# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.2 Defining Properties of Chaotic Systems  
The most important property of chaos is sensitive dependence on initial conditions. It means that a very small difference in the initial condition may lead to a completely different trajectory. To appreciate this property, one may imagine a butterfly flapping its wings sometime on a day in the Amazon rain forest. This contributes to a minor change in the global air currents. If the motion on that day is chaotic, then sunny weather in some city, say Ney York, could have been replaced by a rainy weather not long after the flapping of the butterfly's wings. One may contrast this feature with a the traditional view, largely drawn from the study of linear systems, that small disturbances only produce proportional effects. Under the latter scenario, in order for the motion of the system to be random, the number of degrees of freedom has to be infinite.  
Being the most important property of chaos, sensitive dependence on initial conditions has to be quantified. This is achieved by equating this property with an exponential divergence of nearby trajectories in the phase space. Let \(d(0)\) be the small distance between two arbitrary trajectories at time \(0\), and let \(d(t)\) be the distance between them at time \(t\). Then, for true low-dimensional deterministic chaos, we have  
\[d(t)\sim d(0)e^{\lambda_{1}t} \tag{21}\]  
where \(\lambda_{1}\) is called the largest positive Lyapunov exponent. This property of sensitive dependence on initial conditions of chaos can be conveniently illustrated by the chaotic Logistic map:  
\[x_{n+1}=\mu x_{n}(1-x_{n}), \tag{22}\]  
where \(\mu=4\). We can generate, for example, \(100\) initial conditions by using uniformly distributed random numbers, and iterate the Logistic map to get \(100\) trajectories. We then perturb each of the initial conditions by a small error of \(10^{-4}\) and regenerate the \(100\) trajectories. The evolution of the errors between the original and the perturbed trajectories is shown in Figure 9. Clearly we observe that the logarithm of the errors first increases with time linearly to about a time of \(n=25\), then is saturated. Linear growth in a logarithmic scale amounts to exponential growth. By visual inspection, we can identify that \(\lambda_{1}\) here is close to \(0.7\) (more precisely, \(\ln 2\), which will be explained shortly). That errors very soon saturate is due to the fact that \(x\) defined by the logistic map is in the unit internal, as is the absolute value of the errors.  
Figure 8: Evolution of point clouds in the chaotic Lorenz system: magenta, red, green, and blue correspond to \(t=0,2,4,6\), respectively.  
The largest positive Lyapunov exponent for the Henon map and the chaotic Lorenz system we discussed in Section 2.3.1 can also be conveniently computed based on time series data. This will be discussed shortly.  
The trajectories of a chaotic attractor are bounded in the phase space. This is another fundamental property of the chaotic attractor. The ceaseless stretching due to exponential divergence of nearby trajectories, and folding from time to time due to boundedness of the attractor, make the chaotic attractor a fractal, characterized by  
\[N(\epsilon)\sim e^{-D},\ \ \epsilon\to 0\, \tag{23}\]  
where \(N(\epsilon)\) represents the (minimal) number of boxes, with linear length not larger than \(e\), needed to completely cover the attractor in the phase space. \(D\) is called the box-counting dimension of the attractor. Typically, it is a nonintegral number. For the chaotic Henon and Lorenz attractor, \(D\) is 1.2 and 2.05, respectively.