# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.1 Phase Space and Transformation
The format of the survey data determines that people surveyed in a nation will belong to one of the 12 different categories. To fully utilize the notion of space, we can associate each category with a box. Instead of putting every person belonging to that category at one single point (e.g., the center of the box), we can generate two uniformly distributed random variables as the coordinate of the person in the corresponding box. Please see Figure 6. With such a visualization, one can immediately see the abundance of each category. When WVS data of different waves (times) are used, one can then examine variation of the percentage of people in each category over time for a nation, compare among different nations, deduce functional relationships between these two variables, and classify nations in the world into different clusters. Note Figure 6 may be called phase space ensemble based visualization, where an ensemble amounts to a participant in the survey. Next, let us consider transformations in phase space. A good way to grasp the idea is to imagine the following situation: on a very woody day, a little boy went outside with a sheet of paper in his hand. He grabbed a handful of sand and put it on the paper. Then he released the paper in the air. How would the sand be swept across the sky? One could even think that originally the boy had arranged the sand to resemble the face of a person. How would the face be twisted by the wind? To make this discussion more concrete, we can consider how a unit circle is transformed by the Henon map [61]:  
\[x_{n+1} = 1-ax_{n}^{2}+y_{n},\] \[y_{n+1} = bx_{n}, \tag{19}\]  
where \(a=1.4,b=0.3\). Figure 7 shows the successive (from left to right and top to bottom) images of the unit circle after \(n=1,\cdots,5\) iterations. Note that the fifth image is basically the Henon attractor one can find in textbooks, journal papers, or certain web sites. It is usually obtained by choosing an arbitrary initial condition and iterate the Henon map long enough. If the trajectory does not diverge, then after removing the transient points (which are the first few points here), the remaining trajectory (not connected by lines) will be very similar to the fifth image shown here. In our ensemble scenario, we observe that just after one iteration, the unit circle is already changed to a very different shape, and by the fourth iteration, the shape of the image is already very similar to the Henon attractor. By now, one could easily understand that the Henon attractor can either be readily obtained from an arbitrarily shaped phase space region (discarding initial conditions which lead to the divergence of the iterations) or by iterating a single arbitrary initial condition many times. The equivalence of the two approaches, one based on the evolution of ensembles in the  
Figure 6: Phase space diagram of religious participation vs. happiness for the USA based on wave 7 of the World Value Survey Data. phase space, the other based on long-time iterations, is a clear manifestation of the ergodic property of the Henon map (and more generally, chaotic systems). To enhance our understanding of the materials discussed so far, let us visually observe how chaos manifests itself in the chaotic Lorenz system:  
\[ \begin{array}{l}dx/dt=-16(x-y),\\ dy/dt=-xz+45.92x-y,\\ dz/dt=xy-4z.\end{array}  \tag{20}\]  
For this purpose, let us arbitrarily choose an initial condition, \((-17.3432,\)\(-24.5966,\)\(40.1096)\), perturb it \(2500\) times using standard Gaussian random variables with very small variance, and monitor the evolution of all those points. These initial conditions are shown in Figure 8 as a magenta block centered at our chosen initial condition. After 2 units of time, these initial conditions spread to the points labeled as red in the Figure. After another 2 units of time, the red points further evolve to the points labeled as green. Two more units of time later, the green points become the blue points.