# 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  
In practice, whenever one has a dynamical system model described by discrete maps or differential equations, then the first thing one needs to consider is if the model has a unique fixed point solution, and if yes, if the solution is locally or globally stable. If the model contains some controlling parameter(s), then one also has to consider if the qualitative feature of the solution changes with the parameter(s), and if yes, find out what kind of changes they are. One can also think if any features of the system are shared by systems in other fields. The last point is the universality issue. These considerations make it clear that studies of bifurcations, routes to chaos, and universality are of fundamental importance to the study of dynamical systems. Fixed point solutions are one of the the limiting behaviors of dynamical systems. It turns out the limiting behaviors of dynamical systems are very rich. In order of increasing complexity, they are fixed points, limit cycles, torus, chaos, turbulence, and random motions [38]. Fixed points correspond to motions without any change; limit cycles correspond to periodic motions. We have already mentioned these two in the beginning of this section. Torus corresponds to quasi-periodic motions, i.e., the motion is characterized by two or more independent frequencies. Periodic and quasi-periodic motions may be associated with crystals and quasi-crystals, finding of the latter won Professor [PERSON] a Nobel Prize in Chemistry in 2011. Fixed points, limit cycles, and torus all belong to regular motions. Since chaotic and regular motions appear almost everywhere, we should ask if a chaotic motion may arise from a regular motion, and vice versa. Interestingly, the answer can be found by studying bifurcations and routes to chaos in dynamical systems. Here, it is critical to realize that the qualitative behaviors of the dynamics of a system may change when one or more controlling parameters are changed. The parameter values that cause such qualitative changes are called bifurcation points. To better understand the notion of transitioning from one state to another, let us briefly consider the anti-globalization movement. As often reported in the media, anti-globalization activities are often accompanied with grandeur and truly praiseworthy ideals such as better democratic representation, advancement of human rights, fair trade, and sustainable development. However, this is only part of the story. The more fundamental cause of the anti-globalization movement is the flipping of power ranking among the participating countries--a country afraid of losing competitive edges or even being demoted to a lower position in the power ranking would attribute that to unfair trade, infringement of intellectual property rights, etc. While these concerns are not entirely unfounded, one has to realize that reward to countries participating in economic globalization cannot be linearly proportional to their ranking. As a result, rearrangement of the power ranking surely will occur. Here, the basic parameter controlling the transition from globalization to anti-globalization is associated with the rearrangement of the (relative) power ranking among the participating countries. To understand bifurcations, let us analyze the logistic map described by Equation (22) again. Let us set \(\mu=2\) and iterate the map starting with an initial condition \(x_{0}=0.3\). With simple calculations, we can easily find that \(x_{n}\) soon equals 0.5 after a few iterations. If we choose \(x_{0}=0.5\), then \(x_{1}=x_{2}=\cdots=0.5\). This means that 0.5 is a stable, fixed-point solution. While it is easy to prove this statement rigorously [38], here, let us resort to simulations: For any \(\mu\), where \(\mu\in[2,4]\), we choose an arbitrary initial value of \(x_{0}\), and iterate Equation (22).