# 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  
Among the most promising applications of chaos theory is the exploitation of the short-term deterministic and long-term unpredictable aspects of chaotic behavior for the development of chaos-based communication systems. The actual research in this area goes in two directions. One, started in the early 1990s, is chaos-based secure communications [89]. The other, which is more recent, is to use chaos to rapidly generate random bits in physical devices, for a range of applications in cryptography and secure communication [90; 91; 92; 93; 94; 95; 96; 97; 98; 99]. The potential of each direction is dictated by the role of noise played in the corresponding dynamical systems, which we will explain here. In chaos-based secure communications, the most extensively studied is the scheme exploiting synchronization of chaos in two similar and coupled nonlinear systems [100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 110; 111]. The unpredictable behavior of chaos provides a means of security since chaotic signals are hard to decode by a third party (called an eavesdropper). The chaotic signal is used as a carrier to mask a message in the time or frequency domain. The synchronization of a chaotic receiver with the chaotic emitter is then used to retrieve the message. In mathematical notation,  
* an emitter generates a chaotic signal \(x(t)\),
* a message signal \(s(t)\) is superimposed onto \(x(t)\),
* the signal \(r(t)=x(t)+s(t)+n(t)\) is sent to the receiver through the communication channel,
* a receiver is synchronized to the emitter so that \(y(t)=x(t)\),
* signal \(s(t)\) is retrieved at the receiver by taking the difference between \(r(t)\) and \(y(t)\). Secure chaos communication was first realized in nonlinear electronic circuits [89]. In order to provide higher-speed encryption and be compatible with optical communication networks [112], later efforts have been focused on optical systems. Among the many optical systems studied in the field, the study of chaotic semiconductor diode lasers has been most fruitful. This type of laser, which is the preferred light source in telecommunications, has been an ideal test bed for many fundamental problems in nonlinear dynamics. The state-of-the art cryptosystems using diode lasers are now able to transmit Gb/s messages through a commercial fiber network of size 100 km [113]. The success of secure chaos communications depends on the realization of synchronization in two chaotic systems. While synchronization of periodic oscillators has been well-known since [PERSON] offered a mechanism in the seventeenth century, synchronization of chaotic systems was quite a surprise initially, since most researchers thought the exponential divergence in chaotic systems would prevent two chaotic systems from synchronizing. Amazingly, chaos synchronization can be proven analytically and demonstrated in laboratory experiments. To see the idea, let us consider two diffusively coupled dynamical systems,  
\[x^{\prime} =F(x)+\alpha(y-x)x^{\prime}=F(x)+\alpha(y-x)\] \[y^{\prime} =F(y)+\alpha(x-y)y^{\prime}=F(y)+\alpha(x-y) \tag{58}\]  
Here, \(x\) and \(y\) are both vectors, \(x^{\prime}=F(x)\) is a chaotic system, and \(\alpha\) is the parameter that couples the system \(x\) and \(y\). An invariant subspace of the coupled system is given by \(x(t)=y(t)\). If this subspace is locally attractive, then the two systems can synchronize perfectly. The role of \(\alpha>0\) is to suppress the divergence between the \(x\) and the \(y\) systems: in general, the larger the \(\alpha\), the easier the synchronization. To find the critical \(\alpha\), let us focus on \(v=x-y\). Assuming \(v\) to be small, we can then use Taylor series expansion. Further assuming that higher order nonlinearities can be neglected, we obtain a linear differential equation  
\[v^{\prime}=DF(x(t))v-2\alpha v \tag{59}\]Here, \(DF(x(t))\) is the Jacobian of the vector field along the solution.