# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Essentials of Chaos Theory
#### 2.3.5 Chaotic Time Series Analysis  
In this big data era, data of all kinds, including time series data, have been accumulating explosively. Many techniques developed in the context of chaotic time series analysis will be of tremendous value for the analysis of all kinds of complex time series data whenever linear approaches are not sufficient. Below, we explain briefly but systematically all the main components of chaotic time series analysis. **A. Optimal embedding**  
Often, a complicated dynamical system described by \(d\vec{U}/dt=f(\vec{U})\) lives in a high-dimensional phase space, where \(\vec{U}\) is a vector. In many situations, we may only be able to access a single variable, say \(x\), instead of many components of \(\vec{U}\). In the simplest case, \(x\) is just a component of \(\vec{U}\), say \(U_{1}\). In general, \(x\) may be a function of \(\vec{U}\). From \(x(t)\), how much can we deduce the behavior of the dynamical system? The answer is a lot can be learned from \(x\), thanks to the Takens embedding theorem. The basic procedure is to construct vectors according to the following equation [69; 70; 71],  
\[V_{i}=[x(i),x(i+L),...,x(i+(m-1)L)], \tag{32}\]  
where \(m\) is the embedding dimension and \(L\) the delay time. More explicitly, we have  
\[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\] \[V_{2} =[x(t_{2}),x(t_{2}+\tau),x(t_{2}+2\tau),...,x(t_{2}+(m-1)\tau],\] \[\vdots\] \[V_{j} =[x(t_{j}),x(t_{j}+\tau),x(t_{j}+2\tau),...,x(t_{j}+(m-1)\tau], \tag{33}\] \[\vdots\]  
where \(t_{i+1}-t_{i}=\Delta t\) and \(\tau=L\Delta t\). We thus obtain a discrete dynamical system (i.e., a map),  
\[V_{n+1}=M(V_{n}). \tag{34}\]  
If the original dynamical system has an attractor with a boxing counting dimension \(D\) defined by Equation (23), then so long as \(m>2D\), topologically the dynamics of the original system described by \(d\vec{U}/dt=f(\vec{U})\) are equivalent to that described by Equation (34). In this case, the procedure using the delay coordinates is called an embedding. In proving this theorem, two properties of differential equations play key roles: (1) for any initial condition, a set of ODEs has a unique solution, and this ensures that trajectories corresponding to different initial conditions in the phase space do not intersect in the phase space; (2) a  
Figure 11: Bifurcation diagram for the Henon map. trajectory corresponding to a specific initial condition does not self-intersect in the phase space; when \(m\) is sufficiently large, self-intersection will be fully eliminated. In practical applications, \(m\) and \(L\) have to be determined according to some optimization procedure. To appreciate the issue, let us consider the harmonic oscillator described below, which is among the simplest dynamical systems:  
\[\frac{d^{2}x}{dt^{2}}=-\omega x. \tag{35}\]  
Of course, we can also write it as  
\[\frac{dx}{dt}=y,\ \frac{dy}{dt}=-\omega x, \tag{36}\]  
The general solution is  
\[V_{1} =[x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),...,x(t_{1}+(m-1)\tau],\]0  
Here, the phase space is a 2D plane with coordinates \(x\) and \(y\). Now consider the case that we can only measure \(x(t)\). Using the embedding procedure with \(m=2\), we obtain \(V(t)=[x(t),x(t+\tau)]\). Figure 12 shows embeddings with \(\tau=T/40,\ T/8,\ T/4\), where \(T=2\pi/\omega\) is the period of the oscillation. When \(\tau=T/4\), the difference between the two components, \(x(t)\) and \(x(t+\tau)\), in terms of angle is \(\pi/2\). With this angle difference, the cosine function becomes the sine function. That is, \(x(t+\tau)\) becomes \(y(t))\). Therefore, the reconstructed dynamical system is the same as the original one. In this simple example, the minimal embedding dimension \(m\) is 2, and the optimal delay time \(L\) is 1/4 of the period. The consequence of using this optimal delay time is that the motion in the reconstructed phase plane is the most uniform--the phase velocity is the same everywhere in the case of Figure 12c, but not in those of Figure 12a,b.