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Geometric Analysis and Partial Differential Equations Seminar
May 29, 2013 @ 10:30 am - 11:30 pm
Dr Marianito Rodrigo (University of Wollongong)
Continuous dynamical systems, Laplace transforms and second-degree polynomial equations
A solution trajectory of a linear system of first-order ODEs can be visualised as a curve in R^n (the “time domain”). Its Laplace transform can also be viewed as a curve in R^n (the “frequency domain”). The curve in the time domain can be quite complicated. How does the corresponding curve in the frequency domain look like? In R^2 the curve always lies on a conic section. In R^3 it always lies on the intersection of two quadric surfaces. More generally, in R^n the curve lies on the intersection of second-degree polynomial equations. For nonlinear systems of ODEs, the solution trajectory can be made to lie close to such an intersection of second-degree polynomial equations provided the frequency variable is large enough. I will also show some numerical simulations. This talk is aimed at a general audience