Strange Attractor (1)
A dynamic system is a set of mathematical equations in which one or more variables change according to time. Time is thus the independent variable while the other variables are dependent, their values depend on time. Values that do not change are called constants or parameters. The number of dependent variables is called the degree of the system. Historically, all variables were assumed to be continuous and solutions were found using the calculus of Newton and Leibniz and what are called analytic methods for differential equations. One such analytic method is the concept of linearity. Linearity means that if you multiply the input of a system by a number you get the output multiplied by that number. The same thing happens when you add two inputs, the output is the sum of the inputs.
Additivity: f(x+y) = f(x) + f(y).
Homogeneity of degree 1: f(αx) = αf(x) for all α. (18)
Another concept is time invariance. A system is time invariant if a shift in the start time of the input does not effect the output. Dynamic systems that have the mathematical structure of a line or can be manipulated into that structure and are time invariant can be solved. Today, linear dynamic systems with hundreds of thousands of variables can be solved through the analytic methods of matrix algebra with computers. Given some starting point, the complete time evolution of a system can be solved for both future and past and this doesn’t change if one changes the starting point.
Dynamic systems can also have more than one independent variable, for instance a variable that changes spatially in addition to temporally. These are called partial differential equations (PDEs). While linear differential equations require an initial value, PDEs require boundary conditions, a set of initial conditions. PDEs do not have the rich analytic structure of differential equations so solutions are mostly cookbook. There are also equations that include integrals and can be mixtures of two or all three different types. Modern dynamical theory does some transformation that usually adds more degrees to the system. There is a problem with this, most of these mixed systems are non-linear. Any dynamical system that is not linear is non-linear. This means that some variable has an exponent or is in an exponent or is a trigonometric function or is multiplied by another variable, etc. In 1890, Poincare showed that all non-linear dynamical systems degree 2 or greater could not be solved analytically. In addition, he developed a way to geometrically view the dynamics of such systems. Work has been carried out sporadically until a big push during World War II then the advent of the digital computer changed everything.
Numerical methods to solve linear differential equations were developed during the 19th century during both World Wars teams of calculators (mostly women) would use these methods to solve DEQs. During the 1950s and 60s many numerical algorithms were developed. Note that these used approximations of continuous time and space, not purely discrete variables. In the 1940’s John Von Neumann invented cellular automata, where both space and time are discrete. It should be noted that this is a mathematical construct, whether time and space are physically discrete or continuous is still a debated question. Cellular automata was for years considered a toy system although simple generation rules can create amazing complex patterns, both static and dynamic (Game of Life (17).)
Logistic system, S curve, and logistic map (2)(3)(4)
In the 1970s, Robert May, a mathematical biologist at Oxford University, was studying the logistic model of biological growth, the famous S curve. This dynamic system is non-linear but there is an analytic solution to it. May tried a simple discrete time version of called an iterative or recurrence map. For certain values of a single parameter the equation blew up generating what is now called chaos or a fractal set, an infinite set of points on a line that never repeats.
Lorenz Attractor (5)
During this same time the atmospheric scientist, Edward Lorenz, discovered the Lorenz Attractor, a chaotic system from a third degree non-linear DEQ. Thus, chaos had been found in a one degree discrete system and a third degree continuous system. So is there a general structure to all dynamical systems? Another way to define an independent variable is either stochastic or deterministic. Stochastic means random and stochastic variables can be either continuous or discrete through what is called a markov process. Discrete, continuous; non-linear, linear; stochastic, deterministic; are all of these different ways to define dynamic systems equivalent? Turns out the answer is yes. This is because of the work of the topologist Stephen Smale at UC Berkeley during the 1960s and 1970s. Smale is perhaps one of the greatest mathematicians of the 20th century who founded world class mathematics departments at Berkeley, in Brazil and Hong Kong and is still alive today working at the University of Michigan in Flint, his hometown. Smale is also an activist who was active in the Free Speech Movement at Berkeley and in 1965 helped organize the Vietnam Day Committee’s teach-in with Jerry Rubin (15). What he accomplished mathematically is what is now called the Modern Theory of Dynamical Systems (16), an introduction to which I have. It is a thick dense book.
t must be noted that although continuous, discrete, and stochastic systems are all equivalent topologically, they can give slightly different results. For instance, discrete non-linear systems like the logistic map can have chaotic dynamics with one variable while it takes three variables before a continuous non-linear system shows chaos.
Dynamic models are dynamical systems of all types used to describe some aspect of Nature. Stephen Smale describes the use of models as such:
Good mathematical models don't start with the mathematics, but with a deep study of certain natural phenomena. Mathematical awareness or even sophistication is useful when working to model economic phenomena for example, but a successful model depends much more on a penetrating study and understanding of the economics. (7)
Models also need to be testable by experimental data and the parameters evolution in time and space of the model have to fit experimental data. This is a major critique of model based science. In 1972 the French mathematician Rene Thom published a book titled Structural Stability and Morphogenisis An Outline of a General Theory of Models (12) and in 1975 an English translation was released. An eccentric book to be sure but one full of interesting quotes like this one:
Thought is a continuous orgasm (12)
In 1977 a colleague, Christopher Zeeman published Catastrophe theory: Selected papers, 1972–1977 (11) and outlined a project to transform the sciences using Catastrophe theory. After a bunch of bombastic claims and fawning press coverage a series of strong critiques, including the one above by Smale (7), derailed the momentum of the project. In 1971 another one of Thom’s colleagues, David Ruelle along with Florence Takens published On the nature of turbulence (10) in which they proposed that turbulence in fluid flow is chaotic. It took Ruelle a series of papers and a decade to get the physics community to accept this. A mathematical explanation of the onset of turbulence was one of Hilbert’s 10 problems for 20th Century mathematics to solve. This is still an unsolved problem and is now a 21st Century unsolved problem. This did break the barriers to scientific interest in modern system dynamics. It should be noted that there is nothing wrong with Thom’s mathematics, his project to bring topological ideas into applied mathematics was successful, just not only his ideas. In 1980 Ilya Prigogine, who was a friend and philosophical antagonist of Thom helped found the Sante Fe Institute (6). This organization has turned into a major force for the circulation of dynamic systems modeling into the sciences.
In late 1979 I attended lecture at the University of Arizona Math Dept by Arthur Winfree (20) on the dynamics of biological time. He started the lecture by declaring that he is only working with continuous models and would have nothing to do with time discrete ones. I was a bit shocked but didn’t have the knowledge to challenge him although I was at the time looking at discrete Lidenmayer systems (19) and had found a paper linking them to Lyapunov functions and thus to dynamic system theory. This was unfortunately my last day at the University for almost 30 years and I didn’t realize that he had stayed there until his tragic death from brain cancer right before I returned.
Dynamic modeling has transformed modern science. What is striking is that this has become a computational problem. There are even computer languages like NetLogo (13) for developing what are call Agent Based Models. Dynamics on networks are a major research field. Complex Adaptive Systems are at the pinnacle of this research (14). These are multiple systems that interact and evolve together and include most biological systems and many non-biological ones.
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Wilenski, Uri. “NetLogo Home Page.” Accessed November 15, 2020. https://ccl.northwestern.edu/netlogo/.
Holland, John H. Signals and Boundaries Building Blocks of Complex Adaptive Systems. MIT Press, 2014. https://mitpress.mit.edu/books/signals-and-boundaries.
Wilson, Emily. “Finally Done: A New Biography of Jerry Rubin Fills a Surprising Void.” Cal Alumni Association, April 16, 2018. https://alumni.berkeley.edu/california-magazine/just-in/2018-04-16/finally-done-new-biography-jerry-rubin-fills-surprising-void.
Katok, Anatole, and Boris Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications. Cambridge: Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511809187.
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Lindenmayer, Aristid. “Mathematical Models for Cellular Interactions in Development I. Filaments with One-Sided Inputs.” Journal of Theoretical Biology 18, no. 3 (March 1, 1968): 280–99. https://doi.org/10.1016/0022-5193(68)90079-9.
Winfree, Arthur T. The Geometry of Biological Time. Springer Science & Business Media, 2001.
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