## 2 Materials and Methods
### The Chaotic Lorenz System: Sensitivity, Information Creation & Symbolic Dynamics
Fortunately, these types of algorithms often converge more quickly than the rule-of-thumb indicates (\(\approx\)10,000 points for Lorenz) and usually requires \(\approx\)100 orbits of the attractor. It has been noted that the Lorenz system poses significant challenge due to the ill-defined orbit period [42]. These irregular timings can be observed in the peaks and valleys indicated in the time-series data in Figure 1.  
The magnitude of the exponents indicate a bit-rate for which the system creates or destroys information by translating its dynamics into a quantitative measure using the lens information theory [42, 43]. The units of the exponents for continuous-time flows (differential equations) are \([\frac{\text{bits}}{\text{orbit}}]\) or \([\frac{\text{bits}}{\text{s}}]\) and \([\frac{\text{bits}}{\text{Iteration}}]\) for discrete-time iterated maps (difference equations). The aforementioned local orbit divergence gives an important distinction  
Figure 1: Runge-Kutta (MATLAB ODE45) integration of the Lorenz equations showing (**a**) a partitioned phase space defining symbols from a Poincaré section (red dots) of the chaotic attractor (black) and (**b**) the time series of the \(x(t)\) variable (black) with peaks (red) defining symbols from a Poincaré section. regarding information content produced by the system. A Lyaponuv exponent, in spirit, characterizes the _stretching mechanisms_ of a chaotic system that generates new information from the the micro-scale (precision of initial conditions beyond our measurement/memory capacity or small noise perturbations in a heat bath) and not the _folding mechanisms_ that keep the chaotic system bounded. [PERSON] et. al. reported positive Lyaponuv exponents of the Lorenz system to have a magnitude of \(2.16\left[\frac{\mathrm{bits}}{\mathrm{s}}\right]\) with parameter values \(\sigma=16\), \(\rho=45.92\), and \(\beta=4\). Initial conditions were specified using 20-bit precision (one part per million) and the prediction of the resulting time-series could not be extended beyond 9 s on average yielding \(20\left[\mathrm{bits}\right]/2.16\left[\frac{\mathrm{bits}}{\mathrm{s}}\right]\). This result indicates the Lorenz system will tend to uncover \(\approx\)20 bits of new information from a spherical neighborhood of initial conditions with a radius corresponding to 20-bit precision in \(\approx\)20 orbits. Methods related to the traditional analysis of ODEs and numerical analysis through observing time-series provide invaluable tools for understanding the dynamics of the Lorenz system as well as developing signal processing algorithms for engineering applications. However, abstraction of the system into symbols and the study of the resulting symbolic dynamics provides deep insight that pairs directly with information and communication theory. In practice, these techniques sample a trajectory's state at discrete intervals and study the symbolic sequences obtained through partitioning (or thresholding) rules that coarse-grain the system's time-series. Though, these techniques are not new, they are notably powerful and continue to strengthen an underlying connection between nonlinear dynamics and information theory [2; 21; 24; 44]. The symmetry of the Lorenz attractor has been used to construct partitions that aid in symbolic analysis. A partitioned state space that corresponds to abstracted symbols \(A\) or \(B\) may be studied as symbolic dynamics that represent various underlying behaviors too burdensome to uncover through the traditional analysis of ODEs. Importantly, symbolic dynamics serves as the appropriate tool to study the Lorenz system's periodic orbits. Historically, symbolic dynamics naturally permitted the study of the stable but elusive 'period-5' orbit which traditional methods tend to overlook [45]. In fact, symbolic dynamics has illuminated an encompassing view of the Lorenz system through dissection of the system's periodic orbits [46]. These orbits form the skeleton of the the system's attractor.