4-1 Conservation - Linearization and Conservation Laws...

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Linearization and Conservation Laws Integral Curves and Trajectories April 1, 2008 Linearization and Conservation Laws
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Linearization: the one variable case For a single first order autonomous equation like y 0 = f ( y ) = ay ( M - y ) , critical points (fixed points, equilibrium points) are found by solving 0 = f ( y ) = ay ( M - y ) giving y = 0 or y = M . Near y = 0, we can approximate f ( y ) f (0) + f 0 (0)( y - 0) = 0 + aM ( y - 0) = aMy , and see that for y near 0, the solution of y 0 = ay ( M - y ) is close to the solution of y 0 = amy ,— y ( t 0 ) e aM ( t - t 0 ) . -1 -0.5 0 0.5 1 0 2 4 6 8 10 12 t y y’=y(10-y) y’=10 y y’=-10 y Linearization and Conservation Laws
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For y near M , we approximate f ( y ) f ( M ) + f 0 ( M )( y - M ) = 0 - aM ( y - M ) Let ∆ y = y - M and we get the linearized equation d (∆ y ) dt = d ( y - y 0 ) dt = - aM y with solution ∆ y = ∆ y ( t 0 ) e - aM ( t - t 0 ) . Note: this is equivalent to y = M + ( y ( t 0 ) - M ) e - aM ( t - t 0 ) . -1 -0.5 0 0.5 1 0 2 4 6 8 10 12 t y y’=y(10-y) y’=10 y y’=-10 y Linearization and Conservation Laws
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This note was uploaded on 04/10/2008 for the course MATH 254 taught by Professor Indik during the Spring '08 term at University of Arizona- Tucson.

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4-1 Conservation - Linearization and Conservation Laws...

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