# 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  
Phase space is the arena for the evolution of a dynamical system to unfold. It is spanned by all the variables needed to fully characterize the evolution of the system. To help one to better understand the idea, let us start from a system characterized by only  
Figure 5: Distribution for the ratio between sex offenders and the total population in (**a**) Ohio and (**b**) New York (adapted from [59]). two state variables, \(X_{1}\) and \(X_{2}\). Monitoring the system often amounts to examining the waveforms of \(X_{1}(t)\) and \(X_{2}(t)\). One may instead try to examine the trajectory defined by \((X_{1}(t),X_{2}(t))\), where \(t\) now is treated as an implicit parameter. The space spanned by \(X_{1}\) and \(X_{2}\) is the phase space (or state space) we are discussing. They could be position and velocity, for example. Employing phase space facilitates one to study the dynamics of a complicated system with a geometrical viewpoint. For some dynamical systems, irrespective of initial conditions, the trajectory eventually approaches a single point; this is called a globally stable fixed point solution. Of course, the situation could be more complicated. For example, the trajectory may converge to a closed loop, again irrespective of where the trajectory starts. This is called a globally stable limit cycle. The discrete counter part of a limit cycle is a periodic motion with certain period (say \(N\)): the corresponding attractor consists of \(N\) points, and the trajectory amounts to hopping among the \(N\) points with a definite order. To be more familiar with the concept of phase space, it is useful to examine certain experience in daily life. To illustrate the idea, suppose we were going to a meeting by a taxi. On our way, there was a traffic jam, and the taxi got stuck. Afraid of being late, we decided to call the organizer. How would we describe our situation? Usually, we would tell the organizer where we got stuck and how quickly or slowly the taxi was moving. In other words, we actually have been using the concept of phase space as part of our daily language. Although the concept of phase space is among the most basic in dynamical systems theory, its usefulness in geographical science has yet to be seriously explored [60]. To accelerate the coming of a time that phase space becomes as basic in geographical science as in complexity science, it is helpful to discuss two potential applications of phase space in geographical science. One application is top-down, that is, to systematically think about how many independent variables are needed to fully characterize an interesting and important problem in geographical science, and how each variable can be measured. The other application is bottom-up. It is easiest to illustrate the idea by using some variables in the World Value Survey (WVS, accessed on 17 April 2021, [[http://www.worldvaluessurvey.org/wvs.jsp](http://www.worldvaluessurvey.org/wvs.jsp)]([http://www.worldvaluessurvey.org/wvs.jsp](http://www.worldvaluessurvey.org/wvs.jsp))) as an example. WVS is an interesting project that explores values and beliefs of people around the globe, how the values and beliefs evolve with time, and what social and political implications they may have. Since 1981, researchers have conducted representative national surveys in almost 100 countries. During the survey, a lot of variables have been deduced. We show here that phase space offers a convenient geometrical way to visualize the data and identify co-variations of the variables. For this purpose, we choose a variable that gives three levels of religious participation for people in the nations surveyed. The other variable we choose is happiness, which is given in four levels. How are the two variables related? How different are people in different countries in terms of these two variables? To gain insights into these interesting questions, we can form a phase space spanned by these two variables.