# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 4 Multiscale Analysis with the Scale-Dependent Lyapunov Exponent (SDLE)
However, when we define SDLE using Equation (103), we have not made any assumptions, except \(\Delta t\) being small (usually taken to be the sampling time interval). As we will see, chaos is characterized by a constant \(\lambda(\epsilon)\) over a range of \(\epsilon\). In the computation of SDLE, we first examine which embedding vectors defined by Equation (32) fall within the series of shells defined by Equation (40). Then, the evolution of those vector pairs \((V_{i},V_{j})\) can be monitored, and their average behavior of divergence (not necessarily exponential) can be computed. So far as exponential or power law divergence are concerned, we can exchange the order of taking the logarithm and averaging. Then, Equation (104) becomes  
\[\lambda(\epsilon_{t})=\frac{\left\langle\ln\left\|V_{i+t+\Delta t}-V_{j+t+ \Delta t}\right\|-\ln\left\|V_{i+t}-V_{j+t}\right\|\right\rangle}{\Delta t} \tag{106}\]  
where \(t\) and \(\Delta t\) are measured in terms of the sampling time, and the average, denoted by the angle brackets, is over all indices \(i,j\) with their corresponding vectors satisfying Equation (40). The program for computing SDLE is explained in detail in [202], and can be obtained from the authors. The major scaling laws of SDLE that are most relevant for analyzing complex data are summarized below [189]:  
* For deterministic chaos, \[\lambda(\epsilon)\sim\text{constant},\] (107) Amazingly, this property can even be observed in finite high-dimensional data, including the Lorenz'96 system, which has dimensions close to 30 [193], and in turbulent  
Figure 29: A schematic showing how a small distance between two nearby trajectories grows with time. isotropic fluid with an integral scale Reynolds number reaching 6200 [203]. In such systems, estimation of dimensions is infeasible. * As observational data are always contaminated by noise, it is important to have a scaling law for noisy chaos and noise-induced chaos [82; 118]. The law reads \[\lambda(\epsilon)\sim-\gamma\ln\epsilon,\] (108) The law pertains to small scales, and \(\gamma>0\) controls the speed of information loss. * For \(1/f^{2H+1}\) processes, \[\lambda(\epsilon)\sim He^{-1/H}.\] (109)
* For \(\alpha\)-stable Levy processes, \[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]0 (110)
* For stochastic oscillations, both scaling laws \(\lambda(\epsilon)\sim-\gamma\ln\epsilon\) and \(\lambda(\epsilon)\sim He^{-1/H}\) can be observed when different embedding parameters are used. * When the dynamics of a system are very complicated, one or more of the above scaling laws may manifest themselves on different \(\epsilon\) ranges. It is now clear that with the help of these scaling laws, distinguishing chaos from noise can be readily solved. More importantly, we can now understand very well the nature of each type of behavior of the data by obtaining the defining parameters for that behavior. To illustrate how SDLE characterizes chaotic features and the effect of noise, let us briefly discuss Boolean chaos in a ring oscillator. Boolean chaos normally refers to the continuous time dynamics of a system of interconnected digital gates whose output updates are not regulated by an external clock. Recently, an alternative Boolean architecture for generating chaotic oscillations was proposed by [PERSON] et al. [204]. See \[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]1 Three typical kinds of waveforms for the variable \(v_{3}\) are shown in \[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]2 The chaotic behaviors of the oscillations can be aptly characterized by SDLE, as shown in Figure 32--the Figure actually has shown more than chaos: the chaotic behavior is best defined for the variable \(v_{1}\), and the effect of noise is most clearly visible for the variable \(v_{3}\). The reason is straightforward: in this series circuit, the noise at the third gate is the largest. Among the many properties of SDLE, two make it unique.