## 2 Materials and Methods
### Regularizing Timing for the Lorenz Waveforms  
Regarding Figure 1, the aforementioned 'ill-defined orbit periods' cause irregular timings in the peaks and valleys that define the Poincare section and (in part) establish the symbolic content in these waveforms. These non-uniform intervals create issues for engineering applications as well as extracting basis functions from the Lorenz system. First, the irregular timing is incompatible with engineered systems that rely on regular clocks. Second, the information content embeded in continuous state variables needs to be extracted with a common time origin and time scale. An important pre-processing step is to provide a means to regularize the timing intervals such that the Poincare returns are uniform. Original waveforms with irregular timing are shown alongside waveforms that have been processed to achieve regular timing in Figure 2. This figure displays both time-series and statistical, histogram data of the time spacing between peaks for an \(x(t)\) waveform with \(\approx\)1000 peaks. The algorithm used here was adapted through an example noted by [PERSON] [21] for the case of sampled systems and is valid for simulations with a known time-step. If the time-series for the \(x(t)\) signal is sampled from peak to peak (including both peaks and valleys) with uniform timing, then the number of samples between the peaks will tend to be non-uniform as some orbits tend to return to the Poincare section slower or faster than others. Slow trajectories will tend to contain more samples between peaks, and fast trajectories will tend to contain less samples between peaks. Regular timing may be imposed on the irregular Lorenz waveform by reproducing the sampled \(x(t)\) with a variable clock rate (i.e., using a digital-to-analog converter). The reproduced \(x(t)\) should be time compressed (sped up) proportional to the number of samples between peaks. This method effectively spaces the peak returns in \(x(t)\) uniformly and results in regular sample rate \(T_{s}\) for the symbolic content of the Lorenz waveform. Loosely, one proposed method to accomplish this algorithmically uses the following steps:  
1. [label=()]
2. measure and store the number of samples & indices between two peaks
3. extend each data segment to convenient a sample interval (i.e., 50 samples or 100 samples)
4. pad new \(x(t)\) values and increment new indices
5. define a regeneration sample rate & corresponding period
6. time stretch/compress segment proportionally by original number of samples
7. repeat for each pair of peaks  
and pseudo-code for this algorithm is given by Algorithm 1. We note that this algorithm has not optimized towards any specific performance metric and is intended to serve as an operational guide to illuminate approaches to implement techniques suggested by [PERSON] [21]. Figure 2: (a) Histograms of timing intervals for the unmodified, irregular Lorenz waveform \(x(t)\) (red) showing a broad distribution and for the timing-regularized Lorenz waveform \(x(t)\) (blue) showing a narrow distribution. (b) Time-series waveform for the unmodified, irregular Lorenz waveform \(x(t)\) (black) with Poincaré sections indicated at irregular time intervals (red dots) and waveform for the timing-regularized Lorenz waveform \(x(t)\) (dashed-blue) with Poincaré sections indicated at regular time intervals (blue dots).