### Impact on Engineering Landscapes  
Important engineering prblems demand entropic waveforms such as those produced by the chaotic Lorenz system. These problems include the evaluation of machine learning algorithms as well as the growing density of co-located communication systems and ranging sensors. First, consider early sensors and communication systems that were not densely located. For example, early warning radar systems from the 1940s generally detected targets that initially did not contain electronics competing with signal processing goals. Now, frequency modulated continuous wave (FMCW) radars comprise one of the many sensors in consumer automobiles [4]. A modern issue is that these FMCW waveforms are naturally incompatible in dense sensor networks due to the signal processing errors that arise as these waveforms interfere and cause 'ghost' targets or 'jam' one another [35]. Similarly, a growing number of devices communicate. Modern communication systems must carefully share resources via channel allocation and clever spread spectrum techniques. Many emerging applications and technologies demand communication and sensing with high data rates. Worse, trends such as the Internet of Things (IoT) exacerbate resource-sharing issues as smart devices become prevalent [36]. Personal devices that monitor health and perform real-time sensing, learning and communication see an increasing need for security, authentication and waveform diversity [4; 37]. Chaotic systems naturally solve some of these issues, but often require analysis beyond traditional engineering methods. Unfortunately, chaotic systems almost always lack equations that guide engineering design or quantitative measure. This fact is not surprising as nonlinear, coupled, ordinary differential equations (ODEs) almost always preclude closed-form analytic solutions [38]. Some chaotic behavior may be analyzed using traditional (albeit sometimes advanced) techniques for ODEs [22; 39; 40]. However, many standard signal processing techniques rely on assumptions of linear time-invariant (LTI) systems [26] (such as superposition, convolution, etc.) and are not helpful when nonlinearity is introduced [14]. This restriction complicates the inclusion of chaotic systems in engineering practice. Despite the need for advanced analysis, chaos is a notably mature topic with vast contributions of depth and importance. One especially effective tool set, called symbolic dynamics, takes an information-theoretic approach to understanding chaos by abstracting continuous variables as symbols in discrete partitions [41]. The intersection of symbolic dynamics, communication and ergodic theory gave rise to innovations by [PERSON] who translated chaotic dynamics into engineered waveforms with expressions compatible with standard engineering practice [21]. This previous work paired concepts from symbolic dynamics with conventional communication techniques to extract basis functions from chaotic signals. Surprisingly, these basis functions may be recombined through linear superposition via a convolution sum to produce engineered waveforms that approximate those generated by chaotic systems. This result implied that these waveforms can keep their desirable chaotic properties while allowing optimal detection via a matched filter receiver. These results are not only surprising, but important as traditional signal processing methods for LTI systems [26] are generally considered to be ill-suited and discouraged for use with nonlinear differential equations [14; 38]. Unfortunately, the novelty and multidisciplinary nature of [PERSON] work [21] may tend to impede engineers and dynamicists from adopting methods developed under these previous efforts despite their direct applications to current machine learning and waveform design issues. We note that the cross-pollination of topics that involve chaos, information theory, communication and signal processing still grows in importance today and is compounded by trends in machine learning and IoT as previously discussed.