On Expectation Correlate System and Chaotic Dynamics in Time-Series

This paper suggests a new system of time-series called Expectation Correlate System (ECS) that are good at detecting the behavior of dynamical systems (both deterministic and stochastic systems) and the dependence on initial values. A new measure on sensitivity to initial values can be monitored by the newly defined Lyaponov Correlate, so ECS can be a signal to chaotic property. In a stochastic systems, small shifts in some initial value can lead to error in prediction, this property and a


1-Introduction
In the past few years a large literature on chaos and non-linear science has appeared in economics and many scientists and technologists from diverse disciplines including mathematics (both pure and applied) because chaos is useful as a lens through which to view the word in epidemiology, biology and ecology, not because it helps so much in prediction but because it is suggestive of pathways to complex dynamics [1]. It is associated with complex and unpredictable behavior of phenomena over time and nonlinear science studies stochastic and deterministic dynamical systems that lead to "complex" dynamics. In [6], York who proposed the word 'chaos' as a label for a kind of dynamical behavior characterized by the triad: infinite number of periodic trajectories; uncountable number of no periodic trajectories; hyperbolicity (instability) of all (or the overwhelming majority of; as is proposed here) trajectories in the regime.

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A simple deterministic system may be defined as follows: For a discrete time index, T={0,1,2,…}, consider a time-series {X t ;tT}. Assume that X 0 = x 0 is an initial condition and that X t =F(X t-1 )………………………… (1) for t>1, where X t denoted a state vector in R d , F is a real-vector function (bounded continuous first derivatives ) see [2]. In a deterministic system, it is agreed that the sensitive dependence on initial condition is a typical feature of chaotic system, which is characterized by the well-known Lyaponov exponent [4].
A discrete time stochastic system can be described by the equation Where { t  } is a noise process which satisfies the equation If the noise is additive, equation (2) can be written as follows: Just as in deterministic systems, there has been no general accepted definition of chaos in stochastic systems. Stochastic chaotic system sometimes means a system with a deterministically chaotic skeleton [7].
The plan of the paper is as follow: Section (2) provides a brief sketch of a new system called Expectation Correlate System (ECS) and the correlates of the trajectories are studies. The concept of Lyapunov exponent has been developed to characterize the sensitivity dependence on the initial value of system (1) in section (3). Section (4) presents a quantitative description of how small noise can be amplified rapidly in a ECS if the corresponding system is chaotic. A new simple procedure is suggested to measure the non-linear property in section (5). In section (6) the method is illustrated with example.

2-The Expectation Correlate System and Deterministic Systems.
By the Expectation Correlate System (ECS) of the system (1), mean the suggested system defined as: In a chaotic system, small changes of parameters can change the dynamical behavior from stable periodic cycle or limit point into a strange attracted system [5]. The suggested system (5) will try to study the correlate behavior of the trajectories which dependence on the parameters of the system and how small change in this parameter can change the correlate of the trajectories.
By using the deterministic system (1) (let us consider the one dimension case d=1), starting at the initial points X o =x is the conditional mean given x 0 . For n = 1, we have Here, F (2) (.) means the 2-component of the function F and the overdot denotes the differential operator. From (6) and (7), we have For n>1, the general form as: , a small shift  in the initial value can lead to a considerable divergence in the ECS. This means that the ECS depends on x 0 sensitivity when Can be a signal to the correlate of the trajectories of system (1) and can determine the bifurcation parameter in a dynamical system that leads to chaotic system.

3-Lyapunov Correlate of ECS.
Lyapunov exponent is one of the most popular measures of chaos which is defined as :

4-The Expectation Correlate System and Stochastic Systems:
In this section, the ECS measure another property of nonlinear stochastic systems by comparing the conditional variance (given the initial condition X 0 =x)with the variance t  . Consider the ECS of system (3) when n=1, thus By using Taylor expansion to the function F, we have Then, the general form of ECS as follows:  1, which indicates the dependence of the correlate prediction on initial value, this is a typical feature of nonlinear (but not necessarily chaotic) systems. If the system (3) is stochastically chaotic, equation (13) indicates that small noise can be amplified quickly when the system starts at some initial value, which mean that the n-step prediction based on these initial values could be unreliable even for small n.

5-Non-linear ECS
By using the idea developed in section (4), suppose that  (14), it is easy to see that for n>1 ( ) We now turn the general form of ECS which is consider in equation (4), by using the idea in equation (12) to initial condition means that, the prediction depends on initial condition, which is a typical feature of non-linear but not necessarily chaotic systems. When F(.) is linear, does not depend on x and is monotonically increasing as n increases.

6-Example
Consider one dimensional stochastic system ( )   Table (3) shows the conditional variance and the divergently resulting from a small shift in initial values, also the amplification of noise is shown.