# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.3 A Taste of Analysis  
In order to better understand the key concept of chaotic dynamics, the sensitive dependence on initial conditions, let us engage in some analytic analysis. In practice, if one can identify from the problem a transformation similar to the following map, then one can be more than excited,  
\[x_{n+1}=2x_{n}\mod\ 1, \tag{24}\]  
This is a map on the unit interval, where \(x\) is positive, and \(\mod\ 1\) means that only the fractional part of \(2x_{n}\) is retained as \(x_{n+1}\). The map can also be written as  
\[x_{n+1}=\left\{ \begin{array}{ll}2x_{n},&0\leq x_{n}<1/2\\ 2x_{n}-1,&1/2\leq x_{n}<1,\end{array} \right. \tag{25}\]  
This map in fact acts as a Bernoulli shift [62], or binary shift, since if we represent an initial condition \(x_{0}\) in binary form  
\[x_{0}=0.a_{1}a_{2}a_{3}\cdots=\sum_{j=1}^{\infty}2^{-j}a_{j}, \tag{26}\]  
then  
\[x_{1}=0.a_{2}a_{3}a_{4}\cdots,\]  
\[x_{2}=0.a_{3}a_{4}a_{5}\cdots,\]  
and so on, where each of the digits \(a_{j}\) is either 1 or 0. Now it is clear that when \(x_{0}\) is a rational number, the trajectory is periodic. In fact, we can easily find cycles of any length. Figure 9: Error growth in the logistic map. For example, if \(x_{0}\) is a 3-bit repeating sequence, such as \(x_{0}=0.001001001\cdots\), then the trajectory is periodic with period 3. Since there are infinitely more irrational numbers than rational numbers in \([0,1)\), an arbitrary initial condition \(x_{0}\) will be an irrational number with probability 1, and will almost surely generate an aperiodic, chaotic trajectory. Since after each iteration the map shifts one bit, a digit that is initially very unimportant, say the 80 th digit (corresponding to \(2^{-80}\approx 10^{-24}\)), becomes the first and the most important digit after 80 iterations. This is a vivid example that a small change in the initial condition makes a profound change in \(x_{n}\). Clearly, the largest Lyapunov exponent \(\lambda_{1}\) here is \(\ln 2\). Next, let us re-consider the logistic map with \(\mu=4\). If we make a transformation,  
\[x_{n}=\sin^{2}(2\pi y_{n}) \tag{27}\]  
then the logistic map becomes the Bernoulli shift map discussed above. Therefore, the largest Lyapunov exponent \(\lambda_{1}\) for the logistic map with \(\mu=4\) is also \(\ln 2\), as we already mentioned. Now that we have gained some understanding by considering simple model systems, we can discuss how to characterize general chaotic systems. For a chaotic dynamical system with dimensions higher than 1, first we need to realize that exponential divergence can occur in more than one direction, and possibly in many directions. That means we have multiple positive Lyapunov exponents. We denote them by \(\lambda^{+}\), among them, the largest one is usually denoted as \(\lambda_{1}\). How are these Lyapunov exponents related to the rate of creation of new information, or in other words, loss of prior knowledge, in the system? To find the answer, we may partition the phase space into boxes of size \(\epsilon\), compute the probability \(p_{i}\) that the trajectory visits box \(i\), and finally calculate the Shannon entropy \(I=-\sum p_{i}\ln p_{i}\). For many systems, when \(\epsilon\to 0\), information increases with time linearly [63]  
\[I(\epsilon,t)=I_{0}+Kt \tag{28}\]  
Here, \(I_{0}\) is the initial entropy, and \(K\) is the celebrated Kolmogorov-Sinai (K-S) entropy [16; 17]. Now let us consider the situation that all the initial conditions of the system are confined in a small region in the phase space. In this case, the initial probability in the chosen small region is 1, and 0 in all other regions. Therefore, \(I_{0}=0\). For a chaotic system, because of the exponential divergence, the number of phase space regions visited by the system after a time of \(T\) is \(N\propto\epsilon^{(\sum\lambda^{+})T}\), where \(\lambda^{+}\) are the positive Lyapunov exponents we have already explained.