Here a system is an entity of interest that exists in time.  For comprehension and exploration, an analyst creates an abstraction of the system.  The analyst then refers to this abstraction as “the system.”

The dynamics of the analyst’s system is how that abstraction changes over time. Ideally, the abstract system and the dynamics are correlated with the real system and its observed behavior.

Quantitatively, the abstraction has a set of state variables.  These variables sufficiently describe the state of the abstraction. Typically, the dynamics are expressed as a set of differential equations involving these state variables and variables describing any external inputs to the system.

Example systems include an engine, a car, and an airplane.  However, anything existing in time is a system.  For example, the global economy consisting of individual agents form a system.  The more complex the system the more difficult to find a useful abstraction that sufficiently captures the real system’s complexity.

For Humans, because of the dimension reducing nature of our brains, seeing what is over what is believed is impossible.  Because of this, the abstraction becomes what is believed, and we can’t see what is.

Linear Dynamics are Unicorns

During the 1970s and 1980s, across disciplines, academics wrestled against an ingrained bias towards simple, linear system abstractions.  There was a blindness towards the “messy” but necessary complexity.  In fact, linearity was ingrained deeply in the Engineering curriculum.

The linear engineering problems are solved by applying a series of steps leading to a singular answer.  Any qualitative understanding of the dynamics is ancillary.  Yes, we studied root-locus in Control Systems and we studied Smith charts in E-mag.  But, in my opinion, we students quickly lost sight of the whole racing to the singular answer at the end of the problem solving algorithm.  Problems like this are easy to get right and easy to grade.

BUT THIS FOCUS LEADS TO A VIEW THAT IS INSIDIOUSLY DISTORTED.  Why “insidious”?  This viewpoint blinds us to the real world. Linear Dynamical Systems are unicorns.  They don’t exist.  Sure, linear abstractions were more than useful to learn but not at the occlusion of the real-systems.

In fact, because linear dynamics were desired, analysts abstracted real world system to linear abstractions even knowing the abstractions were a poor representation.

Non-Linear Dynamics Art

Steven Strogatz wrote:

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity)

This book steps through non-linear system dynamics focusing on qualitative understanding leading to a real understand of differential equations (linear and non-linear).

Following Dr. Strogatz’s solutions demonstrates how working analytically with non-linear equations can be more an art than a science.  It reminded me of finding anti-derivatives in calculus.  The vista is holistic.

Not having an algorithm leading to a solution is scary.  However, even if you can’t find a solution, the insight from exploring the dynamics of these example problems is useful in real life.  Assuming your real-life involves making abstractions and understanding the dynamics of these abstractions.