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527centers

# 527centers - 642:527 ORBITS OF CENTERS AND FOCI FALL 2009 A...

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Unformatted text preview: 642:527 ORBITS OF CENTERS AND FOCI FALL 2009 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 λ satisfies λ 2- pλ + q = 0; this equation has solutions λ ± = ( p ± radicalbig p 2- 4 q ) / 2. The origin is a center if the eigenvalues λ ± 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 λ ± = ± iμ with μ = √ 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 = α 2 > 0 and c =- β 2 < 0. Then the equations become x ′ = α 2 y , y ′ =- β 2 x , so that the quantity Q = x 2 α 2 + y 2 β 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 α 2 + y 2 β 2 = γ 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 αγ and that in the y direction has length 2 βγ . These axis lengths of course vary as γ varies, but the ratio of their lengths, α/β , 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: ............................................................................................................................................................ . . . . . . . . . . . ....
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