math4b-17 - Math 4B Lecture 17 Doug Moore Our goal today is...

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Math 4B Lecture 17 May 29, 2013 Doug Moore
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Our goal today is to understand how to draw a rough phase portrait of linear dynamical systems of the form dx/dt = a 11 ( x - x 0 ) + a 12 ( y - y 0 ) , dy/dt = a 21 ( x - x 0 ) + a 22 ( y - y 0 ) . Note that this dynamical system has a constant solution at the point ( x 0 , y 0 ). In fact, we can translate coordinates, setting ˜ x = x - x 0 , ˜ y = y 0 , which relocates the constant solution to the origin: d ˜ x/dt = a 11 ˜ x + a 12 ˜ y, d ˜ y/dt = a 21 ˜ x + a 22 ˜ y. Recall that the phase portrait is simply a sketch of the image of the solution curves in the ( x, y )-plane.
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We will see that if the eigenvalues of the linear system dx/dt = a 11 ( x - x 0 ) + a 12 ( y - y 0 ) , dy/dt = a 21 ( x - x 0 ) + a 22 ( y - y 0 ) , are nonzero, then after linear distortion its phase portrait falls into one of six cases: If the eigenvalues are . . . the point ( x 0 , y 0 ) is a(n) . . . positive real numbers expanding node. negative real numbers contracting node. real numbers with opposite signs saddle point. complex with positive real part expanding spiral. complex with negative real part contracting spiral. purely imaginary center.
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If the eigenvalues are . . . the point ( x 0 , y 0 ) is a(n) . . . positive real numbers expanding node. negative real numbers contracting node. real numbers with opposite signs saddle point. complex with positive real part expanding spiral. complex with negative real part contracting spiral. purely imaginary center.
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