Welcome back. I hope that all of my readers had a restful winter break and are having a fabulous start to their new semester. In this first blog post of 2024, I am excited to share some brief remarks on the rich and growing field of nonlinear dynamics and chaos. Fundamentally, this field is centered around studying nonlinear systems of ordinary differential equations (ODEs) that can behave in very complicated ways. The solutions to these equations evolve along trajectories in abstract mathematical spaces called phase spaces. For the simplest differential equations, these trajectories move neatly along power-law curves, or in loops, or in spirals. However, for the subset of nonlinear equations that are deemed chaotic, these trajectories follow mysterious fractal structures in phase space that have non-integer dimension. These structures are called strange attractors, and they can have beautiful shapes that look like works of modern art. In the next few paragraphs, my goal is to provide a flavor of this branch of mathematics and hopefully inspire further studies.
To begin, we shall review the basics of dynamical systems theory, and we will do this by exploring an illustrative example. Suppose we have a hurricane that is moving across the South Atlantic and is about to make landfall. To improve hurricane preparedness measures, we want to track this hurricane and hopefully make some generalizations about hurricane behavior. To make quantitative measurements, we need a mathematical model of the hurricane, and this mathematical model must balance accuracy with computational tractability. An ambitious student might want to track the motion of every single molecule of air so as to have the most accurate results possible. While this is a noble endeavor, it is computationally intractable on even the fastest supercomputers. A more reasonable model, therefore, might be to track the motion of a few select locations along a hurricane’s structure to give a decent, general overview of its dynamics. To do this, we will create a tall n-dimensional state vector, x, which will include the positions, velocities, and pressures at the select locations along the hurricane that we wish to measure. This state vector will tell us everything we can know about our system (which is the hurricane) at any specific instance in time.
To map the evolution of x, we can use the Navier-Stokes equations, which govern the motion of all fluids, to find a vector field, f, that relates the current value of x to the rate at which it is changing in time. This magic f will give us an n-dimensional dynamical system
dx/dt = f(x, t).
Theoretically, the behavior of all solutions to this system could be visualized by attaching the vector dx/dt to every point x in the phase space, which is the space of all possible state vectors. As the Hurricane follows a physical path along the surface of the Earth, the state vector x will follow a virtual path in the n-dimensional space of all possible states. This virtual path can illuminate the underlying structure of hurricane behavior. If x gets closer and closer to a specific point in phase space, then the hurricane in real life would be entering a steady state. If the trajectory taken by x starts looping around and repeating itself, then the real hurricane would be undergoing an oscillation. In a more complicated circumstance, x could start winding around a high dimensional torus, but never exactly get back to any of its previous positions. In this case, the hurricane would be undergoing quasiperiodic motion. This is motion that almost looks like it repeats itself, but never exactly does. Once we have this sort of insight on how the hurricane behaves, we can better equip ourselves to live in its presence, and perhaps, one day control its behavior.
In 1962, a meteorologist at MIT by the name of Edward Lorenz was performing a similar kind of analysis when he observed behavior that was more complicated than periodic motion or even quasiperiodic motion. The strange trends that he observed were one of the earliest known examples of mathematical chaos. Using previous derivations done by meteorologist Barry Saltzman, Lorenz had found a low-dimensional model of convection in the Earth’s atmosphere that only involved three scalar variables: x, y, and z. The quantity x was related to the intensity of convection, while y was related to the temperature difference between ascending and descending currents of air, and z quantified the nonlinearity of the vertical temperature distribution. This nonlinear model looked like
dx/dt = σ(y-x)
dy/dt = x(ρ – z) – y
dz/dt = xy – βz,
and he had his assistants simulate the evolution of x, y, and z on an old-fashioned computer that took up an entire room. When his assistants started re-running simulations that started at just slightly different initial conditions, they were surprised to find that the results quickly became wildly different. This observation flew in the face of previous intuition, which suggested that trajectories starting close together in a deterministic system should stay close together. Upon further investigation, it became apparent that the equations not only had sensitive dependence on initial conditions, but also that trajectories traced out a coherent, 2.06-dimensional object (now known as a strange attractor) that formed the centerpiece of the dynamics. Aside from having a non-integer dimension, this attractor possessed a dense periodic orbit and exhibited topological transitivity. By dense periodic orbit, we mean that there was a solution trajectory that was periodic but got arbitrarily close to all points on the attractor. By topological transitivity, we mean that given any open neighborhood on the attractor, there was a solution trajectory that started from that neighborhood and passed through all other neighborhoods on the attractor. If we painted the attractor with imaginary fluid of many different colors, then this topological transitivity condition would cause the fluid to mix together, and all of the colors to blend.
This type of behavior has defined what a chaotic system is, and Lorenz’s discovery has inspired decades of research on how to better understand chaotic dynamics. For two excellent introductions to the subject that provide a precise definition of chaos with all of the mathematical terminology, I recommend Steven Strogatz’s Nonlinear Dynamics and Chaos and Guckenheimer and Holmes’ Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Since the time of Lorenz’ discovery, it has become apparent that chaos appears frequently across math and science. Examples include turbulent fluid flows, the double pendulum, the human bloodstream, and numerical schemes such as the Newton-Raphson Method. Therefore, it has become important for engineers to study this math and learn how to work with it when modeling complex systems.
In future blog posts, I look forward to discussing chaos and its associated derivations in more detail. Until then, please take care.