear interconnected structures, as these structures behave as a nonlinear filter that allows synchronization of master and slave systems. Thereby, the target is not to put forward a new nonlinear filter that can replace those designed with the specific purpose of preventing noise in the signal. Thus, the objective of this research is to show how the synchronization phenomenon underlies nonlinear interconnected structure. This kind of structure which is immersed in some chaotic systems could have some useful applications in filtering waves such as those originated from earthquakes and tsunamis, because they can be tuned to produce resonance at certain frequencies. So, the target of this work is to give features of the nonlinear filter as those given by the second and third equations of the Lorenz system in function of the parameters of the input signal 𝑢, which gives us a complementary perspective with respect to the analysis made in [9], where it was only proved and showed that a forced-chaotic system filters constant, sinus, and random signals. To achieve this goal, we make a study on the effect of nonlinear filter based on interconnected nonlinear systems for different input signals 𝑢, like sinusoidal, chaotic, and random signals. Several chaotic systems that have a similar structure to interconnected systems can be analyzed like a nonlinear filtering phenomenon.

The paper is organized as follows: Section 2 contains basic definitions of nonlinear filters based on 𝑛-interconnected systems; Section 3 presents a way how tuning the parameters of an 𝑛-interconnected system in order to the system behaves as a nonlinear filter; in Section 4 is the study of the response to amplitude and frequency of a nonlinear filter; Section 5 shows the response of the filter when the input signal is noise; Section 6 presents a study case of correlation coefficient analysis; in Section 7, we present the relation between nonlinear filters and chaotic systems; finally, conclusions are given in Section 8.

2. Nonlinear Filter Structure

In the theory of linear systems, it is very well known that a first-order low pass linear filter is given as follows: 𝑥̇₁=𝑘₁𝑥₁+𝑘₂𝑢, (4) where 𝑥₁∈ℝ is the output of the filter, and 𝑘₁,𝑘₂∈ℝ are parameters of the filter (4). These parameters control attenuation and amplitude of the input signal, respectively, and 𝑢∈ℝ is the input signal to be filtered. Based on the configuration of low pass filter given by (4), a low pass nonlinear filter is defined as follows.

Definition 1. Let 𝑥₁∈ℝ be an output signal, 𝑢∈ℝ an input signal, and 𝑘₁, 𝑘₂, and 𝑘₃∈ℝ parameters. Thus, a low pass nonlinear filter can be defined as follows: 𝑥̇₁=𝑘₁𝑥₁+𝑘₂+𝑘₃𝑥₁𝑢. (5) The parameter 𝑘₃ in the above definition is used to control the nonlinear term 𝑥₁𝑢. Notice that when 𝑘₃=0, the nonlinear term disappears, 𝑘₃𝑥₁𝑢, and then the nonlinear filter (5) behaves as a linear filter. The structure of nonlinear filters (5) has been used to generate chaos, as it can be seen in the Lorenz system [11], but its states are interconnected. Therefore, a natural question emerges around nonlinear filtering: what are the characteristics of two nonlinear filters if they are interconnected and forced by the same signal 𝑢? An interconnected system via nonlinear filters is given as follows:

Definition 2. Let 𝑥₁,𝑥₂∈ℝ be output signals, 𝑢∈ℝ an input signal, and 𝑘₁, 𝑘₂, 𝑘₃, 𝑘₄, 𝑘₅, and 𝑘₆∈ℝ parameters. So, let us define an interconnected system via low pass nonlinear filters as follows: 𝑥̇₁=𝑘₁𝑥₁+𝑘₂+𝑘₃𝑥₂𝑥̇𝑢,₂=𝑘₄𝑥₂+𝑘₅+𝑘₆𝑥₁𝑢. (6) Thus the system (6) is called two-interconnected systems.

Now, in Definition 2, the parameters 𝑘₃ and 𝑘₆ are coupling parameters for the outputs 𝑥₁ and 𝑥₂, respectively. In general, an 𝑛-interconnected system can be defined by coupled 𝑥_𝑛 to the output 𝑥_𝑛+1 with 𝑛>2. This 𝑛-interconnected system has a core based on low pass linear filters 𝑥̇=𝐾𝐱+𝐵𝑢, where 𝐱∈ℝ^𝑛 is a state vector, 𝐾∈ℝ^𝑛×𝑛 is a matrix, and 𝐵∈ℝ^𝑛×1 is a constant vector. Thus, we can define 𝑛-interconnected system with the following expression: 𝑥̇=𝐾𝐱+𝐵𝑢+𝑓(𝐱,𝑢), (7) where the nonlinear function 𝑓(𝐱,𝑢)∈ℝ^𝑛 is constituted by terms 𝑘_𝑖𝑥_𝑗𝑢, 𝑖∈{1,2,…,3𝑛}, 𝑗∈{1,2,…,𝑛}. An 𝑛-interconnected system (7) is studied in [12] where the authors found that its model produces hyperbolic chaos when it is forced by a sinusoidal wave. On the other hand, system (7) is formed by a linear part 𝐾𝐱, an input signal 𝐵𝑢, and a nonlinear part 𝑓(𝐱,𝑢), which can induce sensitivity to initial conditions (𝑥₁(0),𝑥₂(0),…,𝑥_𝑛(0)). But a filter must act in function of the input signal and not in function of the initial condition. Therefore, it is important to find conditions in order to guarantee that an interconnected system behaves as a filter. Roughly speaking, the response of a filter only depends on the kind of input signal and if the interconnected systems depend on initial condition, this is classified as generalized forced synchronization phenomenon by forced systems [9]. Therefore, according to context, a nonlinear filter is defined in the next way.

Definition 3. Let 𝐱^𝑖(𝑡)=(𝑥^𝑖₁(𝑡),𝑥^𝑖₂(𝑡),…,𝑥^𝑖_𝑛(𝑡))^𝑇 be a vector of output signals given by 𝑛-interconnected system (7) with initial condition 𝐱^𝑖(0)=(𝑥^𝑖₁(0),𝑥^𝑖₂(0),…,𝑥^𝑖_𝑛(0))^𝑇. System (7) is called an 𝑛-interconnected nonlinear filter if it always presents asymptotic behavior: lim_𝑡→∞𝐱⁽¹⁾(𝑡)−𝐱⁽²⁾(𝑡)=0. (8)

In Definition 3, if 𝑡→∞ and the output of the 𝑛-interconnected system is independent of the initial conditions, then system (7) is considered an 𝑛-interconnected nonlinear filter.

3. Tuning of the Parameters

We select the entries of the matrices 𝐾 and 𝐵 in order for condition (8) to be satisfied. Let us start by considering 𝜹(𝑡)=𝐱⁽¹⁾(𝑡)−𝐱⁽²⁾(𝑡), and then 𝛿̇(𝑡)=𝐱̇⁽¹⁾(𝑡)−𝐱̇⁽²⁾(𝑡). Now, by using (7) we deduce that 𝜹̇=𝐾𝜹(𝑡)+𝑓(𝐱⁽¹⁾,𝑢)−𝑓(𝐱⁽²⁾,𝑢) which can be rewritten as follows: 𝜹(𝑡)=𝜹(0)𝑒^𝐾𝑡+𝑒^𝐾𝑡𝑡₀𝑒^𝐾𝑠𝑓𝐱⁽²⁾𝐱,𝑢−𝑓⁽¹⁾,𝑢𝑑𝑠. (9) In order to describe the asymptotic behavior of (9), we state the following theorem.

Theorem 4. If system (7), forced by signal 𝑢, satisfyies the following conditions:(1)there exists a positive constant 𝑘 such that Re{𝜆}≤−𝑘, for every eigenvalue 𝜆 of the linear part of system (7);(2)𝑓(𝐱,𝑢) is a continuous Lipschitz; that is, there exists a positive function 𝐶[𝑢]<∞ such that |𝑓(𝐱⁽²⁾,𝑢)−𝑓(𝐱⁽¹⁾,𝑢)|≤𝐶[𝑢]|𝐱⁽²⁾−𝐱⁽¹⁾|, then, system (7) is a filter for 𝑢, provided that 𝑘>𝐶[𝑢], where 𝐶[𝑢]∶=lim_𝑡→∞1𝑡^𝑡₀𝐶[𝑢(𝑠)]𝑑𝑠. (10)

Proof. Let 𝐱⁽¹⁾(𝑡), 𝐱⁽²⁾(𝑡), and 𝜹(𝑡) defined be before. We want to prove that 𝑘>𝐶[𝑢] implies lim_𝑡→∞|𝜹(𝑡)|=0. From (9) we have the following estimate: |𝜹(𝑡)|≤𝑒^𝐾𝑡|𝜹(0)|+𝑒^𝐾𝑡×^𝑡₀𝑒^{𝐾(−𝑠)}𝑓𝐱⁽²⁾,𝑢−𝑓𝐱⁽¹⁾,𝑢𝑑𝑠≤𝑒^{‖𝐾‖|𝑡|}|𝜹(0)|+𝑒^{‖𝐾‖|𝑡|}×^𝑡₀𝑒^{‖𝐾‖|−𝑠|}𝑓𝐱⁽²⁾,𝑢−𝑓𝐱⁽¹⁾,𝑢𝑑𝑠. (11) Using hypothesis (1), we obtain from the previous inequality the following estimate: |𝜹(𝑡)|≤𝑒^−𝑘𝑡|𝜹(0)|+𝑒^{−𝑘𝑡𝑡}₀𝑒^𝑘𝑠𝑓𝐱⁽²⁾,𝑢−𝑓𝐱⁽¹⁾,𝑢𝑑𝑠. (12) If we multiply by 𝑒^𝑘𝑡, the last expression results in 𝑒^𝑘𝑡|𝜹(𝑡)|≤|𝜹(0)|+^𝑡₀𝑒^𝑘𝑠𝑓𝐱⁽²⁾,𝑢−𝑓𝐱⁽¹⁾,𝑢𝑑𝑠. (13) Using the hypothesis (2), it follows that 𝑒^𝑘𝑡|𝜹(𝑡)|≤|𝜹(0)|+^𝑡₀𝑒^𝑘𝑠𝐶[𝑢]|𝜹(𝑠)|𝑑𝑠. (14) Application of Gronwall's inequality yields |𝜹(𝑡)|≤|𝜹(0)|𝑒^{−𝑡(𝑘−(1/𝑡)∫𝑡0𝐶[𝑢(𝑠)]𝑑𝑠)}, (15) which proves the theorem.

We can infer from the above theorem that all elements of the matrix 𝐾 must be negative; that is, 𝑘_𝑖<0 for 𝑖=1,…,𝑛. On the other hand, due to the fact that (9) does not depend on matrix 𝐵, then its terms are not conditioned and they can take any real value. These will be carried out in the remainder of this paper.

3.1. A Low Pass Nonlinear Filter

The analyzed case is the one-interconnected system given by (7); this case is confined to the particular case of the filter given by (5); where its elements are 𝑥=𝑥₁∈ℝ, 𝐴_1×1=𝑘₁, 𝐵_1×1=𝑘₂, and 𝑓(𝑥₁,𝑢)=𝑢𝑘₃. According to the first condition of the above theorem, we have 𝑘₁<0, and now considering the nonlinear part of the filter, if the Lipschitz condition is satisfied when |𝑓(𝑥₁)⁽¹⁾−𝑓(𝑥₁)⁽²⁾|≤|𝑘₃||𝑢||𝛿(𝑡)|, giving as a result 𝑐[𝑢]=|𝑘₃||𝑢|, therefore the parameter 𝑘₃ can be any real value. Now, the interest is that the one-interconnected system behaves as a nonlinear filter; then Theorem 4 needs to be satisfied by 𝑘₁<−|𝑘₃||𝑢|. If the input signal is a constant, 𝑢=𝜇 (𝜇∈𝑍) and 𝜇≤−𝑘₁/|𝑘₃|, then the response of the filter converges to zero; otherwise it diverges. Another case is when the input signal is 𝑢=𝜇sin(𝜔𝑡), we have the following: |𝛿(𝑡)|≤|𝛿(0)|𝑒^{𝑘1𝑡+|𝑘3|∫𝑡0|𝜇sin(𝜔𝑠)|𝑑𝑠}≤|𝛿(0)|𝑒^{𝑘1𝑡+|𝑘3||𝜇|∫𝑡0𝑑𝑠}, (16) which converges 𝜇≤−𝑘₁/|𝑘₃|. Generally, condition (8) is always satisfied for oscillating functions 𝑢 with |𝑢(𝑡)|≤𝑀. However, for the case that 𝑢 is a polynomial of grade greater than 1, that is, 𝑢=𝑎_𝑛𝑡^𝑛+𝑎_𝑛−1𝑡^𝑛−1+⋯+𝑎₀ for 𝑛>1, condition (8) is not satisfied.

Due to the fact that our interest is to tune the value of the parameters 𝑘₁,𝑘₂, and 𝑘₃, we make a numerical study on the effect of these parameters on the response of 𝑥₁ when the one-interconnected system (7) is being forced by the sinusoidal input signal 𝑢=sin(𝑡). After a transient time, we calculate the maximal range of values of the one-interconnected system (7) by obtaining 𝑅(𝑘_𝑖,𝑘_𝑗)=MAX{𝑥₁(𝑡)}−MIN{𝑥₁(𝑡)} (𝑖≠𝑗), where MAX{𝑥₁(𝑡)} and MIN{𝑥₁(𝑡)} are the maximum and minimum values, respectively, of the response time series 𝑥₁(𝑡) in function of the parameters 𝑘_𝑖,𝑘_𝑗. For example, if we want to calculate 𝑅(𝑘₁,𝑘₂), then we fix the value of 𝑘₃ and simulate different time series of the one-interconnected system (7) for different values of 𝑘₁,𝑘₂. In general, the value of 𝑅(𝑘_𝑖,𝑘_𝑗) is calculated in function of the parameters which are varied and in each case condition (8) is verified if it is satisfied. Figures 1(a) and 1(b) show the graphs of 𝑅(𝑘₁,𝑘₂) for 𝑘₃=−1 and 𝑅(𝑘₁,𝑘₃) for 𝑘₂=1, respectively. These graphs show that the one-interconnected system (7) increments exponentially the amplitude of its response when the parameter 𝑘₃ increases its magnitude; meanwhile, the amplitude of 𝑥₁ is mildly incremented in a linear rate in function of the magnitude of the parameter 𝑘₂.

(a)

(b)

Figure 1 

The maximal range of values of the filter (5) in function of its parameters when it is forced by 𝑢=sin𝑡: (a) 𝑅(𝑘1,𝑘2) and (b) 𝑅(𝑘1,𝑘3).

3.2. Interconnected Nonlinear Filters

Now, the first case is a two-interconnected system (6); the matrices 𝐾_2×2, 𝐵_2×1, and the nonlinear function 𝑓(𝐱,𝑢)∶ℝ²×ℝ→ℝ² are assumed as follows: 𝑘𝐾=¹00𝑘₄𝑘,𝐵=²𝑘₅,𝑓()=𝑘𝐱,𝑢³𝑢𝑥₂𝑘₆𝑢𝑥₁. (17)

System (6) is asked to behave as a nonlinear filter, so the conditions of Theorem 4 are used to select the value of the parameters {𝑘_𝑖}. From the first condition of the theorem, we realize that 𝑘₁,𝑘₄<0, {𝑘₂,𝑘₅}∈ℝ, and if 𝐴>𝐵∫^𝑡₀|𝑢|𝑑𝑠, where 𝐴=max{𝑘₁,𝑘₄} and 𝐵=max{𝑘₃,𝑘₆}, then system (6) is a filter for the signal 𝑢. In order to know more about the characteristics of the values of the parameters, we calculate the eigenvalues of the Jacobian of system (6), which results in the following: 𝜆_1,2=12𝑘₁+𝑘₄±𝑘₁+𝑘₄²−4𝑘₁𝑘₄−𝑘₃𝑘₆𝑢². (18)

Our purpose is that system (6) converges; then Re{𝜆_𝑖} must be negative. Therefore, we need to guarantee that 𝑘₁𝑘₄≥𝑘₃𝑘₆𝑢² is true. Due to 𝑢²≥0, then the parameters 𝑘₃ and 𝑘₆ need to be opposite sign, that is, either 𝑘₃<0, 𝑘₅>0 or 𝑘₃>0, 𝑘₆<0. We choose arbitrarily the first relation of signs.

The effect of the parameters {𝑘_𝑖} on the response 𝑥 of the interconnected nonlinear filters is computed by 𝑅(𝑘_𝑖,𝑘_𝑗) when system (6) is forced by 𝑢=sin(𝑡). Several calculations were made and for each of them we verify that the nonlinear filtering condition (8) was satisfied. Figure 2(a) shows 𝑅(𝑘₁,𝑘₄) when the parameters of the filter (6) are fixed to 𝑘₂=𝑘₅=𝑘₆=1, 𝑘₃=−1. Figure 2(b) shows 𝑅(𝑘₂,𝑘₅) for 𝑘₁=𝑘₄=𝑘₃=−1 and 𝑘₆=1.

(a)

(b)

Figure 2 

The maximal range of values of the filter (17) in function of its parameters when the filter is forced by 𝑢=sin𝑡. (a) 𝑅(𝑘1,𝑘4) and (b) 𝑅(𝑘2,𝑘5).

In the remainder of this paper, we studied the effect of the parameters input signal 𝑢 on the response of the coupled filter (6); that is, we tune the parameters of the filter, vary the parameter's values of the input signal, and observe the effect of the response. Therefore, we need to fix 𝑘_𝑖 (𝑖=1,…,6). Our interest is focus on the study of the response to the input 𝑢 when the linear filters 𝑥₁, 𝑥₂ are coupled in a nonlinear way. Without loss of generality and seeking clarity in our study, we consider 𝑘₅=0. The rest of the parameter values could vary in the intervals 𝑘₁<0, 𝑘₄<0, 𝑘₃𝑘₆<0, {𝑘₂,𝑘₅}∈ℝ, which is a rich variety of values where the filter works. We can select any value in these intervals and produce similar responses; nevertheless, we fix the values to 𝑘₁=−1, 𝑘₂=28, 𝑘₃=−1, 𝑘₄=−2.66, 𝑘₅=1, which will be used in the rest of the paper.

4. Response to the Amplitude and Frequency

In this section, we present a study on the effect of the filter's response (6) as function of parameter's values of the sinusoidal input signal 𝑢(𝑡)=𝜇sin(𝜔𝑡). Figure 3(a) shows the output signals of the 𝑛-interconnected nonlinear filter for the cases 𝜇=1, 𝜔=3 and 𝜇=1, 𝜔=5. One can see that the orbits have a form of a Lissajous curve and the amplitude of the output signal as the frequency of the input signal is increased. In general, to see the effect of amplitude 𝜇 on the filter's response, we calculate the length of the Lissajous 𝐿 in this way 𝐿=^𝑇₀𝑥̇₁²+𝑥̇₂²𝑑𝑡, (19) where 𝑇 is the period of the orbit (𝑥₁(𝑡), 𝑥₂(𝑡)). The effect of the input signal's frequency 𝜔 on the length 𝐿 of the orbit of the filter's response is shown in Figure 3(b). Thus, 𝐿 falls exponentially according to the increment of the frequency 𝜔. System (6) presents a rejection to high frequencies because it is a low pass filter.

(a)

(b)

Figure 3 

(a) The orbits of two Lissajous curves calculated by forcing system (6) with 𝜇=1 and 𝜔=1,3. (b) We can see how the size of the Lissajous curve 𝐿 falls exponentially, as 𝜔 increments its value.

Now, fixing the frequency to 𝜔=1.0 and varying the amplitude 𝜇, we observe that new frequency components appear in the Fourier Transform (FT) of the output signal 𝑥₁. When the amplitude increases, then more peaks of frequency appear as multiples of the input signal's frequency. As is shown in Figures 4(a) and 4(b), 𝜇=1 and 5, respectively. This is a characteristic of a nonlinear filter that linear filters do not present.

(a)

(b)

Figure 4 

Both plots are the FT for the state 𝑥1 of the filter (6) when it is filtering the input signal 𝑢=𝜇sin(𝑡) for the cases (a) 𝜇=1.0 and (b) 𝜇=5. These graphs show that when 𝜇 is incremented, new component frequency peaks appear in the power spectrum of the filter's response.

The nonlinear filters display assorted behaviors, that is, contrary to the example shown in Figure 4, where the amplitude is fixed to 𝜇=10 and frequency 𝜔 is varied. Figures 5(a) and 5(b) show the FT of the response of the filter (6) for 𝜔=1 and 𝜔=2, respectively. Comparing both plots of Figure 5, we see that frequency peaks appear at multiples of input signal's frequency 𝜔.

(a)

(b)

Figure 5 

Both plots are the FT of the state 𝑥1 of the filter (6) when it is filtering the input signal 𝑢=10sin(𝜔𝑡) for the cases (a) 𝜔=1 and (b) 𝜔=2.

4.1. A Theoretical Justification

For the purpose of giving a theoretical justification of the behavior of the filter (6) when it is forced by the sinusoidal signals, we can see in the interconnected nonlinear filter (6) that 𝑥̇₁ and 𝑥̇₂ are linear first-order differential equations which can be rewritten in the next form: 𝑥₁(𝑡)=𝜇𝑒^𝑘1𝑡𝑘₂𝑒^−𝑘1𝑡sin(𝜔𝑡)𝑑𝑡+𝑘₄𝑒^−𝑘1𝑡sin(𝜔𝑡)𝑥₂(𝑡),𝑥𝑑𝑡₂(𝑡)=𝑘₆𝜇𝑒^𝑘5𝑡𝑒^−𝑘5𝑡sin(𝜔𝑡)𝑥₁(𝑡)𝑑𝑡. (20)(21) Without loss of generality and for seeking clarity, we consider the parameter 𝑘₅=0. Applying integration by parts to (20) and considering that the states 𝑥₁(𝑡) and 𝑥₂(𝑡) are functions that depend on time and frequency and that 𝜇 is a constant, we have the following: 𝑥₁(𝑡)=−𝜇𝜔²+𝑘²₁𝑘₂𝑘₁sin(𝜔𝑡)+𝜔cos(𝜔𝑡)+𝑘₃𝑘₁sin(𝜔𝑡)+𝜔cos(𝜔𝑡)𝑥₂(𝑡)−𝑘₄𝑒^𝑘1𝑡∫𝑒^−𝑘1𝑡𝑘₁()(𝑥̇sin𝜔𝑡+𝜔cos𝜔𝑡)₂𝑑𝑡. (22) And for (21) we have 𝑥₂(𝑡)=−𝜇𝑘₆𝜔²+𝑘²₅𝑘₅sin(𝜔𝑡)+𝜔cos(𝜔𝑡)𝑥₁(𝑡)−𝑒^𝑘5𝑡𝑒^−𝑘5𝑡𝑘₅()()𝑥̇sin𝜔𝑡+𝜔cos𝜔𝑡₁𝑑𝑡. (23)

Note that in (22) and (23), the magnitude values of 𝑥₁(𝑡) and 𝑥₂(𝑡) decrease when the frequency 𝜔 increases. Furthermore, since the terms 𝑒^𝑘1𝑡𝑒^−𝑘1𝑡𝑘₁sin(𝜔𝑡)+𝜔cos(𝜔𝑡)𝑥̇₂𝑒𝑑𝑡,_𝑘5𝑡𝑒^−𝑘5𝑡𝑘₅sin(𝜔𝑡)+𝜔cos(𝜔𝑡)𝑥̇₁𝑑𝑡 (24) have factors (𝜔²+𝑘²₅)^𝑛, (𝜔²+𝑘²₁)^𝑛 for 𝑛≤−2, then we approximate (22) and (23) to 𝑥₁(𝑡)𝜇≈−𝜔²+𝑘²₁𝑘₂𝐺₁(𝑡)+𝑘₃𝐺₁(𝑡)𝑥₂(𝑡),𝑥₂(𝑡)≈−𝜇𝑘₆𝜔²+𝑘²₅𝐺₂(𝑡)𝑥₁(𝑡), (25)(26) where 𝐺₁(𝑡)=𝑘₁sin(𝜔𝑡)+𝜔cos(𝜔𝑡), 𝐺₂(𝑡)=𝑘₅sin(𝜔𝑡)+𝜔cos(𝜔𝑡) are periodic functions with period given by the input signal. Substituting (25) in (26), we have 𝑥₁(𝑡)≈−𝜇𝑘₂𝐺₁(𝑡)𝜔²+𝑘²₁+𝜇²𝑘₄𝑘₈𝐺₁(𝑡)𝐺₂(𝑡)𝜔²+𝑘²₁𝜔²+𝑘²₅𝑥₁(𝑡). (27) Now, solving for 𝑥₁(𝑡) from (27), it results that 𝑥₁(𝑡)≈−𝜇𝑘₂𝜔²+𝑘²₅𝐺₁(𝑡)𝜔⁴+𝑘²₁+𝑘²₅𝜔²+𝑘²₁𝑘²₅−𝜇²𝑘₃𝑘₆𝐺₁(𝑡)𝐺₂(𝑡). (28) And for 𝑥₂(𝑡) it results that 𝑥₂(𝑡)≈𝜇²𝑘₂𝑘₆𝐺₁(𝑡)𝐺₂(𝑡)𝜔⁴+𝑘²₅+1𝜔²+𝑘²₅−𝜇²𝑘₃𝑘₆𝐺₁(𝑡)𝐺₂(𝑡). (29) Figure 6 shows the orbit of the solution of (28) and (29) and the orbit of the numerical solution of the filter (6) when 𝜇=1.0 and 𝜔=1.0. The thin Lissajous curve is calculated with (6) and another with (28) and (29). We can see that both orbits oscillate in a Lissajous curve of one knot and the same symmetric form with respect to the 𝑦-axis, but slightly different amplitude.

Figure 6 

Comparison of Lissajouses. Lissajous with solid line corresponds to the solution of system (6) and Lissajous with dashed line corresponds to its approximated solution plotted with the parametric equations (28) and (29).

Now, we analyze the FT of the equations of system (6) which are given as follows: 𝑗Ω𝑋₁(Ω)=𝑘₁𝑋₁(Ω)+𝑘₂FT{𝑢}+𝑘₃FT𝑢𝑥₂,𝑗Ω𝑋₂(Ω)=𝑘₅𝑋₂+𝑘₆FT𝑢𝑥₁(Ω), (30)(31) where FT{𝑥_𝑖}=𝑋_𝑖(Ω) is the FT of the signal 𝑥_𝑖. For the case when 𝑢=𝜇sin(𝜔𝑡), we have Ω𝑋₁(Ω)=−𝑗𝑘₁𝑋₁(Ω)+𝑘₃𝜇2𝑋₂(Ω+𝜔)−𝑋₂(Ω−𝜔)+𝑘₂𝜇𝜋(𝛿(Ω+𝜔)−𝛿(Ω−𝜔)),Ω𝑋₂(Ω)=−𝑗𝑘₅𝑋₂(Ω)+𝑘₆𝜇2𝑋₁(Ω+𝜔)−𝑋₁(Ω−𝜔). (32) Now, solving for 𝑋₁(Ω) and 𝑋₂(Ω), we have 𝑋₁(Ω)=𝜇Ω+𝑗𝑘₁𝑘₂𝜋(𝛿()(+𝑘Ω+𝜔−𝛿Ω−𝜔))₃2𝑋₂(Ω+𝜔)−𝑋₂(,𝑋Ω−𝜔)₂(Ω)=𝜇𝑘₆2Ω+𝑗𝑘₅𝑋₁(Ω+𝜔)−𝑋₁(Ω−𝜔). (33)(34) Equations (33) and (34) show that variables 𝑥₁, 𝑥₂ have the form of low pass filters (as we commented previously). Thus, the terms 𝑋₂(Ω−𝜔) and 𝑋₂(Ω+𝜔) in (33) are as follows: 𝑋₂()=𝑘Ω−𝜔⁶𝜇2Ω−𝜔+𝑗𝑘₅𝑋₁(Ω)−𝑋₁(,𝑋Ω−2𝜔)₂()=𝑘Ω+𝜔⁶𝜇2Ω+𝜔+𝑗𝑘₅𝑋₁(Ω+2𝜔)−𝑋₁(Ω), (35) and using (33) and (35), we obtain 𝑋₁(Ω)=𝜇𝐷Ω+𝑗𝑘₁𝑘₂𝜋(𝛿(Ω+𝜔)−𝛿(Ω−𝜔))+𝜇𝑘₃𝑘₆2×𝑋₁(Ω+2𝜔)2Ω+𝜔+𝑗𝑘₅+𝑋₁(Ω−2𝜔)2Ω−𝜔+𝑗𝑘₅, (36) where 2𝐷=Ω+𝑗𝑘₅²−𝜔²Ω+𝑗𝑘₁2Ω+𝑗𝑘₅²−𝜔²Ω+𝑗𝑘₁+𝜇²𝑘₃𝑘₆Ω+𝑗𝑘₅⋅ (37) Equation (36) has the terms 𝑋₁(Ω−2𝜔) and 𝑋₁(Ω+2𝜔), which are calculated by developing recursively (33), giving as a result 𝑋₁()=𝜇Ω−2𝜔Ω−2𝜔+𝑗𝑘₁𝑘₂𝜋(𝛿()(+𝑘Ω−𝜔−𝛿Ω−3𝜔))₃2𝑋₂(Ω−𝜔)−𝑋₂(,𝑋Ω−3𝜔)₁()=𝜇Ω+2𝜔Ω+2𝜔+𝑗𝑘₁𝑘₂𝜋(𝛿()(+𝑘Ω+3𝜔−𝛿Ω+𝜔))₃2𝑋₂(Ω+3𝜔)−𝑋₂(Ω+𝜔). (38)

Equations (33) and (38) show that 𝑋₁(Ω) have terms which contain factors 𝛿(Ω−3𝜔), 𝛿(Ω−𝜔), 𝛿(Ω+𝜔), and 𝛿(Ω+3𝜔) and if we develop recursively 𝑛 times (33), then new terms of the form 𝜇^|𝑘|𝛿(Ω−(2𝑘+1)𝜔)𝐷_𝑘 for −𝑛≤𝑘≤𝑛 appear in the solution of 𝑋₁(Ω), where 𝐷_𝑘 is a so complex factor which depends on every parameter and variable of (32) and for 𝜇<1.0 the quotient |𝜇^𝑘|/|𝐷_𝑘|≈0. Therefore, when the amplitude 𝜇 is incremented, then the amplitude of the peaks 𝛿(Ω−(2𝑘+1)𝜔) does too. For this reason, the numerical evidence shows that if the parameter input signal 𝜇 is incremented, then apparently new components of frequency appear in the spectrum 𝑋₁(Ω). These frequencies always form part of 𝑋₁(Ω) but with a very small amplitude. They are so small that they look like noise.

5. Response to Random Signals

Now, let 𝑢=𝜅𝜂(𝑡), where 𝜂(𝑡) is a random signal equidistributed on the interval [−1,1] and 𝜅 is its amplitude. When 𝑢=10𝜂(𝑡) is the input signal of the filter (6), the phase space of the response is a limit cycle that has the form of a quarter moon which is shown in Figure 7(a). As a first glimpse the filter's response seems to be a periodic orbit like in the case of the sinusoidal input signal, but the time series 𝑥₁(𝑡) and 𝑥₂(𝑡), which are respectively shown in Figures 7(b) and 7(c), are aperiodic signals.

(a)

(b)

(c)

Figure 7 

Plots of the filter's response (6) when a random input signal 𝑢=10𝜂(𝑡) is filtered. (a) The phase space (𝑥1,𝑥2). The time series (b) 𝑥1(𝑡) and (c) 𝑥2(𝑡).

The following remarks are pertinent. (i)The output signal of the filter has a normal (or Gaussian) distribution when the input signal is random with uniform distribution. For the case 𝑢=𝜂(𝑡), Figures 8(a) and 8(b) show the histogram and the FT of the 𝑥₁ signal, respectively, where one can see that the FT of the response has an exponential decay.(ii)The filter presents saturation after a threshold value of the amplitude of input signal 𝑢=𝜅𝜂(𝑡). Figure 9 shows the range of values 𝑅(𝜅)=MAX{𝑥₁}−MIN{𝑥₁} of the response 𝑥₁ versus the amplitude of the parameter 𝜅. To justify the exponential decay of the filter's response when it is forced with random signal, we consider FT{𝑢}=𝐾₁, where 𝐾₁ is a constant which depends on 𝜅, the amplitude of the random signal, and 𝑥₃≪FT{𝑢}. Thus the following terms of (30) are 𝑘₂FT{𝑢}=𝑘₂𝐾₁ and FT{𝑢𝑥₂}≈𝐾₂, where 𝐾₂ depends on 𝜅 too. So, the FT results are 𝑗Ω𝑋₁(Ω)≈−𝑋₁(Ω)−𝐾₂+𝑟𝐾₁, (39) and for the magnitude of 𝑋₁(Ω) we have 𝑋₁(Ω)≈𝑟𝐾₁+𝐾₂1+Ω². (40) Figure 8(b) shows the form of the curve |𝑋₁(Ω)| which corresponds to (40).

(a)

(b)

Figure 8 

Response of the filter (6) when it is forced by the random signal 𝑢=𝜂(𝑡). (a) Graphs of the histograms of the state 𝑥1 (continuous line) and histogram of the external input signal 𝑢 (dotted line). (b) TF of the component 𝑥1.

Figure 9 

Range of values of the state 𝑥1 in function of the 𝜅 when system (6) filters random signal.

6. Correlation Coefficient

Because a filter acts in function of the input signal, another way to characterize the effect of the input signal on the filter's response is to compute the correlation between the input and output signals. If we have time series of 𝑁 data, the correlation coefficient 𝐶_𝑥𝑦 of series 𝑥(𝑖) and 𝑦(𝑖) is defined as 𝐶_𝑥𝑦=∑^𝑁_𝑖=1(𝑥(𝑖)−𝑥)𝑦(𝑖)−𝑦(𝑁−1)𝑆_𝑥𝑆_𝑦, (41) where 𝑥, 𝑦 are the means and 𝑠_𝑥, 𝑠_𝑦 are the standard deviations of the series 𝑥(𝑖) and 𝑦(𝑖), respectively. We calculate correlation coefficients for different input signals and the output signals 𝑥₁ and 𝑥₂ which are given in Table 1. The first two cases in Table 1 correspond to a periodic and chaotic signals, 𝐶_𝑢𝑥1≈0.745 and 𝐶_𝑢𝑥2≈0.0, and the last case considers a random signal as an input signal.

Table 1 

Coefficient correlations between different input signals and the responses 𝑥₁, 𝑥₂ of the filter (6).

We make a further numerical study of the coefficient correlation in function of the parameters of the sinusoidal input signal. In Figure 10(a), we can see that the correlation between sinusoidal external signals is not constant and that it depends on both parameters 𝜇 and 𝜔. On the contrary, Figure 10(b) shows that the absence of correlation between the response 𝑥₂ and their respective forces is independent of almost the whole range of values of the input signal parameters.

(a)

(b)

Figure 10 

Correlation between the response of the filter (6) and the input signal 𝑢=𝜇∈(𝜔𝑡): (a) 𝐶𝑢𝑥1 and (b) 𝐶𝑢𝑥2.

On the other hand, the last row in Table 1 shows that 𝐶_𝑢𝑥1≈0.1160 and 𝑅_𝑢𝑥2≈0.0630. This means that the filter's response (6) does not have correlation when the input is a random signal. Despite the fact that the amplitude of the parameter 𝜅 is increased, the output signal of the filter does not have correlation with input signal (random), as shown in Figure 11. Therefore, the correlation between the input random signal and its filter's response cannot be induced by incrementing the value of its amplitude. A similar result occurs for the sinusoidal signal and its response 𝑥₂.

Figure 11 

Correlation in function of the parameter of the random input signal 𝜅. Dashed line and continuous line correspond to 𝐶𝑢𝑥1 and 𝐶𝑢𝑥2, respectively.

7. Low Pass Filters in Chaotic Systems

The Lorenz system is a very well-known third-order chaotic system [11] which is defined as 𝑦̇₁=−𝜎𝑦₁+𝜎𝑦₂,𝑦̇₂=𝑟𝑦₁−𝑦₂−𝑦₁𝑦₃=−𝑦₂+𝑟−𝑦₃𝑦₁,𝑦̇₃=𝑦₁𝑦₂−𝑏𝑦₃=−𝑏𝑦₃+𝑦₂𝑦₁, (42) where 𝐲=(𝑦₁,𝑦₂,𝑦₃)^𝑇 is the state vector and 𝜎, 𝑟, and 𝑏 are parameters. Each component of system (42) is defined based on low pass filters. For example, the first equation is a linear low pass filter, where the state variables 𝑦₁ and 𝑦₂ are the input and output signals, respectively. The second and third equations comprise a 𝑡𝑤𝑜-interconnected nonlinear filter whose outputs are 𝑦₂ and 𝑦₃, and the input signal is 𝑢=𝑦₁. The 𝑡𝑤𝑜-interconnected nonlinear filter of the Lorenz system (42) can be rewritten in terms of (7) as follows: 𝑟0,𝑓()=𝐾=−100−𝑏,𝐵=𝑥,𝑢𝑢𝑦₃𝑢𝑦₂, (43) where 𝑥₁=𝑦₂, 𝑥₂=𝑦₃, and 𝑢=𝑦₁. The projection of the Lorenz attractor onto the plane (𝑦₁,𝑦₂) is shown in Figure 12(a) for the following parameter's values: 𝜎=10, and 𝑏=2.66, and 𝑟=28.0. In order to study the response of the interconnected system (43) when the input signal is the state variable 𝑦₁, we have calculated 𝑅(𝜎) which is the maximal range of values of the time series 𝑦₂. In Figure 12(b), we can see that 𝑅≈0 for 0≤𝜎≤5; this means that the solution converges to a fixed point. For 𝜎∈[5,7], there is a transition period in the behavior of the filter, and for 𝜎≥7 the range of values of the filter's response is approximately constant.

(a)

(b)

Figure 12 

(a) Orbit of the filter (43) when the external input is 𝑢=𝑦1 of system (42). (b) The maximal range of response of system (43).

When calculating the correlation coefficient between the components of the 𝑡𝑤𝑜-interconnected system (43), it results that 𝐶_𝑦1𝑦2=0.9022, 𝐶_𝑦1𝑦3=−0.0384, and 𝐶_𝑦2𝑦3=−0.0424. The high correlation between the state variables 𝑦₁ and 𝑦₂ occurs because of 𝑦₁ is a simple low pass lineal filter for the signal 𝑦₂ without any nonlinear term included in its equation (see 𝑦̇₁ in (42)). On the other hand, the time series 𝑦₁ has no correlation with 𝑦₃ which could be induced by the nonlinear term settles in the equation 𝑦̇₃ of system (42). Now, we put a parameter to control the amplitude of the input signal such that 𝑢=𝜅₁𝑦₁ and calculate 𝐶_𝑦1𝑦2 and 𝐶_𝑦2𝑦3 as a function of the parameter 𝑘₁. In Figure 13, it is shown that only 𝐶_𝑦1𝑦2 is affected when the value of 𝜅₁ is incremented, while the absence of correlation between states 𝑦₂ and 𝑦₃ remains for all values of 𝜅₁. For 𝜅₁>32, the trajectories of system (43) diverge.

Figure 13 

Correlation between state variables against the parameter 𝜅1. The continuous line and the dashed line correspond to 𝐶𝑦1𝑦2, 𝐶𝑦1𝑦3, respectively.

We have shown by means of numerical experiments in previous sections that the absence of correlation between the external force 𝑢 and the state variable 𝑥₂ of the filter response generally does not depend on the parameters of the input signal. Similar result occurs when the Lorenz autonomous system is considered like a 𝑡𝑤𝑜-interconnected system which is forced by the signal 𝑢=𝑦₁. As shown in Figure 13, there exists a high correlation between the state variables 𝑦₁ and 𝑦₂, and an absence of correlation between state variables 𝑦₂ and 𝑦₃. But can the correlation between the state variables 𝑦₁ and 𝑦₂ be incremented by changing the value of the parameters of the Lorenz system? With the purpose of answering this question, we have calculated 𝐶_𝑦1𝑦2, 𝐶_𝑦1𝑦3 as a function of the autonomous Lorenz system parameter 𝑟 keeping the value of the parameter 𝑏 constant and vice versa. Figure 14(a) shows that for 𝑟<15 approximately, correlations 𝐶_𝑦1𝑦2, 𝐶_𝑦1𝑦3 have a transition period, and for 𝑟≥15, the correlations return to their previously showed behavior: 𝐶_𝑦1𝑦2≈1.0 and 𝐶_𝑦1𝑦3≈0. Figure 14(b) shows that the parameter 𝑏 does not affect the correlation between 𝑦₁ and the responses 𝑦₂, 𝑦₃.

(a)

(b)

Figure 14 

Correlations between the state variables of the Lorenz system (42) against the parameters (a) 𝑟 and (b) 𝑏. Each continuous line and dashed line correspond to 𝐶𝑦1𝑦2, 𝐶𝑦1𝑦3, respectively.

If Lorenz system parameters are tuned so that the Lorenz system behaves as a filter and can be used as a slave system, then generalized synchronization of master and slave systems always appears.

8. Conclusions

An 𝑛-interconnected nonlinear system given by (6) always behaves as a filter if the condition given by Theorem 4 is satisfied. Despite of the nonlinearities in the structure of 𝑛-interconnected nonlinear system, its dynamical behavior does not depend on the initial conditions but it does as a function of the input signal. The nonlinear filter's response to sinusoidal input signal presents several interesting phenomena such as (i) frequency components at multiples of the input signal's frequency (ii) when the amplitude of the input signal increases, the number of peaks of the filter's response increases (iii) the amplitude of the filter's response falls exponentially as a function of the input signal's frequency.

The Lorenz system is described in terms of low pass filters which consists of a linear low pass filter and a 𝑡𝑤𝑜-interconnected nonlinear low pass filter. This gives us the possibility to describe all the systems that conform the Lorenz family and others with similar structure in terms of low pass filters.

In several studies of chaos synchronization, specifically in forced systems 𝐱̇=𝑓(𝐱,𝑢), 𝐱∈ℝ^𝑛, it has been found that generalized synchronization of response systems occurs for specific external signals 𝑢, but all these cases satisfy condition (8). Then, the general synchronization phenomenon and the phenomenon of nonlinear filter are seen to be the same. Nevertheless, condition (8), in generalized synchronization phenomenon, could not be satisfied, because of the trajectories generated with different initial conditions asymptotically could go to a different basin of attractions. This characteristic is different to the phenomenon of nonlinear filter which by definition condition (8) needs to be always satisfied if a system behaves like a nonlinear filter. Several chaotic systems have a similar structure to 𝑛-interconnected system, so that we conjecture that this study can help to distinguish between generalized synchronization behavior and nonlinear filtering behavior of an 𝑛-interconnected system.

Acknowledgments

The authors would like to thank Emanuel Rodríguez-Orozco for reading the paper and giving constructive comments. E. Jiménez-López is a doctoral fellow of CONACYT in the Graduate Program on Applied Science at UASLP, and IPICYT for the hospitality during his sojourn in DMAp-IPICYT. E. Campos-Cantón acknowledges CONACYT for the financial support through Project no. 181002.