Archive for the ‘Nonlinear Models’ Category

Nonlinear Dynamics and Chaos:

Thursday, April 9th, 2015

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Steven H. Strogatz by Daniel Lathrop.

For someone with Steven Strogatz‘s track record on lucid explanations, it comes as no surprise that Latrop says:

In presenting the subject, the author draws from the past 30 years of developments that have advanced our understanding of dynamics beyond the linear examples—for instance, harmonic oscillators—that permeate current physics curricula. The advances came from theoretical and computational scholars, and the book does a great job of acknowledging them. The methods and techniques that form the bulk of the book’s content apply useful concepts—bifurcations, phase-space analysis, and fractals, to name a few—that have been widely adopted in physics, biology, chemistry, and engineering. One of the book’s biggest strengths is that it explains core concepts through practical examples drawn from various fields and from real-world systems; the examples include pendula, Josephson junctions, chemical oscillators, and convecting atmospheres. The illustrations, in particular, have been enhanced in the new edition.

The techniques needed to understand the behavior of nonlinear systems are inherently mathematical. Fortunately, the author’s excellent use of geometric and graphical techniques greatly clarifies what can be amazingly complex behavior. For example, in carefully working through the development and behavior of the Lorenz equations, Strogatz introduces a simple waterwheel machine as a model to help define terms and tie together such key concepts as fixed points, bifurcations, chaos, and fractals. The reader gets a feel for the science behind the differential equations. Moreover, for each concept, the mathematics is accompanied by clear figures and nicely posed student exercises.

Rate this one as must buy!

Ahem, I guess you noticed that political science, sociology, psychology, sentiment, etc. aren’t included in the title?

Blake LeBaron of Brandeis University penned an answer to “Has chaos theory found any useful application in the social sciences?,” which reads in part:

One of several key ideas in chaos is that simple models can generate very rich (and random-looking) dynamics. Implicit in some early work in the social sciences was a hope that simple chaotic models of social phenomena could be matched up with many of the near-random and difficult-to-explain empirical patterns that are observed. This early goal has proved elusive.

One problem is that determining if a time series was generated by deterministic chaos is not easy. (A time series is a data set showing the state of a system over a period of time–a sequence of voting results, for instance, or the fluctuating price of gold.) There is no single statistic capable of being estimated that indicates what is going on in a social system. Also, many common time-series problems (such as seasonality and trends) can confuse most of the diagnostic tools that people use. These complications have led to many conflicting results. Building an easy-to-use test that can handle the intricacies of a real-world time series is a tough problem, one which will probably not be solved anytime soon.

A second difficulty is that most of the theoretical structure in chaos is based on purely deterministic models that have no noise, or at most just a very small amount of noise, affecting the dynamics of the system. This approach works well in many physical situations, but it does not offer a very good picture of most social situations. It is hard to look at social systems isolated from the environment in the way that one can analyze fluid in a laboratory beaker. Once noise plays a major role in the dynamics, the problems involved in analyzing nonlinear systems become much more difficult.

A great illustration of why “low noise” techniques have difficulty providing meaningful results when applied to social systems. Social systems are noisy, very noisy. That isn’t to say you can’t ignore the noise and make decisions on the results, but you could save the consulting fees and consult a Ouija Board instead. Social engineering programs, both liberal and conservative in the United States suffer from a failure to appreciate the complexity of human interaction.

PS: I did see in the index that Steven cites Romeo and Juliet but I will have to await the arrival of a copy to discover what was said.

Nonlinear Dynamics and Chaos

Tuesday, May 27th, 2014

Nonlinear Dynamics and Chaos – Steven Strogatz, Cornell University.

From the description:

This course of 25 lectures, filmed at Cornell University in Spring 2014, is intended for newcomers to nonlinear dynamics and chaos. It closely follows Prof. Strogatz’s book, “Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering.” The mathematical treatment is friendly and informal, but still careful. Analytical methods, concrete examples, and geometric intuition are stressed. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. A unique feature of the course is its emphasis on applications. These include airplane wing vibrations, biological rhythms, insect outbreaks, chemical oscillators, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. The theoretical work is enlivened by frequent use of computer graphics, simulations, and videotaped demonstrations of nonlinear phenomena. The essential prerequisite is single-variable calculus, including curve sketching, Taylor series, and separable differential equations. In a few places, multivariable calculus (partial derivatives, Jacobian matrix, divergence theorem) and linear algebra (eigenvalues and eigenvectors) are used. Fourier analysis is not assumed, and is developed where needed. Introductory physics is used throughout. Other scientific prerequisites would depend on the applications considered, but in all cases, a first course should be adequate preparation.

Storgatz’s book “Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering,” is due out in a second edition in July of 2014. First edition was 2001.

Mastering the class and Stogatz’s book will enable you to call BS on projects with authority. Social groups are one example of chaotic systems. As a consequence, the near religious certainly of policy wonks on outcomes of particular policies is mis-guided.

Be cautious with those who response to social dynamics being chaotic by saying: “…yes, but …(here follows their method of controlling the chaotic system).” Chaotic systems by definition cannot be controlled nor can we account for all the influences and variables in such systems.

The best you can do is what seems to work, most of the time.