# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.4 Bifurcations, Routes to Chaos, and Universality
After discarding the initial iterations so that the solution of the map has stabilized, we retain a large number (say, 100) of the value of the iterations, and form a scatter plot of those values with \(\mu\). When the map has a globally attracting fixed-point solution, then the recorded values of \(x_{n}\) will all become the same since the transients have been discarded. In this situation, one only observes a single point with the horizontal axis being the chosen \(\mu\) and the vertical axis being the converged value of \(x_{n}\). For a periodic solution with period \(m\), one can observe \(m\) distinct points on the vertical axis. When the motion becomes chaotic, one observes on the vertical axis as many distinct points as one records (100 in our example). Figure 9(a) shows the bifurcation diagram for the logistic map--the interesting structure is the celebrated period-doubling bifurcation to chaos. Figure 10a embodies more structures than one could comprehend by a simple glance. For example, if one enlarges Figure 10a the small rectangular region containing the period-3 window, then one obtains Figure 10b. We have again observed a period-doubling route to chaos! (To truly understanding the presentations here, it is beneficial for readers new to chaos theory to write a simple program to reproduce Figure 10a,b). Having been observed in many diverse fields, period-doubling bifurcation to chaos is one of the most studied and most celebrated routes to chaos [65]. To better comprehend this universality, it is worth noting that it also underlies the bifurcations in the Henon map (see Figure 11) and the Lorenz system. In fact, the notion of universality can be quantified for the period-doubling bifurcation to chaos, through the Feigenbaum constant defined by  
\[\delta=\lim_{k\to\infty}\frac{\mu_{k}-\mu_{k-1}}{\mu_{k+1}-\mu_{k}}=4.669201 \cdots. \tag{31}\]  
Other routes to chaos also exist. They include the well-known quasi-periodicity route [66] and the intermittency route [67]. The former refers to when a controlling parameter is changed, the motion of the system changes from a periodic motion with one basic frequency, a quasi-periodic motion with two or more basic frequencies, to chaotic motions. This route has been observed in many mechanical and physical systems, including fluid systems. A bit surprisingly, this route has also manifested itself in the Internet transport dynamics (concretely, a variable amounting to the round-trip time of a message transmitting through the Internet can change from periodic and quasi-periodic motion to chaos when the congestion level increases [68]). The third classic route to chaos, intermittency, refers to the behavior that the motion of the system alters between smooth and chaotic modes, again when a controlling parameter is changed. This route to chaos is very relevant to many nonstationary phenomena in nature, including river flow dynamics, which are very different in wet and dry seasons. Figure 10: Bifurcation diagram for the logistic map; (**b**) is an enlargement of the little rectangular box indicated by the arrow in (**a**).