On the Importance of Visualization in Nonlinear Dynamics: From Huygens' Clocks to Hurricanes

Many systems in science and engineering are governed by nonlinear evolution equations. Starting with the familiar case of the simple pendulum, I will discuss some of the ways that nonlinear systems differ from linear ones and discuss how nonlinearity can lead to unexpected behavior. One famous example is the so-called "butterfly effect" where small differences in initial conditions can lead drastically different long term behavior. Another consequence of nonlinearity is that systems can have multiple solutions for a given set of parameters. The competition between solutions often leads to very complicated spatial and temporal dynamics. As it turns out many of the most time honored mathematical techniques used in the analysis of linear systems fail miserably in the analysis of the complex systems, forcing us to resort to more qualitative and graphical approaches. As an example of this, I will discuss recent advances in identifying a special class of solutions of the Navier-Stokes equations (the governing equations for fluid flows) and how these are being used being used to shed light on one of the oldest open problems in mechanics, the study of hydrodynamic turbulence. Subject: Science Colloquium Series Run Length: 00:53 Author: Daniel Borrero Publisher: Linfield College Copyright: 2015