Symbolic Dynamics Methods for Low Dimensional Dynamics

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Proposal: DMS-0407110 PI: Divakar Viswanath Institution: University of Michigan Ann Arbor Title: Symbolic dynamics methods for low-dimensional dynamics ABSTRACT Symbolic dynamics is an important and central concept in the mathematical description of low-dimensional chaos. The leading principle of symbolic dynamics is the construction of symbol sequences that encode segments of trajectories. As nearby trajectories of a chaotic system diverge rapidly, even long segments of trajectories will have short symbolic codes. Furthermore, these symbolic codes explicitly indicate various recursive relationships between families of trajectories. In earlier work, the investigator used symbolic dynamics to obtain explicit plots of the fractal structure of the well-known Lorenz attractor, which was inferred by Lorenz in 1963 but was not exhibited explicitly prior to the investigator's work. The investigator develops mathematics and computational algorithms to apply the symbolic dynamics method to low dimensional Hamiltonian systems. A particular focus of this project is on physically interesting instances of the three-body problem. In particular, this project studies resonance transitions in the three-body problem with a view towards the motion of celestial bodies and spacecrafts. In collaboration with other researchers, the investigator explores new methods that apply to dissipative partial differential equations with low dimensional attractors. The investigator uses results from this project and related research to transmit the excitement of research to an interdisciplinary audience of undergraduate and graduate students. Climactic changes, weather prediction, the motion of planets and other celestial bodies, spacecraft trajectories, and the flow of hydrogen through the channels of a fuel cell, to consider but a few examples, present quite different scientific challenges. Yet all these diverse phenomena are described by the same type of mathematical objects called nonlinear differential equations. The use of mathematics leads to concepts of quite great generality that can illuminate very diverse phenomena. An example is the concept of resonance, which means that the ratio of the period of motion of one part of the system to that of another part is a simple fraction such as 1/2, 2/1, or 1/1. When two parts of a system are in resonance, the two parts interact much more strongly than otherwise leading to unexpected results, one of these unexpected results being the loss of predictability. The investigator studies resonances related to the motion of planets, satellites, and spacecrafts. The investigator develops methods that make accurate computations possible in spite of the loss of predictability. The key to these methods is a concept that utilizes the loss of predictability to give compact descriptions of changing and unpredictable phenomena. This concept is called symbolic dynamics.