This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 642:527 ORBITS OF CENTERS AND FOCI FALL 2007 A. Centers Consider a linear system z = A z , with z = bracketleftbigg x y bracketrightbigg , A = bracketleftbigg a b c d bracketrightbigg , (4.1) for which the origin of the phase plane is a center, that is, for which the eigenvalues of A are pure imaginary. Recall that if p = a + d = Tr A and q = ab cd = det A then any eigenvalue satisfies 2 p + q = 0; this equation has solutions = ( p radicalbig p 2 4 q ) / 2. The origin is a center if the eigenvalues are pure imaginary, which requires that Tr A = p = a + d = 0 , (4.2a) det A = q = ad bc > . (4.2b) In the remainder of this section we assume that (4.2) holds, so that A has eigenvalues = i with = ad bc . 1. A special case. Consider first the special case in which a = d = 0. Then (4.2a) is satisfied and (4.2b) requires that c and d have opposite signs. Suppose, for example, that b = 2 > 0 and c = 2 < 0. Then the equations become x = 2 y , y = 2 x , so that the quantity Q = x 2 2 + y 2 2 (4.3) satisfies Q = 0, i.e., Q is conserved : it is constant during the motion. The trajectories or orbits of the system are thus given by the level curves of the function Q , so that these orbits have the form x 2 2 + y 2 2 = 2 . (4.4) But (4.4) is just the equation of an ellipse whose axes lie along the coordinate axes; the axis of the ellipse in the x direction has length 2 and that in the y direction has length 2 . These axis lengths of course vary as varies, but the ratio of their lengths, / , is the same for all the elliptical orbits. If b > 0, as we have assumed above, then in the upper half plane, where y > 0, x = by > 0 so that x increases as t increases: the trajectories circle the origin clockwise. Here is a typical picture of some trajectories in the phase plane: ............................................................................................................................................................ . . . . . . . . . . . ....
View Full
Document
 Fall '07
 SPEER

Click to edit the document details