## 2 Adaptive and Generalized Synchronization
### Synchronization between Systems Implemented in Different Ways
**Preliminary 3**. : _Consider the system \(\tilde{F}:P\oplus X\to X\):_  
\[\dot{\mathbf{x}}(t)=\tilde{F}(\mathbf{p},\dot{\mathbf{x}}(t)),\]  
_where \(\mathbf{p}\in P\). Let \(|\cdot|\) be a metric in a function space. Then, \(|F-\tilde{F}|>0\)._  
A distorted master system \(\tilde{F}\) can differ in both parameters and nonlinearities from \(F\). **Preliminary 4**. : _Consider a discretization operator over \(Y\)_  
\[L_{h}:G(\mathbf{y}(t))\to\Gamma(\mathbf{\hat{y}}(n))\]  
_and \(P\)_  
\[L_{h}:C_{g}(\mathbf{g}_{s}(t))\to\hat{C}_{g}(\mathbf{\hat{g}_{s}}(n))\]  
_with a local error \(\mathbf{e}_{y}(h,n)\):_  
\[\mathbf{y}(t_{n})-\mathbf{\hat{y}}(n)=\mathbf{e}_{y}(h,n),\]  
\[\mathbf{g}_{s}(t_{n})-\mathbf{\hat{g}_{s}}(n)=\mathbf{e}_{gs}(h,n),\]  
_where \(h\) is a discretization period, and \(\lim_{h\to 0}\varepsilon_{h}(n)=0\)._  
The trajectory \(\mathbf{y}(n)\) of the discrete system \(\Gamma:P\oplus Y\to Y\) mapped from \(G\) with the operator \(L_{h}\) is defined only in discrete points \(t_{n}\) and in those points is close to the trajectory of \(G\), if the discretization error is low. A similar case is true for \(\mathbf{g}_{s}(n)\), which is a trajectory of the system \(\hat{C}_{g}\) mapped from the system \(C_{g}\). The discrete systems \(\Gamma\) and \(\hat{C}_{g}\) preserve all parameters and nonlinearities of tge original systems. **Preliminary 5**. : _There is a coupling channel with a key between the continuous master system \(\tilde{F}\) and the discrete slave system \(\Gamma\), which transforms the system \(\tilde{F}\) continuous trajectory \(\tilde{\mathbf{x}}(t)\) into the discrete series \(\{\hat{\mathbf{x}}(n)\}\) with a discretization period \(h\). For clarity, \(\{\hat{\mathbf{x}}(n)\}=\tilde{\mathbf{x}}(t_{n})\)._  
From these preliminaries, a conjecture follows. **Conjecture 1**. : _If \(|F-\hat{F}|<\delta\) and \(\mathbf{e}_{g}(h,n)<\eta\) and \(\mathbf{e}_{g}(h,n)<\xi\), where \(\delta,\eta,\xi>0\) are certain small positive numbers, generalized synchronization between \(\hat{F}\) and \(\Gamma\) occurs, and the trajectory of the error \(\mathbf{e}(n)=(\hat{\mathbf{y}}(n)-\hat{\mathbf{x}}(n);\hat{\mathbf{g}}_{ \hat{\mathbf{e}}}(n)-\mathbf{g}_{m})^{\top}\) lies within a bounded set \(\mathcal{A}\subset Y\oplus P\). Its \(L_{1}\)-volume \(V_{L_{1}}(\mathcal{A})\) satisfies the limit property:_  
Figure 1:0  
We propose that this would be true in case the rate of exponential decay of error dynamics \(\mathbf{e}(n)\) between \(G\) and \(F\) is faster than the growth of errors caused by the dissimilarity between the original synchronized systems \(F\) and \(G\) and their perturbed counterparts \(\hat{F}\) and \(\Gamma\). A rigorous proof is beyond the scope of this article. The following corollaries are of practical importance. **Corollary 1**. : _Parameter identification via adaptive synchronization between two different systems is possible with the error \(\mathbf{e}_{g}(n)\)._  
In practice, while the maximal values of \(\mathbf{e}_{g}(n)\) can be rather high, the mean or median value might be rather small, allowing its use for identification and measurement. **Corollary 2**. : _The synchronization error between master and slave systems \(\mathbf{e}_{g\delta}(n)\) depends on the integration method and is lower for more precise methods._  
Both corollaries are illustrated in Section 3.