# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Estimation of fractal dimension and Kolmogorov entropy
#### 2.3.6 Chaos-Based Communications and Effect of Noise on Dynamical Systems
When \(\alpha=0\), we have  
\[u^{\prime}=DF(x(t))u, \tag{60}\]  
since the dynamics are chaotic, we have  
\[\|u(t)\|\leq\|u(0)\|e^{\lambda_{1}t}, \tag{61}\]  
where \(\lambda_{1}\) denotes the largest positive Lyapunov exponent of the isolated system. Now letting  
\[v=ue^{-2\alpha t}, \tag{62}\]  
we obtain  
\[\|v(t)\|\leq\|u(0)\|e^{(-2\alpha+\lambda_{1})t} \tag{63}\]  
therefore, the critical coupling strength is  
\[\alpha_{c}=\lambda_{1}/2. \tag{64}\]  
In general, when \(\alpha>\alpha_{c}\), and higher-order nonlinear terms in the Taylor series expansion can indeed be ignored, then the coupled system will exhibit complete synchronization. In building chaotic secure communication systems, the coupling is usually unidirectional, and the two systems are called drive and response (or master and slave) systems--in the example discussed here, if the term \(\alpha(y-x)\) is dropped in the \(x\) system, then the \(x\) system is the drive system, and the \(y\) system is the response system. To better understand the potential of chaotic secure communications, it is important to examine the effect of noise on dynamical systems. There are two types of noise, one is measurement noise. In chaotic secure communications, the channel noise is a type of measurement noise. The other type of noise is dynamical noise. It is in the equations governing the dynamics of the system. The channel noise becomes part of the dynamical noise for the response system (which can have additional dynamical noise sources). For two chaotic systems to synchronize, dynamical noise in the response system has to be small. This means the signal \(s(t)\) has to be small compared with the chaotic signal \(x(t)\). As a consequence, power consumption in chaotic secure communications is larger than traditional communication systems. This may be considered a cost for achieving better security. Although in most situations noise is detrimental in chaotic secure communications, there are a few fortunate situations where noise is beneficial. This is enabled by an interesting phenomenon, the noise-induced chaos. The existence of the phenomenon can be demonstrated via a driven nonlinear oscillator [114], or the noisy logistic map [115], or other systems [116; 117]. A mechanism for the phenomenon has also been developed [82; 118]. The phenomenon is still a hot topic today, see for example [119; 120]. Here we explain the basic properties of and the mechanism for noise-induced chaos via the noisy logistic map:  
\[x_{n+1}=\mu x_{n}(1-x_{n})+P_{n},\hskip 14.226378 pt0<x_{n}<1, \tag{65}\]  
Here, \(\mu\) is the bifurcation parameter, and \(P_{n}\) is a zero-mean Gaussian random variable with standard deviation \(\sigma\). When \(P_{n}=0\), the map generates periodic orbits with periods 8, 6, 5, and 3 at parameter values \(\mu=3.55\), 3.63, 3.74, and 3.83, respectively. The period-8 motion at \(\mu=3.55\) is on the main \(2^{n}\) cascade, and the period-3 motion at \(\mu=3.83\) is on the period(3)-doubling cascade (see Figure 10). For the case of \(\mu=3.55\), with a fairly large noise of \(\sigma=0.01\), the noisy trajectory is still very similar to the clean period-8 trajectory, as one can clearly see from Figure 16a. The case of \(\mu=3.74\) is very different. With \(\sigma\) as small as 0.0003, the noisy trajectory is already completely different from the original clean period-5 trajectory, as shown in Figure 16b. In fact, this noisy trajectory is chaotic, as shown by the time-dependent exponent curves shown in Figure 17c. In contrast, the noisy dynamics at \(\mu=3.55\) are definitely not chaotic, as shown by Figure 17a. The noisy dynamics at \(\mu=3.63\) and \(3.83\) are also chaos-like, though not as well defined as at \(\mu=3.74\). The mechanism for noise-induced chaos can be found by examining how a small amount of noise affects the dynamics. In general, the noisy dynamics when noise is very small is a diffusion characterized by Equation (44).