Conditional Lyapunov Exponent Criteria in terms of Ergodic Theory

The conditional Lyapunov exponent is defined for investigating chaotic synchronization, in particular complete synchronization and generalized synchronization. We find that the conditional Lyapunov exponent is expressed as a formula in terms of ergodic theory. Dealing with this formula, we find what factors characterize the conditional Lyapunov exponent in chaotic systems.


I. INTRODUCTION
The conditional Lyapunov exponent is defined for investigating chaotic synchronization [1][2][3][4][5][6][7][8][9][10][11][12][13][14], in particular Complete synchronization (CS) and Generalized synchronization (GS). Transitions from desynchronization to synchronization of trajectories occur when the conditional Lyapunov exponent changes from positive to negative [15][16][17]. Although it is widely known that the chaotic synchronization occurs in many systems, it is not clearly known why the conditional Lyapunov exponent changes. For example, it has not completely been clarified why the CS occurs in chaotic systems. A report [10] showed that an external forcing input in CS may change the dynamical system to another one. The forcing input changes the balance between phase contraction or expansion, and the CS occurs when such contraction dominates. Although the explanation is well considered, there could be another reason why the conditional Lyapunov exponent may change. Furthermore, they have focused on the mean of external forcing inputs for the CS. The study [11] showed that they got the transversal Lyapunov exponent with changing the noise distribution. These may be important. However we expect that more precise information on phase space can lead to more relevant analysis. Therefore, we would like to clarify the relation between the conditional Lyapunov exponent and the chaotic synchronization in CS.
We have two claims through this research. The first one is about a formula to express the conditional Lyapunov exponent in terms of ergodic theory, and the second is what factors influence the conditional Lyapunov exponent in chaotic dynamical systems. Although existing research offered some mechanisms of chaotic synchronizations by focusing on mean amplitudes or variances of common input signals ( See [12,13] for example), we find that such mean amplitudes or variances of common input signals are not imperative. Instead, we reveal that a distribution characteristic of input signals is the most important factor. We also reveal that this characteristic determines transitions between synchronization and desynchronization, and this does not depend on whether input signals are chaotic or noisy. Thus, although the second claim can easily be derived from the first one, we emphasize that our second claim is physically important.
In Section 2, we describe the definition of conditional Lyapunov exponent in chaotic systems. Furthermore, we describe two main claims in our research.
In Sections 3 -5, we construct solvable chaotic dynamical systems, and confirm the our claims by analysis for such systems.
In Appendix A, we summarize a short introduction of ergodic theory.

II. DEFINITION AND MAIN CLAIM
To discuss the conditional Lyapunov exponent, we consider the following simple one-dimensional unidirectional coupling system: where t is time, x(t) is a response, f is a chaotic mapping, and ξ(t) is an external forcing driver. We define P X (X) as the probability distribution of variables X for the dynamical system, and also define R X as the range in which variables X are defined. We define the CS of the system as follows [8][9][10][11]15]. We consider two different trajectories x 1 (t) and x 2 (t) in Eq. (1) with different initial points. To judge whether the response system exhibit CS, we introduce the conditional Lyapunov exponent. The occurrence of CS implies that the difference between x 1 (t) and x 2 (t) decreases as time increases. The CS is said to occur when the following condition about synchronization error e ′ (t) ≡ |x 2 (t) − x 1 (t)|: is satisfied for almost all initial points. Here |...| expresses Euclidean norm of the argument. The interpretation of (2) is that the state in the large t limit is independent of any initial state for almost all initial points. The infinitesimal synchronization error to the projection to x axis e(T ) in system (1) is defined as: When CS occurs, the equation lim t→∞ e ′ (t) = lim T →∞ |e(T )| = 0 is satisfied. It is noted here that ξ(t) in Eq. (1) affects the time-evolution of x(t) and contributes to e(T ). The conditional Lyapunov exponent for variables x is defined by Clearly λ expresses the stability of the state x 1 (t) = x 2 (t). In this paper, the CS is said to occur when λ < 0, and we do not use Eq.
(2) directly. The conditional Lyapunov exponent exists when the system (1) is ergodic with respect to x and ξ according to Oseledets' multiplicative ergodic theorem for autonomous dynamical systems [18].
Here, we assume that the system (1) can be seen as a two-dimensional autonomous dynamical system of x and ξ. When the system (1) is ergodic with respect to the absolutely continuous invariant measure (physical measure) with respect to x and ξ, the conditional Lyapunov exponent λ is expressed as the ensemble average: where, P(x, ξ) is the invariant distribution of the system (1). We can generalize Eq. (4) to higher dimensional dynamical systems.
We define the following twodimensional dynamical system with an external forcing input: If the two-dimensional dynamical system (5) is ergodic with respect to the absolutely continuous invariant measure in terms of x 1 and x 2 variables, the conditional Lyapunov exponent λ ij (i = 1, 2, j = 1, 2), which is defined by the infinitesimal synchronization error for ψ j to the projection to x i axis, is defined by In what follows, however we assume that the system (1) has the one-dimensional limiting distribution given by the absolutely continuous invariant ergodic measure µ(dx) = P x (x)dx, (x ∈ R x ) with some P x , and P(x, ξ) is a continuous density function in x and ξ satisfying Rx R ξ P(x, ξ)dξdx = 1 for simplicity.
Then, we give our two main claims. Firstly, the conditional Lyapunov exponent is expressed by the following equation. where Note that although ξ 0 is originally a constant value, we consider the distribution of ξ 0 in the above equation.
Then, we show how we get this claim provided the ergodicity in the system (1). The Lyapunov exponent λ Ψ (ξ 0 ) can be obtained by the following equation with the ergodicity in the system (9), where P(y|ξ 0 ) is the unique conditional probability distribution in dynamical system (9) with each constant value ξ 0 . To obtain Eq. (8) we use the marginalization about random variables a and b in the probability theory: Then, substituting (11) into (7), we have Note that λ =λ is satisfied when the invariant distribution P y (y|ξ) exists. This is satisfied when the system (9) is ergodic for any ξ 0 . Therefore, our first claim Eq. (6) is derived from the mixing property of (1) and the ergodicity of ξ and the system (9). Figure 1 illustrates this concept. Here, ... expresses the ensemble average with respect to the ergodic invariant measure. Note that if we have two-dimensional dynamical systems, the conditional Lyapunov exponent λ ij (i = 1, 2, j = 1, 2), which is defined by the infinitesimal synchronization error to the projection to x i axis, is expressed as: As for the second claim, with the first one, we can derive that the conditional Lyapunov exponent in a system (1) is characterized by only two factors, a dynamical system and a distribution of external forcing input.

III. EXAMPLE1 FOR THE FIRST CLAIM
In order to confirm the first claim, we would like to show its relevance by analyzing solvable chaotic synchronization systems [19,20]. Here, a solvable chaotic synchronizing system is such that an invariant measure, the conditional Lyapunov exponent and the threshold between synchronization and desynchronization are analytically obtained for the unidirectional coupling system.

A. Definition of our system
As for the first claim, we need three steps to show it. Firstly, we analytically calculate the conditional Lya-punov exponent λ in our solvable chaotic dynamical system in accordance with the definition (4). Secondly, we analyze the auxiliary dynamical system similar to Eq. (9), and get the exponentλ accordance with Eqs. (9) - (12). Thirdly, we confirm that the results obtained by the two types of analytical methods coincide.
We study the following dynamical system where ε is a coupling parameter, and εζ(t) correspond to an external forcing input ξ(t). Firstly we prepare the following solvable chaotic dynamical system [21][22][23]: where the function g is associated with the double-angle formula given by cot 2θ = 1 2 cot θ − 1 cot θ [20]. The mapping associated with g is a chaotic mapping which has the mixing property [20], and its Lyapunov exponent is ln 2. The invariant measure of the system x(t + 1) = g(x(t)) is the standard Cauchy distribution as follows: where C(x) is Cauchy distribution defined as: with c being a median and γ a scale parameter. Therefore, the distribution of variable x in the dynamical system x(t + 1) = g(x(t)) follows C(x; 0, 1) [20], and the distribution of external forcing ξ follows also C(x; 0, 1). This is our first solvable chaotic dynamical system. We see that the conditional Lyapunov exponent changes when the coupling parameter is varied. We calculate the conditional Lyapunov exponent of this dynamical system by the definition (4) in accordance with the first step.

B. Conventional ergodic theoretical approach
We express the conditional Lyapunov exponent of the system (14) as λ g (ε) since it will turn out that the conditional Lyapunov exponent crucially depends on ε. The conditional Lyapunov exponent λ g (ε) is expressed as: where P x,ξ (x) is the invariant distribution for the variable x in the superposed dynamical system x(t + 1) = g(x(t)) + εζ(t) with a parameter ε. Note that the superposed distribution is expressed as P x,ε (x) since ζ is given and the external forcing ξ is only dependent on the coupling parameter ε.
Then we firstly calculate the invariant distribution P x,ε (x), secondly the conditional Lyapunov exponent λ g (ε), and thirdly the threshold between synchronization and desynchronization analytically.
We can calculate the invariant distribution P x,ε (x) by using the following three properties, the mixing property for the mapping g (Property 1), the property of preserving the form of Cauchy distributions in terms of the Perron-Frobenius equation (PF equation) for g (Property 2), and the property of the stable property for Lévy distributions (Property 3). Then showing these three properties, we explain how these properties play roles in order to get P x,ε (x)dx.
As to Property 1, it is already known that the mapping g has mixing property [20].
As for Property 2, this property implies that g is the mapping which changes an input Cauchy distribution into another Cauchy distribution with a different median and scale parameter. We consider the PF equation for the equation z = g(x) where the input variables x follows C(x; c, γ). Since the g is a two-to-one mapping, P z (z) satisfies the following PF equation (Probability Preservation Relation): P z (z)|dz| = C(x 1 ; c, γ)|dx 1 | + C(x 2 ; c, γ)|dx 2 | where x 1 and x 2 (x 1 > x 2 ) are the solutions of the quadratic equation z = g(x). They satisfy the following, With these relations, the probability distribution P z (z) is obtained, as the following rescaled Cauchy distribution: As to Property 3, this stable property has already widely been known. When distributions of two independent variables X 1 and X 2 obey a Lévy distribution family, the distribution of the variable aX 1 + bX 2 (a, b ∈ R) also obeys the same family. These three properties play roles in order to get the invariant distribution P x,ε (x).
We can prove that variables x and εζ in the system (14) obey Cauchy distributions respectively with three properties. In Appendix B, we describe these three properties in more detail. By taking into account the Properties 2 and 3, the distribution of the variable x is changed to a Cauchy distribution with a different median and scale parameter in every iteration if the initial input follows a Cauchy distribution. Hence, we get the following selfconsistent recurrence equations about a median c and a scale parameter γ per iteration, as We get the following convergence values c * and γ * as the stable fixed point of Eq. (15) for t → ∞: c * g = 0 γ * g = |ε| + ε 2 + 1. Hence, we analytically obtain P x,ε (x) as the fixed point of the recurrence equation Eq. (15): We should emphasize the following. Although we assume the initial input follows a Cauchy distribution in the above analysis, we do not need any restriction for a distribution of initial point for the system (1) because of the mixing property (Property 1). Therefore, the invariant distribution P x,ε (x) is always obtained regardless of any initial points.
With P x,ε (x), we can calculate the conditional Lyapunov exponent λ g (ε) as follows: Note that λ g (0) = ln 2. We can also get the threshold ε * g between the synchronization and desynchronization. This is the solution of λ g (ε * g ) = 0, which is ε * g = 1. Figure 2 illustrates that these analytical results coincide with the results of numerical simulation with the initial condition x 0 = √ 2 and ξ 0 = √ 3.

C. Proposing ergodic theoretical approach
As above, we can calculate λ g (ε) by Eq. (4). The invariant distribution P(x, ξ) is expressed as: because P ξ (ξ) is derived from Property 3 for the variable ζ that obey C(ζ; 0, 1). Then in accordance with the second step, we would like to calculateλ g (ε) by Eqs. (9) - (12). We prepare the following dynamical system by Eq. (9): We firstly need to confirm that this system is ergodic for every ξ 0 , which is the sufficient condition for that the conditional distribution P y (y|ξ 0 ) exists for every ξ 0 . Hence, we sufficiently calculate three factors, the conditional distribution P y (y|ξ 0 ), and the unique Lyapunov exponents λ Ψ (ξ 0 ), the distribution of external forcing inputs P ξ (ξ 0 ). Firstly, we calculate the conditional distribution P y (y|ξ 0 ). The invariant measure P y (y|ξ 0 )dx is also obtained with Properties 1 -3.
As for unique Lyapunov exponents λ Ψ (ξ 0 ), we can calculate them as follows: Thus, we can calculate the exponentλ g (ε) according to Eq. (12): As above, we confirm that our first main claim is certainly satisfied. This claim signifies that a conditional Lyapunov exponent is expressed as the ensemble average of the set of unique Lyapunov exponents of the auxiliary dynamical system. Figure 3 illustrates that two different trajectories with different initial points (x 1 (0) = 0.46, x 2 (0) = 1.3) partially synchronize. The coupling parameter ε is 0.8. Note that although the conditional Lyapunov exponent is positive in FIG. 3, the partial synchronization occurs since it is the averaged factor of unique Lyapunov exponents. When the conditional Lyapunov exponent is negative, the infinitesimal synchronization error converges to 0 for t → ∞ (see FIG. 4).

IV. EXAMPLE2 FOR THE FIRST CLAIM
Our analysis is not restricted to the above example. We consider another solvable chaotic dynamical system, and can also confirm that the conditional Lyapunov exponent in the system is obtained by Eq. (6). This dynamical system is defined as follows: The chaotic mapping h is associated with the double formula tan 2θ = 2 tan θ 1−tan 2 θ [20]. Its Lyapunov exponent is ln 2, and the invariant measure obeys the standard Cauchy distribution. Then, the invariant distribution P x,ξ (x) and the conditional Lyapunov exponent λ h (ε), and the threshold ε * h are given as: where γ * h is the positive number which satisfies the following equation:

V. EXAMPLE FOR THE SECOND CLAIM
Here, as for the second claim, the conditional Lyapunov exponent for a system (1) is characterized by only two factors, a dynamical system associated with f and a distribution of external forcing input ξ.
In order to show this claim we consider six different dynamical systems. Each of them has a different chaotic mapping f or different system for ξ as follows: where Crand(c, γ) is a set of random numbers which follow C(ζ; c, γ). The algorithm to get Crand(c, γ) follows: where U(−1 : 1) are uniform random numbers on the interval (−1 : 1). We compare the conditional Lyapunov exponent of these systems with Table I. As we can see from the Table I, we can find that some factors do not influence the conditional Lyapunov exponent such as the property of time series in external forcing input. Then, we show that the conditional Lyapunov exponent is characterized by the original mapping f and the distribution of external forcing input ξ. The original dynamical system is changed to the system which has an attracting fixed point because of the external forcing input. The conditional Lyapunov exponent is changed by the existence of attracting fixed points. Figures 5 and 6 illustrate that the mappings G and H are changed to ones that have attracting fixed points by external forcing input respectively. Attractors of systems depend on how attracting fixed points are generated. The created attractor leads to the ξ-dependence of λ Ψ (ξ 0 ). Figures 7 and 8 show the set of Lyapunov exponents λ G (ξ 0 ) and that of λ H (ξ 0 ), respectively. Hence, the original mappings determine how attracting fixed points are generated. Furthermore, with Eq. (6) from the first claim, the conditional Lyapunov exponent is calculated by the set of Lyapunov exponents {λ Ψ (ξ 0 )|ξ 0 ∈ R ξ } and the distribution of the external forcing input P ξ . As above, the conditional Lyapunov exponent in system (1) is uniquely characterized by only two factors, the original dynamical system and the distribution of external forcing input.