## 2 Adaptive and Generalized Synchronization
### Synchronization between Systems Implemented in Different Ways  
In any experimental study, we are restricted in terms of implementation accuracy since each method of implementation imposes several errors on the system. Therefore,any experimental study deals not with an ideal continuous system \(F\) but with its distorted counterpart \(\hat{F}\). Figure 1 shows the main error sources in analog-to-numeric synchronization. Each kind of error in chaotic system implementation leads to the non-predictable disruption of system behavior due to the sensitive dependence of system dynamics on parameter values and nonlinearity types. Circuit implementation suffers from non-ideal components and analog noise. For example, typical values of resistor tight tolerance are within 0.5-5%, which substantially limits the accuracy of the parameter setting. The analog integrators usually implemented with operational amplifiers are not strictly linear. For instance, the high-quality operational amplifier OPA2134 has a total harmonic distortion at 1 kHz equal to \(8\times 10^{-5}\%\)[25], which is relatively small but restricts the integration accuracy to values much lower than achievable by a digital computer. Temperature drift and noise of various natures also affect the circuit implementation accuracy. A computer model can be implemented much more accurately but also suffers from errors. In floating-point arithmetic, round-off noise sets the theoretical lower bound of simulation result error at the level of the machine epsilon, equal to approximately \(2.2\times 10^{-16}\) for double precision [26]. Truncation error plays a notable role if the order of the integration method is low. An additional source of error in synchronization between analog and numeric systems is contained in a coupling channel. The time-domain resolution of this channel is restricted by at most 1 GHz, but usually is much lower. Besides, the ADC and DAC have a limited level of resolution--e.g., 12 or 24 bit--which makes the use of a full double precision scale impossible. Under all of the abovementioned conditions, it is reasonable to ask whether the synchronization between analog and numerical systems is possible. Our previous works [22; 23] provide some experimental evidence of its possibility but also report on valuable restrictions in error elimination. Theorem 1 establishes the theoretical bounds on synchronization accuracy in this case. Definition 1: The \(L_{1}\)-volume of an attractor \(\mathcal{A}\) is the \(L_{1}\)-norm of the main diagonal of the hypercube circumscribed around the attractor. In other words, if the trajectory is denoted as \(\mathbf{x}(t)\), then \(L_{1}\)-volume is  
\[V_{L_{1}}(\mathcal{A})=\sum_{j}(\max_{l}(\mathbf{x}_{j}(t))-\min_{l}(\mathbf{ x}_{j}(t))).\]  
The concept of the \(L_{1}\)-volume might be especially useful for estimating some attractor properties, as a conventional volume of a strange attractor may be equal to zero. Figure 1: Sources of error in analog-to-numeric synchronization. **Preliminary 1**. : _Let \(F:P\oplus X\to X\) be an autonomous continuous chaotic system, defined by the equation_  
\[\dot{\mathbf{x}}(t)=F(\mathbf{g}_{m},\mathbf{x}(t))\]  
_in the Banach space \(X\), and \(\mathbf{g}_{m}\in P\) is a parameter vector. Denote this system as a master system._  
**Preliminary 2**. : _Let the slave system \(G:P\oplus Y\to Y\) be defined by the equations_  
\[\dot{\mathbf{y}}(t)=G(\mathbf{g}_{m},\mathbf{y}(t))\]  
_Consider the particular case of diffused coupling and automated parameter matching:_  
\[\dot{\mathbf{y}}(t)=F(\mathbf{g}_{s}(t),\mathbf{y}(t))+C_{F}(\mathbf{g}_{s}(t),\mathbf{x}(t),\mathbf{y}(t)),\quad\mathbf{g}_{s}(t)=C_{g}(\mathbf{x}(t), \mathbf{y}(t)),\]  
_where \(\mathbf{g}_{s}\in P\), \(C_{F}:P\oplus X\oplus Y\to Y\) is the synchronization law and \(C_{g}:X\oplus Y\to P\) is the parameter adaptation law, and \(Y,P\) are also Banach spaces._  
Consider the case when the systems \(F\) and \(G\) are completely synchronized; i.e., conditions (3) and (6) are satisfied. For simplicity, assume \(X=Y\).