Our son is home from college for the Thanksgiving holiday. In his backpack were his Organic Chemistry books and a library book from school. Fractals and Chaos: Simplified for the Life Sciences. This book has guidance for understanding chaos and avoiding the mistakes of using populist descriptions for systems found in nature - in his case celluar biology and organic chemistry students. (Yes a double major)
One popular misunderstanding is that chaotic systems are not deterministic. The common - but wrong - example is the double pendulum. This book describes the background for the actual fact that Chaotic Systems are Deterministic.
Chaos is defined as a complex output that mimics random behavior that is generated by a simple, deterministic system.
This is worth repeating - the chaotic system MIMICS random behavior. This is the error in seeing the double pendulum pattern as random, but not realizing this pattern is generated by a deterministic set of equations - the Lagrangian solution to the equations of motion.
Randomness is different from chaos, but it is hard to tell. Although sequences of values or behavior appear similar but underlying these numbers or behavior are two distinctly different processes. We can only tell the difference when we look at the phase space view of the numbers.
In chaotic systems- dynamical - there are only a small number of independent variables, but the output of a chaotic systems is complex. But it is not random. The sequence of values of measured data can be transformed into an object in space. The Phase Space. This object is the Phase Space Set. Some properties of these numbers is easier to analyze in this phase space.
If there is a algorithm for a chaotic system and we run that algorithm with different starting conditions we get different outcomes. This is the double pendulum. But that outcome is NOT random. It is deterministic, but different. The deterministic equations of motion for the double pendulum are simple algebra and trigonometry. The outcome is dependent on the starting conditions, but the output is NOT random, within certain constraints. There is an angle above which the result is random and chaotic.
The next book that came home is A Bee in a Cathedral. This is a book about using analogies to convey information about analogies and their connection with other topics. Unlike the poor analogy of an untended garden can be an analogy emergent system - well it is emergent, an emergent weed patch. This book uses analogies to convey understandings about things like scaling factors. Or a bowling ball on a rubber sheet as an analogy for general relativity and gravity. Or unshuffling a deck of cards can not be done no how many times you try - entropy in action. Or an eye opener, a troop of Chimps has more genetic diversity than the entire human population. This is because we are very young compared to chimps.
So What's the Point?
When you read something about complexity, about analogies, about a process or a method and there is not a reference to the source of the statement that has itself been sourced, be suspect. And absolutely most important, that those sources are peer reviewed in some way. This eliminates most populist approaches to solution to complex problems.
Both books are worth reading if only to show how organized thought processes are based on more organized thought processes, whose underlying concepts - even if they are written in a populist style - can be directly traced to the underlying equations. The double pendulum to the Lagrania of the equations of motion in a gravitational field.
