Studying the Stability of a Non-linear Autoregressive Model (Polynomial with Hyperbolic Cosine Function)

In this paper we study the statistical properties of one of a non-linear autoregressive model with hyperbolic triangle function(polynomial with hyperbolic cosine function)by using the local linearization approximation method to find the stability of the model (singular point and its stability conditions and the stability of limit cycle).Where we started by the model of lower order (first and second and third order) and generalized the idea, and we tried to apply these theory results by using some of examples to explain one of important truth that says (if the model has unstable singular point, then it, maybe, has a stable limit cycle).


Introduction
In the field of discrete time non-linear time series modeling, there are many different types of a non-linear models which are considered by the researchers such as bilinear model (Priestley (1978), Rao (1977)) exponential autoregressive model (Ozaki and Oda (1977) ) [5] and threshold model ( Tong (1990) ) [8].
In (1985) Ozaki proposed the method of local linearization approximation to find the stability of a non-linear exponential autoregressive models [7].
In (1986) Tsay R.S. studied the stability of non-linear time series [9]. In (1988) Priestley M.B. studied the non-stability and non-linear time series [9]. In (1990) Tong H. studied the dynamical system with stability of non-linear time series [8].
In this paper, we study the statistical properties of one of a non-linear autoregressive model with hyperbolic triangle function(polynomial with hyperbolic cosine function)by using the local linearization approximation method to find the stability of the model (singular point and its stability conditions and the stability of limit cycle) and we give some examples to explain this method. , Where F is a given function, n is some positive integer, and k = 0, 1, 2, …... ., [4].

Definition 2.2:
A time series is a set of observations measured sequentially through time. These measurements may be made continuously through time or be taken at a discrete set of time points. Then, a time series is a sequence of random variables defined on probability space multi variables refer by index (t) that back to index set T , and we refer to time series by } , ); ( ,......., 1 1 are the parameters of the model [5].

Definition 2.5:
The bilinear model of order (p,q,m,s) satisfies the equation , where q is a positive integer. Isolated means that every trajectory beginning sufficiently near the limit cycle approaches either for we call it stable limit cycle and if it approaches it for − → t we call it unstable limit cycle [5].

Theorem 1:
Let   t x be expressed by the exponential autoregressive model . proof (see [5]).
The proposed model A non-linear autoregressive model (polynomial with hyperbolic cosine function) of order p is defined by

3.The Stability of the Proposed Model
In this section, we shall study the stability of a non-linear autoregressive model with hyperbolic cosine function with low order such that p=1,2,3 and, then we generalized this idea to the general model of order p by using the local linear approximation method that consists of the following three steps: Step(1):find the singular point of the model.
Step (2):study the stability condition of the singular point.
Step (3):find the stability condition of a limit cycle if it exists .

Singular Point
Consider the following model The singular points of the model in equation (4) are . Therefore, we get a third order algebraic equation and by using reference [1], we have a, b and c are real constants such that The singular points of the model in equation (6) are as follows:

The Stability of Singular Point:
We will find the stability condition for the non-zero singular point as follows : Put for all s=t,t-1 , in equation (2) (when p=1), and also suppose that the white noise is not an effect, then we have: ,then, Equation (10) , then the root must be bigger than one, that is meaning 1 We will find the stability condition for the non-zero singular points of equation (4) (when p=2) as follows : Then, from the compare between the roots of the equation (11) and it's coefficients we get We will find the stability condition for the non-zero singular points of equation (6) (when p=3) , as follows : The characteristic equation of linear model in equation (12) is Then,   3  2  1  3  3  2  3  1  2  1  2  3  2  1  1 ), ( , The stability condition is that 3 , 2 ,

The General Form:
Let the model in equation (1) be given, that is we will find the stability condition for the non-zero singular points of equation (1) The characteristic equation of the given model is defined as: 0 ...
. The stability condition of singular point of equation (1) is the absolute values of the characteristic roots of equation (13) are all less than one, that means

Limit cycle:
We find the stability condition for the limit cycle (if it exists) as follows: Let the limit cycle of period q of the proposed model in the equation (2) has the form  (15) is a linear difference equation with a periodic coefficient, which is difficult to solve analytically what we want to know whether t  of (15) converges to zero or not, and this can be checked by seeing whether t q t   + is less than one or not [7].

4.Examples
In this section, we give two examples to explain how to find the singular points of the proposed model and the conditions of stability of singular points and limit cycle.  . Then, the model has a stable limit cycle.This means that the model has unstable non-zero singular point but have a stable limit cycle.  ,Then, the model has not a stable limit cycle.Therefore, the model has two non-zero singular points(one of them is a stable and the other is unstable)and also has unstable limit cycle.

Conclusion
The conclusions of this paper are as follows: 1-We find the non-zero singular point of the proposed model of order one and two and three. 2-We find the stability conditions of the non-zero singular point of the proposed model of order one and two and three and the general model. 3-We find the stability conditions of the limit cycle of the proposed model of order one and two and three and the general model. 4-We explain the stability conditions of a non-zero singular point and the stability conditions of the limit cycle in two examples and find that the model of order one example (1) is not stable singular point and a stable limit cycle and find that the model of order two example(2) have a two complex singular points 1  , 2  one of them 1  is a stable and the other 2  is unstable, and not a stable limit cycle.