Stability Conditions for a Nonlinear Time Series Model

. Tis research aims to determine whether the proposed time series model is stable or not. To achieve this, Ozaki’s approximation method was utilized, where the nonlinear part is a function that approaches zero, similar to Ozaki’s imposed function. Te study produced examples of both stable and unstable models, and it was discovered that the stability and instability of the model are afected by the enforced optional constants. Te researchers used the approximation method to obtain an asymptotic linear model that satisfed the singular point of the proposed model. Te study frst identifed the singular points of the suggested model and then focused on determining the stability conditions, which was the main objective of the study. Last, the stability conditions of the limit cycle were established.


Introduction
It is of much importance to view the studies that have been done on Ozaki's method of approximation within the last few years. Te method itself has been used diferently by diferent scholars as it will be mentioned later. As for Ozaki himself, '"he used the local linearization method on the nonlinear exponential autoregressive model" [1]. Both "Mohammad and Salim used the same method on the logistic autoregressive model" [2]. Within the same, later "Salem and Ahmad used it on a nonlinear exponential autoregressive model" [3]. Sometimes later, "Mohammad and Mudhir used the same method but on the exponential (GARCH) models" [4]. After that "Khalaf and Mohammad used it on the Burr X autoregressive model" [5]. "Youns and Salim used this method on the nonlinear model with hyperbolic secant function" [6]. "Youns used this method on the nonlinear time series models with fractional functions" [7]. Also, it has been seen that "Noori and Mohammad used the same method on GJR-GARCH (Q, P) Mode l" [8] and "Hamdi et al. used it too but on stability conditions of Pareto autoregressive model " [9]. "Salim et al." applied it in [10]. "Chen et al. studied the stability, estimation, and applications for the generalized exponential autoregressive models of nonlinear time series" [11]. "Gan et al. studied the local linear RBF dependent AR model of nonlinear model" [12]. Linear model is stable if the roots of the deference equation are in the unit circle [13].
In essence, the Ozaki method relies on transforming the proposed nonlinear model into a linear autoregressive model that depends on the singular point which satisfes the nonlinear model. Consequently, the stability of the nonlinear model in the study is heavily infuenced by the stability of the singular point.
Tis study addressed three examples that showcased the method of local linearization and illustrated the stability or instability of the singular point of a proposed nonlinear model of order one through graphing the model's orbits. Example 1 demonstrated the stable singular point, Example 2 demonstrated the unstable singular point, and Example 3 showcased both the unstable singular point and the unstable limit cycle of an order one model. Orbits were graphed for models with various initial values.

Ozaki Method
"Te nonzero singular point and the limit cycle terms" are defned in references [2,3].
2.1. Defnition. "Te exponential nonlinear autoregressive equation is defned as follows: (1) ε t is a white noise, u 1 , · · · , u p ; v 1 , · · · , v p be the parameters (constants) for this model; then, it is said that the model is an autoregressive exponential model of the p order that symbolized with the symbol EXP AR (p)" [1,13].
Theorem 1 (see [13]). "Suppose that p � 1 at the above variation equation. So, we have y t � (u 1 + v 1 e − y 2 t− 1 )y t− 1 + ε t . Let q is a positive integer, the limit period of the period q. Ten, y t , · · · , y t+q of an equation Theorem 2 (see [13]). "Assumed that Terefore, y t , · · · , y t+q is a limit period of the period q. Wherever q is a positive integer, the previous equation is orbital stable, if the eigenvalues |λ i | for a matrix A, where � A q .A q− 1 · · · A 2 .A 1 , have the utter values smaller than one, |λ i | < 1, so that

Study
Model. Te suggested research model is defned in the p order such that where w 1 , · · · , w p ; v 1 , · · · , v p indicate the parameters for the proposed research model and ε t is the white noise.

Nonlinear Research Model Observations.
Regarding the research model, the following is noted clearly: (1) When y t− 1 approaches to positive infnity, the equation (3) model is transformed into a linear regression model: (2) When y t− 1 approaches to minus infnity, the equation (3) model is transformed into a linear regression model: (3) When y t− 1 equal to zero, the equation (3) model is transformed into a linear regression model:

Stability Conditions of the Proposed Study Model
Te approximation technique was used to identify the stability for research model study when the order i � 1, 2, . . ., p in that portion of a publication. Te study model is 3.1. Finding the Singular Point Z. Assume you have a model of order one: Ten, the unique point Z can be located: where (Z ≠ 0), (v 1 ≠ 0), and (w 1 ≠ 1). Terefore, Te condition for the existence of a single point Z in If the model's order is two, Consider the following: International Journal of Mathematics and Mathematical Sciences Te singular point Z for (12) is Te singular point of a non-linear model of an equation Te singular point of equation (3) was found by using similar method to (10), such as To obtain Terefore, As a result, when the root of (16) is inside the unit circle, (9) is a stable model of order one. (20) Te following represents the stationary conditions of singular point of (10): we have that via making use of the Taylor expansion series. Ten, Terefore, Te distinguishing equation Tus, a 1 � (r 1 + r 2 ), a 2 � − r 1 r 2 .
Ten, the roots of v 2 − a 1 v − a 2 � 0 are r 1 , r 2 . Hence, the stationary condition is |r i | < 1; ∀i � 1, 2. Te singular point in a nonlinear model of an equation (3) has the following condition: Te stability of the singular point is defned as follows: International Journal of Mathematics and Mathematical Sciences Ten, when the roots' absolute values |r i | for v p − a 1 v p− 1 − a 2 v p− 2 − a 3 v p− 3 − · · · − a p � 0 , must become situated within a unitary circle, a conditional of stability singular point to the suggested model is obtained.
3.3. Te Limit Cycle. Te limit cycle of the period q of (6) has the following form: y t , y t+1 , · · · , y t+q � y t ; then, all y s replaced by y s � y s + Z s and then substitute y t by y t + Z t , and y t− 1 by y t− 1 + Z t− 1 , ε t � 0.
Te formula for the research model (12) is y t , · · · , y t+q � y t .
Ten, the points y s approaching the limit cycle are y s � y s + Z s ; ∀s, Ten, Z t � w 1 e y t− 1 + 1 + w 1 y t− 1 e y t− 1 + v 1 − e y t− 1 y t + w 2 e y t− 1 y t− 2 e y t− 1 + 1

International Journal of Mathematics and Mathematical Sciences
As a result, similar to Teorem 1, calculate a value |Z t+q /Z t | < 1 [13].

International Journal of Mathematics and Mathematical Sciences
where m 1 � e y t+i− 1 + 1; m 2 � e y t+i− 1 .

Examples
Regarding the examples presented in this study, Examples 1 and 2 demonstrate the method of identifying and locating the true singular point of the model under study, as well as satisfying the stability condition for the singular point and graphing the model's orbits. Meanwhile, Example 3 clarifes the concept of an unstable limit cycle.
Since the root of the preceding equation is in the unit circle, a singular (solitary) point for this case is found stable. Te following shape Figure 1 illustrate the model's stability with varying beginning values.
Since Z � 0.8938 and applied (19) to reach Z t � 1.0571Z t− 1 . Ten, Z � 0.8938 of Example 2 is not stable because the root r � 1.0571 is located outside the unit circle.
Since Z � 0.8938 and Z t � 1.0571Z t− 1 is deduced, the model of Example 2 is unstable since the root is outside the unity circle.
Since Z � 1.2 and by applying (19), then Z t � 1.1(Z t− 1 ) was obtained. As a result, Example 3 is an unstable model simply because the root r � 1.1 lies outside the unity circle. Te limit cycle is unstable in the period q � 2, which include 1.5; − 1.5; 1.5 { }, where equation (31) was employed.
Te following fgures show the paths inception from various starting values: Figure 5 displays the trajectory of the model with an initial value y(0) � 0.1, Figure 6 displays the trajectory of the model with an initial value y(0) � − 0.1, and Figure 7 displays the trajectory of the model with an initial value y(0) � 0.001.

Conclusions
Te stability criteria for the suggested nonlinear time series model were established, along with the limit cycle stabilization condition for the model, and numerical examples that met these conditions were identifed. It was found that the stability factors of the model depended on the values of its arbitrary constants and the selected nonlinear function.
Furthermore, if the singular point is stable, then the proposed research model is also stable, and the opposite is also true. Te proposed model is a unique case among all other general cases as its nonlinear component represents a decreasing function.
It is also possible to use other methods in the future to analyze and fnd stability conditions for diferent nonlinear models.

Data Availability
No data supporting this study are made availabe.

Conflicts of Interest
Te authors declare that they have no conficts of interest.