# 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
The normal diffusion with \(\alpha\approx 0.5\) corresponds to Brownian motions around the periodic orbit (or limit cycle), which is clear from Figure 16a. The case of super diffusion with \(\alpha>0.5\) is the very condition for noise-induced chaos to occur. This is shown in Figure 18 and can be readily understood as follows: chaos, which amounts to exponential divergence, can be more easily approached through larger \(\alpha\), especially when \(\alpha\) is larger than \(1\), for a tiny amount of noise. Figure 16: Clean (open triangles) and noisy (filled circles) trajectories for (**a**) \(\mu=3.55\) and (**b**) \(\mu=3.74\) (reproduced from [118]). \[y^{\prime} =F(y)+\alpha(x-y)y^{\prime}=F(y)+\alpha(x-y) \tag{58}\]0 Time-dependent exponent curves for the noisy Logistic map: (**a**) \(\mu=3.55\) and \(\sigma=0.01\); (**b**) \(\mu=3.63\) and \(\sigma=0.005\); (**c**) \(\mu=3.74\) and \(\sigma=0.002\); and (**d**) \(\mu=3.83\) and \(\sigma=0.005\). Six curves, from the bottom up, correspond to shells \((2^{-(i+1)/2},2^{-i/2})\) with \(i=7\), \(8\), \(9\), \(10\), \(11\), and \(12\) (reproduced from [118]). \[y^{\prime} =F(y)+\alpha(x-y)y^{\prime}=F(y)+\alpha(x-y) \tag{58}\]1 Logarithmic displacement curves illustrating the mechanism for noise-induced chaos. Each group actually consists of three curves, corresponding to shells \((2^{-(i+1)/2},2^{-i/2})\) with \(i=12\), \(13\), and \(14\). They basically collapse on each other. The parameters for the four groups are (**a**) \(\mu=3.74\) and \(\sigma=0.0003\); (**b**) \(\mu=3.83\) and \(\sigma=0.001\); (**c**) \(\mu=3.63\) and \(\sigma=0.0003\); and (**d**) \(\mu=3.55\) and \(\sigma=0.0005\). To separate these four groups (**a–d**) of curves from each other, they are shifted by \(2\), \(1\), \(-0.5\), and \(-0.2\) units, respectively, where a positive number indicates shifting upward, and a negative number indicates shifting downward. All four groups of curves are well modeled by \(\ln k^{\alpha}\) with \(\alpha=1.5\), \(1.0\), and \(0.25\) (reproduced from [118]). Let us now come back to chaotic secure communications. Although noise-induced chaos can help with chaos synchronization, and thus chaos communication, the noise level has to be small. Otherwise, chaotic systems will desynchronize, and we will not be able to have any kind of communication at all [82]. In the beginning of this subsection, we have mentioned that recently there is a strong interest in using chaos to rapidly generate random bits in physical devices, for use in cryptography and secure communication. For this purpose, noise is always beneficial. The key here is to test whether a generated sequence of 0's and 1's is truly random. The usual tests for randomness, such as the widely used Statistical Test Suite for random number generator of NIST SP 800-22, basically test whether the distributions of 0's and 1's in the entire and the sub-sequences, as well as recurrences of certain patterns, are consistent with certain random distributions. The degree of divergence of nearby trajectories characterized by the time-dependent exponent curves offer additional information [109]. This is best understood by referring to \[y^{\prime} =F(y)+\alpha(x-y)y^{\prime}=F(y)+\alpha(x-y) \tag{58}\]2 the noise-induced chaos at \(\mu=3.74\) and \(3.83\) is more suitable to be used as fast physical random bit generator than at \(\mu=3.63\). The normal diffusion-like process at \(\mu=3.55\) will not pass the randomness test of NIST SP 800-22 since the dynamics are periodic-like. Finally, as a side comment, we note that the pioneering works on chaos synchronization [100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 110; 111] are not cited evenly. Rather, some were only cited a few times, while the largest citation goes to [100], which is over 12,000 times. To better appreciate this somewhat astonishing behavior, we have listed these works in the reference not chronologically, but in descending order of the citations.