# Dynamic Mathematical Models’ System and Synchronization

Authors: Malik Bader Alazzam; Abdulsattar Abdullah Hamad; Ahmed S AlGhamdi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We created the equilibrium, which includes sickness outcomes, health and risk behaviors, environmental factors, and health-related assets and delivery systems, and it should be incorporated in system Dyc (dynamic) modelling of chronic disease prevention. System Dyc has the ability to model a variety of interconnected illnesses and dangers, as well as the interaction between delivery systems and afflicted people, as well as state and national policies. This paper proposes a unique idea. Hybrid synchronization utilizes four positive LYP (Lyapunov) exponents based on state feedback management with two identical systems of the Lorenz system 6D HYCH system.


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## Body

## 1. Introduction

In 1963, a Chaos, a fascinating occurrence in a dynamic, nonlinear system, encourages us to be prepared for anything. 3D Lorenz, an American meteorologist, was the first to notice these phenomena. LYP. Rössler introduced a new three-dimensional Dyc system in 1976. He had, at the period, six terms and only one polynomial nonlinearity. Many three-dimensional chaotic systems’ ideas may be found in [1–4]. The first four-dimensional model (4D) was proposed in 1979 and provided a system with two positive LYP formulas and real variables; since then, numerous 4D chaotic schemes have been identified in [5, 6]. The size of a HYCH system is proportional to the number of positive LYP chaotic system architectures which is four [7]. Even though a chaotic system only has one positive LYP opposite, a HYCH system has several positive LYP inverses [8–10]. The system's size must be quadrupled to maximize the number of positive LYP derivations. Building 5D HYCH systems with three large LYP exponents has attracted a lot of attention since the Hu system [11–15]. The HYCH system with a multidimensional space is more effective and precompiled than the low dimension, and it surpasses typical 3D, 4D, and 5D systems due to its enhanced unpredictability and randomness. There have been an increasing number of articles on the development of new high-dimensional (9D) [16–20] systems with four basic components LYP exponents, as well as several papers on the development of new strong (6D) [21–25] systems with four test LYP coefficients. In [26–30], a six-dimensional HYCH system with four positive LYP exponents is constructed: LEA 1 = 0.5311, LEA 1 = 0.3100, LEA 3 = 0.1300, LEA 4 = 0.0780, LEA 5 = -0.0001, and LEA 5 = -12.5224, consisting of 14 terms; three terms are described. In [31], a new 7D HYCH system is built by A. A Hamad et al. in which points, stability, and LYP exponents are all important elements of a novel mechanism:(1)u˙1s=au2−u1+u4+ru6,u˙2s=cu1−u2−u1u3+u5,u˙3s=−bu+u1u2,u˙4s=du4−u1u3,u5s=−hu2+u6,u˙6s=k1u1+k2u2,where u1s,to,u6sT∈R6. Tiny changes in the starting values cause small variances in the sheer randomness of chaotic complex systems, according to this multidisciplinary hypothesis. With variable in system (1) and a,b,d,h≠0, a,b,andc are constraints and d,h,r,k1,andk2 are control.The 6-D sports’ system, which represents the driving system, is(2)u˙1=au2−u1+u4,u˙2=cu1−u−uu3+u6,u˙3=−bu+uu,u˙4=hu−uu3−u,u˙5=qu2−pu5−gu1,u˙6=gu2−qu6.It can be written as(3)u˙1u˙2uu˙4u˙5u˙6=−aa0100c−1000100−b000000h−10−gq00−p00g000−qu1u2uuu5u6+000100010001000000−uu3uu2−uu3,A=−aa0100c−1000100−b000000h−10−gq00−p00g000−q,B=000100010001000000,C=−u1u3u1u2−uu3.The response system is as follows:(4)vv˙2v˙3v˙4v5v˙6=A11v1v2v3v4v5v6+B−v1v3v1v2−v1v3+u1u2u3u4u5u6.The error is calculated for the Dyc system using the previous relationship as follows:(5)e˙1e˙2e˙3e˙4e˙5e˙6=A11e1e2e3e4e5e6+F+BD+u1u2u3u4u5u6,F=−2au2−2u42cu1−2v1u3−2u1u2−2u5−2v1u3−2qu20,D=−v1e3+u3e1e1e2−u2e1+u1e2−v1e3+u3e1,e˙1=ae2−ae1−2au2+e4−2u4+l1,e˙2=ce1+2cu1−e2−v1e3+u3e1−2vu3+e6+l2,e˙3=−be3+e1e2−u2e1+u1e2−2u1x2+l3,e˙4=he4−v1e3+u3e1−2v1u3−e5−2u5+l4,e˙5=qe2−2qx2−pe5−ge1+l5,e˙6=−qe6+ge2+u6.Using the method of linear approximation for Dyc system error (1), postcontrol is(6)λ6+33015λ5−101110λ4−4242115λ3−27960730λ2−270804λ+73793=0,λ1=−53,λ2=10.8034,λ3=0.2616,λ4=−1.6352,λ5=−8.0034,λ6=−22.8264.Now, we design several controllers based on the Lebanov methods and linear approximation.


## 2. Result and Disscussion

Theorem 1.
If system U control is (1), the following design is taken:(7)u1=2au2+2u4−ce2−u3e2+u2e3−u3e4+ge5,u2=−ae1−2cu1+2v1u3−u1e3−qe5−ge6,u3=v1e2+2u1u2−e1e2+v1e4,u4=−e1−2he4+2v1u3+2u5,u5=e4+2qu2,u6=−e2.It is possible to synchronize system (2) with system (3) in two ways, namely, Lebanov and linear approximation.Proof.
After (5) control compensation in Dyc system error (4), we obtain(8)e˙1=ae2−ae1+e4−ce2−u3e2+u2e3−u3e4+ge5,e˙2=ce1−e2−v1e3+u3e1+e6−ae1−u1e3−qe5−ge6,e˙3=−be3−xu2e1+u1e2+v1e2+v1e4,e˙4=−he4−v1e3+u3e1−e5−e1,e˙5=qe2−pe5−ge1+e5,e˙6=−qe6+ge2−e2.
First, using the linear approximation method,(9)λ6+33015λ5+569810λ4+35369060λ3+5527045240λ2+327631880λ+93485130=0,λ1=−53,λ2=−7.0017,λ3=−1.5007−i,λ4=−4.5979+19.0319i,λ5=−4.6979−19.0319i,λ6=−1.5002+1.2442i.
It is clear that the linear approximation method achieves a hybrid synchronization between the two systems (4).
Second: using the Lebanov Method,(10)Vei=12∑i=16ei2=eiTPei,P=daig0.5,0.5,0.5,0.5,0.5,0.5.
By differentiating, we obtain(11)V˙e=e1ae2−ae1+e4−ce2−u3e2+u2e3−u3e4+ge5+e2ce1−e2−v1e3+u3e1+e6−ae1−x1e3−qe5−ge6+e3−be3−u2e1+u1e2+v1e2+ve4+e4−he4−v1e3+u3e1−e5−e1+e5qe2−pe5−ge1+e4+e6−qe6+ge2−e2,V˙ei=−ae12−e22−be32−he42−pe52−qe62=−eiTQ54ei.
The Lebanov formula achieved a hybrid synchronization between the two systems:(12)Q54=diaga,1,b,h,p,q.
Figure1 illustrates the results numerically depending on the initial values 14,3,0,−1,−3,0−13,−10,−8,5,0,−5, respectively.Figure 1
Hybrid synchronization.Theorem 2.
Let us have the nonlinear U control of Dyc system error (7) as follows:(13)u1=2au2+2u4−ce2−u3e2+u2e3−u3e4+6e5,u2=−ae1−2cu1+2v1u3−u1e3−ge6,u3=v1e2+2u1u2−e1e2+v1e4,u4=−e1−2he4+2v1u3+2u5+e5,u5=−qe2+2qu2,u6=−e6.System (4) can achieve the phenomenon of hybrid synchronization with system (5) in two ways.Proof.
By substituting control (8) in system (7), we obtain(14)e˙1=ae2−ae1+e4−ce2−u3e2+u2e3−u3e4+6e5,e˙2=ce1−e2−v1e3+u3e1+e6−ae1−u1e3−ge6,e˙3=−be3−u2e1+u1e2+v1e2+v1e4,e˙4=−he4−v1e3+x3e1−e1,e˙5=−pe5−ge1,e˙6=−qe6+ge2−e2.
The first method is linear approximation:(15)λ6+34615λ5+1015320λ4+27127960λ3+19215815λ2+23405320λ+146385=0,λ1=−83,λ2=−0.4057,λ3=−1.0001,λ4=−8.0012,λ5=−5.4965+17.5492i,λ6=−5.4965−17.5492i.
The hybrid synchronization of the two systems was achieved using the linear approximation approach.
The Lebanov technique is the second method. The Lebanov derivative with control (10) is as follows:(16)V˙e=−ae12−e22−be32−he42−pe52−qe62+e1e56−g=−eTQ64e.
And, the resulting matrix is(17)Q64=a000g−62001000000b000000h00g−62000p000000q.
The matrixQ64 is nondiagonal.
That is, the matrix is negatively defined, and we can check this by looking for inequalities in the above matrix's determinants.(18)1.a>0,2.b>0,3.h>0,4.q>0,5.a>g−624p.
Because the fifth inequality is faulty, the control system was unable to establish hybrid synchronization between the two systems.Q64 is negatively defined. Now, we update the P-matrix with the same control as the following:(19)P64=diag12,12,12,12,6,12.
The derivative of Lebanov becomes as follows:(20)V˙ei=−ae12−e22−be32−he42−12pe52−qe62=−eTQ74e,which leads to Q74=diag9.5,1,5/3,1,4.8,8; it is a positively specified matrix, resulting in hybrid synchrony between system (8) and system (7) with the control unit (7), as shown in Figures 1 and 2.Figure 2
Convergence of the system.Figure 3
Hybrid synchronization between the two systems (u1,…u6 and v1,…….,v6).Figure 4
Convergence of the system (healthcare).


## 3. Conclusion

Utilizing four positive LYP coefficients and state feedback control, this work creates a unique hybrid synchrony between two identical Lorenz system 6D HYCH systems (Figures3 and 4). Equilibrium point, stability, and LYP coefficients are all evaluated as significant properties of a new mechanism. According to computer modelling, the new system shows complex Dycal characteristics such as chaotic, stochastic, and periodic.



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