wp-plugin-hostgator domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114ol-scrapes domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/flow-proof-helps-mathematicians-find-stability-in-chaos-20230615\/#comments<\/a><\/br> As with so much in mathematics, the proof started with coffee. In September 2019, Kathryn Mann<\/a> of Cornell University visited Kingston, Ontario, to give a guest lecture at Queen\u2019s University. Afterward, she sat down with her host, Thomas Barthelm\u00e9<\/a>, for what was supposed to be a quick cup of coffee. He wanted to get her opinion on a problem he was working on involving mathematical models called dynamical systems, which describe how phenomena as simple as a pendulum\u2019s back-and-forth motion or as complex as the weather evolve over time.<\/p>\n Before they knew it, hours had passed. \u201cWe were just sitting in this coffee shop, drawing pictures, each of us trying to figure out what the other was trying to say,\u201d Mann said. \u201cAt the beginning, I was like, this guy makes no sense.\u201d But as they learned to speak each other\u2019s mathematical languages, both became more optimistic about their chances of finding a solution.<\/p>\n Mann didn\u2019t always like mathematics \u2014 as a child, she wasn\u2019t good at arithmetic \u2014 but it was precisely this kind of conversation that ultimately led her to study it. Although initially interested in pursuing a career in philosophy, she realized it wasn\u2019t the right fit. For philosophers, \u201ca productive discussion means testing your position against someone else\u2019s,\u201d she said. \u201cMath is the opposite. You talk to someone, and you\u2019re both on the same team from the get-go. If someone\u2019s like, \u2018That doesn\u2019t work that way,\u2019 you\u2019re like, \u2018Oh, tell me more.\u2019 I found that mode of discourse much better.\u201d<\/p>\n Barthelm\u00e9 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.<\/p>\n Each of those properties arises in dynamical systems that mathematicians have wanted to understand for centuries, from the motion of planets around the sun to the spread of disease through a population. But in such systems, disentangling these different features gets hopelessly complicated. Anosov flows were defined as objects of study in their own right in the 1960s because they exhibit these important behaviors to an extreme degree, making them easier to analyze. \u201cYou hope that once you have a perfect picture of this case, you can turn back to the messy real world and approach it with fresh eyes,\u201d Mann said.<\/p>\n \u201cIn the beginning of the theory of dynamical systems, Anosov flows were like lighthouses telling you the direction where you should go,\u201d said \u00c9tienne Ghys<\/a> of the \u00c9cole Normale Sup\u00e9rieure in Lyon, France.<\/p>\n Nevertheless, they have been incredibly difficult to grasp. Though mathematicians have made a lot of progress over the past 60 years, they are still far from achieving the perfect picture Mann spoke of: a classification of all the different kinds of Anosov flows.<\/p>\n Now, in a series of<\/a> recent papers<\/a>, Barthelm\u00e9 and Mann, together with Steven Frankel<\/a> 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. (Their result also holds for related but more general dynamical systems called pseudo-Anosov flows.) Hidden in all that chaos, they found structure.<\/p>\n The mathematicians have already applied their result to address key questions about Anosov flows \u2014 namely, how to construct them, and how many of them there could be. \u201cSometimes results are important because they really provide a new point of view on the subject,\u201d said Rafael Potrie<\/a>, a mathematician at the University of the Republic in Uruguay. \u201cThat\u2019s what happens here.\u201d<\/p>\n Since the late 19th century, when Henri Poincar\u00e9\u2019s work in celestial mechanics jump-started the modern theory of dynamical systems, mathematicians have thought about dynamics through the lens of geometry.<\/p>\n Consider a pendulum. Left alone, it hangs vertically, unmoving. But if you lift it and let go, it will swing back and forth. At a given moment, the state of the pendulum can be captured by two pieces of information: its angle from the vertical hanging position and its velocity. As a result, you can represent all possible states of the pendulum as points on a plane, which is known as the state space.<\/p>\n If you start at any one of those points, a differential equation (based on Newton\u2019s laws of motion) tells you how the pendulum\u2019s angle and velocity will change over time. This motion is captured by a curve, or \u201ctrajectory,\u201d winding through the state space. If you change the starting angle and velocity of the pendulum, you\u2019ll get a different trajectory through the state space.<\/p>\n You want to study all such trajectories as a single mathematical object. This geometric way of encoding your dynamical system is called a \u201cflow.\u201d Instead of thinking about the pendulum carving out arcs through the air, you can study its behavior by analyzing the flow. In this case, the flow consists of nested ellipses (with each ellipse representing a way the pendulum can oscillate back and forth), as well as curves above and below those ellipses (representing scenarios in which the pendulum spins rapidly like a pinwheel).<\/p>\n But flows can get a lot more complicated, involving convoluted, higher-dimensional state spaces. Take 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. (To describe just one particle\u2019s state, you need six pieces of information: three numbers for its position, and three for velocity. So a description of all 100 particles at once requires 600 numbers.)<\/p>\n To develop tools for studying these more complicated systems, mathematicians needed the right testing grounds \u2014 systems simple enough to make sense of, but complicated enough to reflect the properties they were actually interested in.<\/p>\n That\u2019s where Anosov flows came into play.<\/p>\n Even before Poincar\u00e9\u2019s 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.<\/p>\n A geodesic flow describes all possible ways a particle can move when not subject to any outside forces. The state space, in general, is four-dimensional: Two of these dimensions correspond to the particle\u2019s physical position, while the other two correspond to how fast it\u2019s moving. But if you\u2019re willing to throw away some information \u2014 if, for instance, you don\u2019t care how fast the particle is moving but only which way it\u2019s facing at any moment in time \u2014 you could describe its motion using a three-dimensional state space. Two dimensions describe its position, and the third dimension represents the direction it\u2019s facing. That\u2019s how mathematicians often think about geodesic flows: as a collection of trajectories through a three-dimensional state space.<\/p>\n These flows piqued mathematicians\u2019 interest for a number of reasons. Since the late 1800s, they have allowed geometers and topologists to better understand the structure of very complicated surfaces \u2014 surfaces that are too difficult to study directly, but which become more tractable when you look at how particles move on them. And they\u2019re just as crucial for dynamicists, because many mechanical systems in physics can be represented as geodesic flows.<\/p>\n By the dawn of the 20th century, the French mathematician Jacques Hadamard had started to investigate geodesic flows on \u201cnegatively curved\u201d surfaces \u2014 that is, surfaces that look like a saddle at any given point. (This is perhaps impossible to visualize, but mathematically useful.) He found that these kinds of geodesic flows are always chaotic: If you tweak a particle\u2019s initial position by a tiny amount, it will end up on a vastly different trajectory, meaning that you can\u2019t predict the system\u2019s long-term behavior. (The geodesic flow on the surface of a sphere, on the other hand, does not have this property.)<\/p>\n \u201cIn some ways, these systems \u2026 are maximally chaotic,\u201d said Andy Hammerlindl<\/a> of Monash University in Australia.<\/p>\n In the 1960s, building on Hadamard\u2019s work, 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. Under typical circumstances, such structural stability is by no means guaranteed \u2014 but it is for these kinds of geodesic flows. \u201cThe tag line is \u2018global stability, local chaos,\u2019\u201d Mann said. \u201cIn dynamics, you\u2019re really interested in this confluence of stability and chaos.\u201d The two coexist in many dynamical systems, striking a subtle and crucial balance that mathematicians have been trying to disentangle since Poincar\u00e9\u2019s work on our solar system.<\/p>\n Anosov discovered that both chaos and stability arise automatically in a geodesic flow because its trajectories converge and diverge like lines drawn on a piece of taffy as it is squeezed together and stretched out.<\/p>\n Anosov generalized the concept of a geodesic flow on a negatively curved surface by writing down particular mathematical conditions for such taffy-like behavior. These generalized flows now bear his name. Any geodesic flow on a negatively curved surface is one. But Anosov thought there might be a whole litany of dynamical systems that act this way.<\/p>\n Although this definition might seem oddly specific, such stretching and compressing can be found in many dynamical systems. But it\u2019s not as obvious or ubiquitous in those contexts. In an Anosov flow, the taffy-like behavior appears everywhere \u2014 making it an extreme case and therefore a particularly good model system for the development of new tools and insights.<\/p>\n Just as scientists might try to learn about gene expression in a fruit fly before moving on to humans, mathematicians have proved results about topological, statistical and other properties in Anosov flows and then extended that work to other dynamical systems. For example, in the 1970s, mathematicians used what they knew about Anosov flows (and related systems) to formulate a conjecture about what kinds of flows can exhibit structural stability. In the 1990s, Shuhei Hayashi<\/a> of the University of Tokyo proved that it was true<\/a>.<\/p>\n 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. (Most of these are flows in a three-dimensional state space. Higher-dimensional Anosov flows remain poorly understood.)<\/p>\n<\/div>\n <\/br><\/br><\/br><\/p>\n
\nFlow Proof Helps Mathematicians Find Stability in Chaos<\/br>
\n2023-06-16 21:58:43<\/br><\/p>\nA Geometric Perspective<\/strong><\/h2>\n
Global Stability, Local Chaos<\/strong><\/h2>\n
Shortcut to Identification<\/strong><\/h2>\n