### 4.1. Dynamics analysis of controlled Douglas's MCA algorithm

As is mentioned in Section 2, Jacobian matrix is a powerful approach to judge the non-convergence phenomena of dynamical system [24]. A dynamic system is unstable under the condition that each eigenvalue absolute of the Jacobian matrix is larger than 1. Lv and Zhang [8] has found that a lot of chaotic behaviors are represented in the interval λ ∈ [1, 2.32588]. Accordingly, we use STM to modify the eigenvalue of Jacobian matrix of Equation (5) under the condition and get the controlled MCA Equation (8) without changing the value and location of unstable fixed points.

From Function (3) and (5)

\begin{array}{c}{x}_{k+1}={x}_{k}+\lambda D[f({x}_{k})-{x}_{k}]\\ w(k+1)=w(k)-\eta {w}^{5}(k)+\eta {w}^{3}(k))\end{array}

Hence, the new dynamic equation based on STM method is described as Equation (6):

w\left(k+1\right)=w\left(k\right)+\eta \lambda D\left[{w}^{3}\left(k\right)-{w}^{5}\left(k\right)\right]

(6)

when *D* = *I*, the controlled MCA Equation (6) is presented as following:

w\left(k+1\right)=w\left(k\right)+\lambda \eta \left[{w}^{3}\left(k\right)-{w}^{5}\left(k\right)\right]

(7)

*Proof*. we define a point *w** ∈ *R*^{n}is called an equilibrium of (7), if and only if

{w}^{*}={w}^{*}+\lambda \eta \left[{\left({w}^{*}\right)}^{3}-{\left({w}^{*}\right)}^{5}\right]

Clearly, the set of all equilibrium points of (7) is 0,1, -1.

For each equilibrium, the eigenvalues of Jacobian matrix at this point is computed.

Let

G=w\left(k\right)+\lambda \eta \left({w}^{3}\left(k\right)-{w}^{5}\left(k\right)\right)

the Jacobian matrix of (7) is shown as following:

J=\frac{dG}{dw\left(k\right)}=1+\lambda \eta \left(3{w}^{2}\left(k\right)-5{w}^{4}\left(k\right)\right)

There are three cases:

As for equilibrium *w** = 0

\frac{gG}{dw(k)}|{}_{0}=1.

Therefore, 0 is unstable point.

As for equilibrium *w** = 1

\frac{dG}{dw(k)}|{}_{1}=1-2\lambda \eta

When 0<\lambda <\frac{1}{\eta}, it holds that \left|\frac{dG}{dw\left(k\right)}\right|<1.

As for equilibrium *w** = - 1

\frac{dG}{dw(k)}|{}_{-1}=1-2\lambda \eta

When 0<\lambda <\frac{1}{\eta}, it holds that \left|\frac{dG}{dw\left(k\right)}\right|<1.

The proof is completed.

Consequently, in the new Jacobian matrix (8) of Equation (7), each of eigenvalue is less than 1 if 0<\lambda <\frac{1}{\eta}.

In summary, we can control chaotic behavior in the original system if 0<\lambda <\frac{1}{\eta}, and the absolute of eigenvalue of formula (7) is less than 1 when 0<\lambda <\frac{1}{\eta}. This means that the dynamic system can converge, and the unstable system is transferred to a stable system by using STM.

Furthermore, according to the Lyapunov exponent method [19], we can justify and confirm the results by using STM with the illustration of Lyapunov exponent. As mentioned in Section 2.1, when Lyapunov exponent *LE* < 0, the system trajectory is stable corresponding to the periodic motion or a fixed point;when *LE* > 0, it denotes that the system has dynamic behaviors and presents the chaotic phenomena of strange attractor. The Lyapunov exponent's transition from negative to positive indicates the change of periodic motion to chaos.

Figures 4 and 5 present the scenarios in which Lyapunov exponent of original MCA algorithm and the Lyapunov exponent of the controlled Douglas's MCA dynamic system by STM separately. In Figure 4, in some intervals of *η*, the Lyapunov exponent *LE* is less than 0, while in some intervals, *LE* is larger than 0 in which the chaotic solutions of MCA algorithm occur. In Figure 5, *LE* < 0 is presented, which means the chaotic behavior of Douglas's MCA dynamic system has been controlled, and the expected convergent solution of Douglas's MCA is caught.

### 4.2. Chaos control of Douglas's MCA for STM

In this section, case studies of using the STM are illustrated and the time series results of Douglas' MCA from different starting points are shown in Figures 6, 7, 8, 9, 10, and 11. For each iterative map *w*, simulated results of an original system are given to be compared with those using STM. It is evident that the chaotic behaviors of the original dynamic system have been controlled by the STM, the unstable fixed points have been transferred to stable points, and the convergence results have been reached in the original chaotic interval.

Figure 6 illustrates that, when *w* = 1.15548, *η* = 1.3, the original MCA system appears the periodic-4 solutions. Moreover, compared with Figures 4 and 6, when *η* = 1.3, periodic-4solutions appears clearly. On the other hand, in Figure 4, when *η* = 1.3, Lyapunov exponent *LE* > 0, periodic oscillate must occur. Concurrently, the absolute of each eigenvalue of the Jacobian matrix \left|\overline{J}\right|<1. Hence, Lyapunov exponent and the numerical simulation conducted from Jacobian matrix can justify each other. Figure 7 exhibits that when λ = 0.1, the periodic oscillation of controlled Douglas's MCA algorithm by STM is controlled and a convergence solution is achieved.

Figure 8 shows when *w* = 1.0783, *η* = 1.93, the original Douglas's MCA system appears chaotic solutions. Figure 9 presents that when λ = 0.1, the chaotic behavior of Douglas's MCA algorithm is controlled.

Figure 10 demonstrates that when *w* = 0.75187, *η* = 2.25, the original Douglas's MCA system appears chaos phenomena. Figure 11 describes for λ = 0.1, the chaotic behavior of the system is controlled.

In addition, the bifurcation diagrams of Douglas's MCA algorithm corresponding to different starting points *w*(0) = 0.6 and *w*(0) = -0.6 are shown in Figures 12 and 13, respectively.

Further, applying the STM to the original MCA system, the control results of MCA algorithms with respect to Figures 12 and 13 are exhibited in Figures 14 and 15.

It is found that STM can obtain the stable convergence solutions of Douglas's MCA algorithm, and control the numerical instability of periodic oscillation, bifurcation and chaos. Besides, it is worth mentioning that, Figures 12, 13, 14, and 15 also has odd function properties which present symmetric attractors.