## 5 Applications in Control Theory
### Control of Chaos
#### 5.2.1 Stabilization of Chaos  
Orbits of a dynamical system originating near an unstable fixed point remain unstable throughout the evolution of the system. To obtain regular motion from such initial point, the method used by Saha and Kumra [111] is to modify the value of the parameters \(a\) and \(b\). As an example, they considered the case of \(a=1.6\) and \(b=0.5\) for which there exists an unstable fixed point \(P^{*}=(0.47619,0.238095)\). Taking the initial point \(P=(0.5,0.3)\) near \(P^{*}\), they performed search using their previously defined method of asymptotic stability [112]for the new parameter values, for which the solution is periodic with the same initial point. This method is based on the computation of the Lyapunov exponents. They found that \(a=1.52381\) and \(b=0.57619\).  
Gokce et al. [113] investigated the stability and tracking performance of discrete-time chaotic systems in the presence of external disturbance and noise. They developed a neural network control scheme on the basis of a novel adaptive learning rate to stabilize the chaotic motion of discrete-time chaotic systems to a fixed point as well as to track the desired reference trajectory. The effectiveness of the proposed method was investigated through simulation studies on the Lozi map and a performance comparison was made with the backstepping control strategy. The backstepping method based on the Lyapunov stability is a recursive procedure that breaks the full system into a sequence of small subsystems. Simulation studies demonstrated that the proposed method can successfully stabilize the chaotic trajectory of the system to the fixed point and track the desired reference trajectory. It has been shown that the proposed method is less sensitive to noise and disturbance and has a more robust property than the backstepping control scheme if the neural network is trained properly with accurate training parameters.  
Joshi et al. [42] studied the control of a complex fractional-ordered Lozi map (see Section 2.5.2). They used the Pyragas method [78], which is a feedback method that can be attempted without any detailed knowledge of the dynamical system under consideration. There are two ways in which delay can be introduced in the system, are formulated as follows:  
\[x(n)=\gamma x(j-2)+x(0)+\frac{1}{\Gamma(v)}\sum_{j=1}^{n}\frac{\Gamma(n-j+v)}{ \Gamma(n-j+1)}[-\alpha|x(j-1)|-x(j-1)+1+bx(j-2)], \tag{73}\]  
and  
\[x(n)=x(0)+\frac{1}{\Gamma(v)}\sum_{j=1}^{n}\frac{\Gamma(n-j+v)}{\Gamma(n-j+1 )}[-\alpha|x(j-1)|-x(j-1)+1+bx(j-2)+\gamma x(j-3)], \tag{74}\]  
where \(\gamma\) is the strength of the delay. For their numerical simulations, they set \(\gamma=0.33\) for both (73) and (74).  
They found that in both cases there is a reduction in the range over which chaos is observed. Period 2 is stabilized instead.  
In Case 1 (73), Period 2 has an even more stable range, and the stability range of Period 1 is reduced, whereas for Case 2 (74) the range of Period 1 is significantly increased and that of Period 2 spans over a smaller range of parameters. Thus, chaos control can be achieved using the Pyragas method even in complex-order fractional maps.