Short History of Nonlinear Dynamics

A paper I wrote skimming the surface of nonlinear dynamics

Introduction

Linearization of functions has been a main method for dealing with curves for all of mathematical history. Turning a circle into an n-gon, approximating functions based on the differential, etc. However, in the study of dynamical systems linearization actually hides certain phenomena. Chaotic behavior doesn't exist in linear systems, that's why we choose to make things linear, but we miss out on very interesting math when we do so.

Poincaré

In his work New Methods in Celestial Mechanics, Poincaré formalized the study of systems and demonstrated stability theory for fixed points and for periodic orbits. Stability in dynamical systems can be demonstrated through Poincaré maps. A Poincaré map is effectively an \((n-1)\)-dimensional slice of \(n\)-dimensional phase space through which a periodic orbit travels transversally, with points marked as the orbit passes through the slice (called the Poincaré section). To better illustrate, let \(S\) be a Poincaré section of space passing through the Earth and the Sun, oriented perpendicular to the orbit which the Earth takes around the Sun. Every time the moon occupies a point on \(S\) it is plotted, as if it were piercing a piece of paper. These "piercing holes" will be called \(x_n\) with n being the order which the point appears on S. Let this continue for a few iterations, the Poincaré map is be the mapping which sends each point on S to the place it next shows up on S. So it would map \(x_{n}\) to \(x_{n+1}\) so we can say \(x_{n+1} = P(x_{n})\). After sufficient iterations a pattern of points may begin to clearly emerge, depending on the stability of the orbit being measured. Poincaré was able to demonstrate this orbital stability by using fixed points on the Poincaré map. A point on the Poincaré map is fixed if \(x_{n+1} = x_{n}\) and stable if points near it stay near to it forever. If other points tend toward this fixed point, it can be called asymptotically stable. Using this map method Poincaré proved that an orbit passing through the Poincaré section is stable if and only if the point it passes through is stable on the map. Similarly, the entire orbit is asymptotically stable if and only if the point passed through is asymptotically stable.

Attractors

To begin discussion of attractors it is prudent to discuss them using manifolds. Stable and unstable manifolds formalize the general idea of attractors and repellers respectively. A stable manifold is where for a fixed point of p, all positions in the manifold converge to p after enough iterations. Drawing on discussion of Poincaré maps, the stable manifold would be the set of points for which \(p\) is asymptotically stable. One could imagine sweeping a Poincaré section around until each portion of the phase space had been covered and using those slices of the 2D to draw out the 3D stable manifold of the orbit (like a circle becoming a cylinder). Formally, the stable set (the list of elements stable to \(p\)) can be described as follows. \[ W^{s}(f,p) = q \in X: f^{n}(q) \longrightarrow p \,\, as\,\, n \longrightarrow \infty \] An unstable manifold for the fixed point \(p\) is the manifold for which all positions diverge from \(p\). This can be demonstrated by inverting the iterated function which describes the stable manifold. Effectively stating: the unstable set of \(p\) is the set of all points which converge to \(p\) when iterating backwards. \[ W^{u}(f,p) = q \in X: f^{-n}(q) \longrightarrow p \,\, as\,\, n \longrightarrow \infty \] An attractor can be a point, a set of points, a line, a manifold, or can have a fractal structure. Introduced initially by Poincaré in 1881, a limit cycle is a closed loop shaped attractor in 2D space.

_{The two outer orbits converge toward the limit cycle.}

Concerning fractal attractors, we can imagine throwing a dart at the Mandelbrot set and if it lands in the set, moving one way, if it lands outside, moving a different way. Because fractals are self similar, when close to the edge of a fractal structure, arbitrarily close points might not both be on the structure. A strange attractor is also often chaotic, meaning that points on the attractor may end up arbitrarily far from each other on the attractor. In this way the system holding a strange chaotic attractor would be globally stable (points settle into consistent subspaces called attractors) but locally unstable, another way of saying points on these attractors will not generally follow paths through phase space based on approximate location on the attractor. A strange attractor is not always chaotic, but non-chaotic strange attractors have been known to be difficult to find in nature. The two features of self-similar borders between attractors and broader phase space, and the chaotic nature of attractors themselves is what leads to the sensitive dependence on initial conditions we now call the butterfly effect, but this name wouldn't be spoken until Edward Lorenz.

Lorenz

By the time Lorenz was nearing chaos theory, he was working at MIT, and had recently been put in charge of a project that ran simulations of weather models used to evaluate statistical forecasting techniques. This was about the time he started becoming skeptical of linear statistical models in meteorology. Working on a simple computer in 1961, Edward Lorenz and his associate Ellen Fetter wanted to get a second look at some weather predictions, and had a printout of the initial conditions modelling the movement of air currents. However after running the simulation again the results varied drastically from what they had originally been. The computer had only printed out the initial variables up to 3 decimal places. For instance \(.56133\) would become \(.561\). These findings took a while to spread to other disciplines though, as Lorenz wasn't one to self-promote. It wasn't until 1972 after Lorenz was finally convinced to speak at a conference that the first biologist noted it, Robert May suggested that populations of species fluctuate in a chaotic fashion. Now chaotic behavior is found in a large variety of systems, like heartbeats and riverbed erosion. Lorenz knew this would change how meteorological prediction was thought of, as he stated, "When our results concerning the instability of non-periodic flow are applied to the atmosphere, which is ostensibly non-periodic, they indicate that prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly." (Lorenz) Indeed, adapting to these discoveries, meteorologists now use ensemble forecasting, making large amounts of predictions based on slight variations in initial conditions, and comparing data between them.

Conclusion

Now that the study of nonlinear systems has found its way into mathematics it has been expanded greatly. Some mathematicians like Stephen Smale have expanded on attractors in nonlinear systems, while others like Ali H. Nayfeh applied it to engineering for use in construction of cranes, buildings, spacecraft and other machines and structures. Much more can be said on the topic but this hopefully lays a good outline of the larger concepts and events in nonlinear systems.

References

Lorenz, Edward Norton. “Deterministic nonperiodic flow.” Journal of the Atmospheric Sciences 20 (1963): 130-141.

Teschl, Gerald. "Ordinary Differential Equations and Dynamical Systems." American Math Society 140 (2012): 196-200

That's it for this one! Thanks for reading.
~Bone🦴