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Unformatted text preview: Linear Planar Systems Math 246, Fall 2009, Professor David Levermore We now consider linear systems of the form (1) d d t parenleftbigg x y parenrightbigg = A parenleftbigg x y parenrightbigg , where A = parenleftbigg a 11 a 12 a 21 a 22 parenrightbigg . Here the entries of the coefficient matrix A are real constants. Such a system is called planar because any solution of it can be thought of as tracing out a curve ( x ( t ) , y ( t )) in the xy-plane. Of course, we have seen that solutions to this system can be expressed analytically as (2) parenleftbigg x ( t ) y ( t ) parenrightbigg = e t A parenleftbigg x I y I parenrightbigg , where parenleftbigg x (0) y (0) parenrightbigg = parenleftbigg x I y I parenrightbigg , and e t A is given by one of the following three formulas that depend upon the roots of the characteristic polynomial p ( z ) = det( z I A ) = z 2 tr( A ) z + det( A ). If p ( z ) has simple real roots with negationslash = 0 then (3a) e t A = e t bracketleftbigg I cosh( t ) + ( A I ) sinh( t ) bracketrightbigg . If p ( z ) has conjugate roots i with negationslash = 0 then (3b) e t A = e t bracketleftbigg I cos( t ) + ( A I ) sin( t ) bracketrightbigg . If p ( z ) has a double real root then (3c) e t A = e t [ I + ( A I ) t ] . While these analytic formulas are useful, you can gain insight into all solutions of system (1) by sketching a graph called its phase-plane portrait (or simply phase portrait ). As we have already observed, any solution of (1) can be thought of as tracing out a curve ( x ( t ) , y ( t )) in the xy-plane the so-called phase-plane . Each such curve is called an orbit or trajectory of the system. The existence and uniqueness theorem implies that every point in the phase-plane has exactly one orbit that passes through it. In particular, two orbits can not cross. You can gain insight into all solutions of system (1) by visualizing how their orbits fill the phase-plane. Of course, the origin will be an orbit of system (1) for every A . The solution that starts at the origin will stay at the origin. Points that give rise to solutions that do not move are called stationary points . In that case the entire orbit is a single point. Orbits Associated with a Real Eigenpair. If the matrix A has a real eigenpair ( , v ) then system (1) has special solutions of the form (4) x ( t ) = ce t v , where c is any nonzero real constant. These solutions all lie on the line x = c v parametrized by c . This line is easy to plot; it is simply the line that passes through the origin and the point v . There are three possibilities. If = 0 then every point on the line x = c v is a stationary point, and thereby is an orbit....
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This note was uploaded on 01/24/2011 for the course MATH math246 taught by Professor Levermore during the Spring '10 term at Maryland.
- Spring '10