## 2 Adaptive and Generalized Synchronization
### Terms and Definitions  
Let us consider a master system  
\[\dot{\mathbf{x}}(t)=F(\mathbf{x}(t)), \tag{1}\]  
and a slave system  
\[\dot{\mathbf{y}}(t)=G(\mathbf{x}(t),\mathbf{y}(t)), \tag{2}\]  
where \(\mathbf{x}\in\mathbb{R}^{n}\), \(\mathbf{y}\in\mathbb{R}^{m}\) and \(t\in\mathbb{R}^{1}\). For \(n=m\) the systems (1) and (2) are said to be completely synchronized if  
\[\lim_{t\to\infty}\|\mathbf{x}(t)-\mathbf{y}(t)\|=0. \tag{3}\]Equation (3) shows that the dynamics of the master and slave systems become identical over time irrespective of the different initial conditions they had. One way to completely synchronize two similar systems is to introduce a control law \(C_{F}\) and design the slave system as follows:  
\[\dot{\mathbf{y}}(t)=F(\mathbf{y}(t))+C_{F}(\mathbf{x}(t),\mathbf{y}(t)). \tag{4}\]  
If there is a vector of uncertain parameters \(\mathbf{g}\) in the master system, a case of adaptive synchronization is considered. A parameter update law for the slave system \(C_{g}\) can be designed, leading to an overall master-slave configuration of  
\[\begin{cases}\dot{\mathbf{x}}(t)=F\big{(}\mathbf{g}_{m},\mathbf{x}(t)\big{)},\\ \dot{\mathbf{y}}(t)=F(\mathbf{g}_{s}(t),\mathbf{y}(t))+C_{F}(\mathbf{g}_{s}(t),\mathbf{x}(t),\mathbf{y}(t)),\\ \mathbf{g}_{s}(t)=C_{g}(\mathbf{x}(t),\mathbf{y}(t)).\end{cases} \tag{5}\]  
The control law \(C_{g}\) is designed to ensure that the vector of the slave parameters \(\mathbf{g}_{s}\) matches the vector of the master parameters \(\mathbf{g}_{m}\) over time:  
\[\lim_{t\to\infty}\|\mathbf{g}_{m}-\mathbf{g}_{s}(t)\|=0. \tag{6}\]  
The technique to obtain \(C_{F}\) and \(C_{g}\) is as follows. Denote the error between system phase variables and parameters as \(\mathbf{e}(t)=(\mathbf{y}(t)-\mathbf{x}(t);\mathbf{g}_{s}(t)-\mathbf{g}_{m})^ {\top}\). Then, consider the Lyapunov candidate function as  
\[V(t)=\frac{1}{2}\mathbf{e}^{\top}P\mathbf{e}, \tag{7}\]  
where \(P\) is a positive definite matrix; e.g., an identity matrix. The Lyapunov candidate function derivative \(\dot{V}(t)\) is then algebraically expanded, and \(C_{F}\) and \(C_{g}\) are found analytically from equations, providing \(\dot{V}(t)<0\) for all \(t>0\). If it is impossible to find such functions, a weaker condition is considered, \(\dot{V}(t)\leq 0\). In this case, it should be ensured that  
\[\lim_{t\to\infty}\int_{0}^{t}\|\mathbf{e}\|^{2}dt=\alpha<\infty.\]  
If this condition is satisfied, then, according to [PERSON]'s lemma, \(\lim_{t\to\infty}\|\mathbf{e}\|^{2}=0\). Section 3 gives the particular examples of \(C_{F}\) and \(C_{g}\) for the Rossler system with uncertain parameters.  
Two different systems (1) and (2) are said to be in generalized synchronization if a nonlinear function \(\phi\) exists such that  
\[\mathbf{x}(t)=\phi(\mathbf{y}(t)). \tag{8}\]  
In the case of a generalized synchronization between systems with similar dimensions, the error in dynamics \(\mathbf{\Delta}(t)=\mathbf{x}(t)-\mathbf{y}(t)\) persists. The error trajectory \(\mathbf{\Delta}(t)\) lies within an attractor \(\mathcal{A}_{c}\subset\mathbb{R}^{n}\) with non-zero amplitude.  
There is a simple approach to detect generalized synchronization between systems (1) and (2) using an auxiliary system:  
\[\dot{\mathbf{z}}(t)=G(\mathbf{x}(t),\mathbf{z}(t)), \tag{9}\]  
where \(G\) is similar in both slave and auxiliary systems. If GS occurs, then, due to the existence and uniqueness theorem, the error between \(\mathbf{y}(t)\) and \(\mathbf{z}(t)\) vanishes over time. In fact, in this case, we deal with two identical slave systems driven by one master system, and we know nothing about function \(\phi\), which is convenient in many applications.