Stability in Chaos

“Although initially interested in pursuing a career in philosophy, she realized it wasn’t the right fit. For philosophers, “a productive discussion means testing your position against someone else’s,” she said. “Math is the opposite. You talk to someone, and you’re both on the same team from the get-go. If someone’s like, ‘That doesn’t work that way,’ you’re like, ‘Oh, tell me more.’ I found that mode of discourse much better.””

“Barthelmé was interested in particular dynamical systems called Anosov flows, which crop up naturally in many areas of mathematics and act as important toy models.”

“These systems showcase seemingly paradoxical properties all in one place: chaos and stability; rigidity and flexibility; the presence of intrinsic geometric structure amid an underlying topological wilderness”

“Now, in a series of recent papers, Barthelmé and Mann, together with Steven Frankel of Washington University in St. Louis, have taken a striking step toward that elusive goal. By translating questions about motion and shape into the language of algebra, they showed that relatively little data is needed to completely and uniquely determine a given Anosov flow”

“Since the late 19th century, when Henri Poincaré’s work in celestial mechanics jump-started the modern theory of dynamical systems, mathematicians have thought about dynamics through the lens of geometry”

“you can represent all possible states of the pendulum as points on a plane, which is known as the state space”

“You want to study all such trajectories as a single mathematical object. This geometric way of encoding your dynamical system is called a “flow.” Instead of thinking about the pendulum carving out arcs through the air, you can study its behavior by analyzing the flow”

“100 particles moving and interacting in space. The flow that captures their behavior is a collection of infinitely many trajectories through a 600-dimensional state space”

“Even before Poincaré’s 19th-century work changed the way dynamical systems were studied, mathematicians were interested in systems where a particle takes the shortest path available: a so-called geodesic. On a plane, particles follow a bunch of straight lines; on the surface of a sphere, they travel along great circles. The topology, or global shape, of the surface affects what these paths look like”

“A geodesic flow describes all possible ways a particle can move when not subject to any outside forces”

“the Russian mathematician Dmitri Anosov observed that if you slightly adjust the equation that defines the geodesic flow, all the trajectories shift just a bit: You can wiggle your original flow into the new one without changing its overall structure”

““The tag line is ‘global stability, local chaos,’” Mann said. “In dynamics, you’re really interested in this confluence of stability and chaos.” The two coexist in many dynamical systems, striking a subtle and crucial balance that mathematicians have been trying to disentangle since Poincaré’s work on our solar system”

“Anosov could only come up with one other family of systems that fit his criteria. But since then, mathematicians have uncovered a sprawling zoo of examples”

“Imagine an Anosov flow as a complicated tangle of infinitely many trajectories, which together fill up a three-dimensional state space like yarn. This state space is what’s known as a manifold. If you zoom in on any part of it, it will look like regular three-dimensional space, but globally, it can have a very complicated structure, full of holes and other strange features”

“The three mathematicians proved that for most Anosov flows (as well as pseudo-Anosov flows), knowing just the closed, or “periodic,” trajectories allows you to completely determine the entire system”

“The fundamental group is effectively a list of loops on the manifold (and all their combinations) that encode information about the manifold’s shape”

“Every periodic trajectory in a given Anosov flow corresponds to a class of loops represented in the fundamental group. According to Barthelmé, Frankel and Mann, for most Anosov (and pseudo-Anosov) flows, knowing this subset is enough to allow you to reconstruct the entire flow”

“Just as the contours of a riverbank affect the possible ways the water in a river can flow, the structure of a manifold affects what sorts of dynamical flows are possible”