Asymptotic Stability Analysis Applied in Two and Three-Dimensional Discrete Systems to Control Chaos

Asymptotic stability analysis applied to stabilize unstable fixed points and to control chaotic motions in two and threedimensional discrete dynamical systems. A new set of parameter values obtained which stabilizes an unstable fixed point and control the chaotic evolution to regularity. The output of the considered model and that of the adjustable system continuously compared by a typical feedback and the difference used by the adaptation mechanism to modify the parameters. Suitable numerical simulation which are used thoroughly discussed and parameter values are adjusted. The findings are significant and interesting. This strategy has some advantages over many other chaos control methods in discrete systems but, however it can be applied within some limitations. KeywordsAsymptotic stability, Control parameter, Chaos, Lyapunov exponents.


Introduction
Orbits of dynamical system, originating nearby an unstable fixed point, may remain unstable throughout the evolution of the system. The evolutionary motion starting from such positions often turn to be chaotic. To obtain regular motion, one has to stabilize the initial point by changing values of certain parameters of the system. For this, there should be a well-defined procedures and asymptotic stability method can be considered as one of the most effective one. This technique proved to be useful, as it could regulate the chaos to achieve desirable results.
Chaos management refers to the technique of modifying and regularizing the chaotic motion displayed in nonlinear systems. Several interesting reports on chaos control, (Auerbach et al., 1992;Braiman et al., 1995;Carroll & Pecora, 1993;Garfinkel et al., 1992;Litak et al., 2007;Ott et al., 1990;Pecora & Carroll, 1990;Pyragas, 1992;Shinbrot et al., 1993) have shown that If properly applied to chaotic and dynamic systems, management of disorder can be of great benefit. Chaos control techniques like OGY and feedback control method with a good effect in a relative short period are being used in many nonlinear dynamical systems (Bilal Ajaz et al., 2020;Wang et al.,

Description of the Method
Dynamics of the actual map 1  n X and that of the desired map 1  n Y can be explained by following mapping: Also, the neighborhood dynamics of 1  n X and 1  n Y can be represented by the relation: A A B B can be obtained from the following: Let a, b are two parameters of the system and (x u , y u ) be any unstable fixed point of above system for given values of a and b. Then, our objective is to obtain two new suitable values for a and b so that this unstable point becomes stable. For this, we need the Jacobian matrices defined by The control input parameter matrix * p can be given by Then, using (1), (2) & (3), one obtains the following error equation: Note that in equation (3) and (4)

(ii) Food Chain Model: (F -C Model)
Next, we have considered three dimensional food chain model, (Elsadany, 2012)  Bifurcation diagrams of the above F -C map (9) along x-axis, for three ranges of parameter c are obtained and given by Figure 3.  Thus, for fixed value of b = 3.7, c= 3.0, we proceeded above calculations with input and obtained At these new parameter values of a, d and r, we obtain the phase plot and the plot of Lyapunov exponents as shown in Figure 5 below. Clearly phase plot on XY plane shows finite number of points. Similar is the case with also other two planes. Also, LCE plot shows the Lyapunov Exponents all are negative. Hence, the system is no more chaotic and chaos is controlled.

(iii) 3-D Arneodo-Coullet-Tresser (ACT) Map
Further, let us consider an another 3-dimensional map, known as the Arneodo-Coullet-Tresser (or ACT) Map, (Arneodo et al., 1981(Arneodo et al., , 1982,written as For parameter values a = 0.6, b = 0.5, c = 0.41, d = 1, e = 1, k =3, ACT map (10) has an unstable fixed point given by (0.640312,0.0128062, 0.525056) and thus, the neighboring points this point (0.55, 0.01, 0.5) is also unstable. Time series graphs and chaotic attractor obtained for orbit originating from this neighboring point are shown in Figure 6. a 4.1 p= d = 3.5 r 3.8    with these new parameter values, a = 0.585951, b = 0.59804, c = 0.0201784 together with d = e = 1, k = 3, x=0.55, y = 0.01, z = 0.5 the chaotic system shown earlier is controlled and evolve regularly. A phase diagram and the plot for Lyapunov exponents for this case are given in Figure  8. Clearly phase plot on XY plane shows a single point and same is the case with other two planes. Also, the values of Lyapunov Exponents all are negative. This shows that the system is no more chaotic and showing one periodic regular case. Hence, chaos is controlled.

Limitation
The method of asymptotic analysis to control chaos has some limitations & it does not work for all dynamical systems. Suppose we desire to have a solution of system of equations AX = B, where A and B are known matrices of order, m × n and m × r respectively and X is unknown matrix of order n × r which is to be determined. X can be written as X (X1, X2…Xr) and B as B (B1, B2, Br) where Xi (i = 1,2,3, r) are n × 1 columns and Bi (i = 1,2,3…, r) are m × 1 columns.
The system A X = B will be consistent if and only if Rank of A = Rank of augmented matrix (A, Bi),  i =1, 2, 3, r Equations (12), (13) and (14) are consistent if and only if each one of them satisfies condition(s) giving by equation (11).For example (Saha et al., 2004), consider the model: Fixed points for this map are given by (  n , 0). The fixed point (0, 0) for k=1 is unstable & proceeding above calculations one finds matrix BD = 0, hence Rank (BD) = 0.
But, Rank (BD: BR) is not equal to zero. So, the equation BR = BD CM is inconsistent in this case. Similarly, there are many more systems where this technique fails.

Conclusion
We observed that the asymptotic stability method can be used appropriately to have chaos control for maps with n dimensions and m parameters such that m  n. One can fix the values of some mn parameters in case m > n, with appropriate considerations. This method could be applied in systems of 2, 3 or higher dimensions. However, it does have its drawbacks and it cannot be used for every chaotic system. Therefore, there is no evergreen strategy for treating chaos management. In order to have stability, different methods are needed in different non-linear systems.

Declaration of Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.