# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 4 Multiscale Analysis with the Scale-Dependent Lyapunov Exponent (SDLE)
One is its skill of dealing with nonstationarity, including detecting intermittent chaos from models as well as observational data [155; 195]. To understand intermittency, it is useful to consider the evolution of river flow dynamics over 1 year. With some thinking, one can readily realize that the time period may be divided into two periods, wet and dry, where the wet season may be associated with frequent rain and snow melting, and the dry sea may be largely associated with no or little rain, and constant evaporation. The river flow dynamics must be very different in these two periods. Since standard methods for detecting chaos assume the existence of a single chaotic attractor, those methods are ill positioned to unambiguously determine whether river flow dynamics are chaotic or not. To illustrate how intermittent chaos can be detected by SDLE, Figure 33 shows an example of the Umpgua river in Oregon. The exponential divergence is evidently shown by the linear \(\ln\epsilon(t)\) vs. \(t\) curve for \(t\) going from about 20 days to about 100-150 days. Consequentially, there are well-defined plateaus of SDLE, i.e., a constant SDLE, shown in Figure 33a2 (the blue curves). It is also interesting to note the scaling law of Equation (108) on small scales. This is caused by the faster-than-exponential growth of small distances in the initial period (less than 20 days), and it is mainly due to stochasticity, i.e., randomly driven by snow melting, rain, etc., besides measurement noise. The chaotic and the noisy dynamics depicted in Figure 33 can be improved by using the adaptive algorithm discussed earlier. The results using the filtered data are shown in Figure 33 as the red curves. \[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]3 (**a**) A three inverter ring oscillator. (**b**) A ring oscillator driven by an external periodic signal. The resistor-capacitor stages may represent either discrete components or the finite bandwidth of non-ideal inverters. \[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]4 Typical oscillations displayed by an experimentally implemented ring oscillator. (**a**) Self oscillations occur with the input held constant above the threshold. (**b**) Slow driving produces periodic bursts of self oscillation. (**c**) Faster driving produces an irregular oscillation. The other unique property of SDLE is that it provides a unified framework to understand other complexity measures. Concretely, the values of other complexity measures can be inferred from the values of SDLE at specific scales. This statement is best appreciated by using signals with phase-transition-like changes (or regime changes). Because of this, let us use electroencephalography (EEG) data with epileptic seizures. A typical result is illustrated in Figure 34, where we observe that the temporal variations of the Lyapunov exponent, the correlation dimension, the correlation entropy, and the Hurst parameter are similar to the values of SDLE either on smaller or on larger scales. In fact, the list of the complexity measures can be expanded to include the permutation entropy, the LZ complexity, and the energy of the EEG waves such as \(\alpha,\beta,\delta,\theta\). For the details, we refer to [38]. While here these connections are illustrated using EEG data, the issue is relevant to many other situations, including paleoclimatological data and fMRI data analysis. This  
\[\epsilon_{t+\Delta t}=\epsilon_{t}e^{\lambda(\epsilon_{t})\Delta t}, \tag{103}\]5 Intermittent chaos in the Umpqua river. Shown in (**a1,a2**) are the error growth curves and SDLE curves, respectively. The blue curves are for the original data, while the red curves are for the filtered data. The embedding parameters used in the computation are \(m=6\), \(L=3\). Three different shells specified by Equation (40) are used. These curves collapse on each other, except when \(t\) is small. This highlights that the computational results are essentially independent of the initial shells chosen.