PPbeam - Phase Plane Autonomous first order systems with...

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Unformatted text preview: Phase Plane Autonomous first order systems with two dependent variables March 25 and 27, 2008 Phase Plane Systems of differential equations are more general than scalar differential equations. In systems of ordinary differential equations you can have more than one dependent variable, though you still have only one independent variable. Systems of equations can typically be put into standard form dy 1 dt = F 1 ( t , y 1 , y 2 , . . . , y n ) dy 2 dt = F 2 ( t , y 1 , y 2 , . . . , y n ) . . . . . . dy n dt = F n ( t , y 1 , y 2 , . . . , y n ) Here I have assumed that the number of equations and the number of dependent variables are equal the case where we have a good existence and uniqueness theorem. Phase Plane Autonomous systems If the right hand sides of the equations do not have any explicit dependence on the independent variable, we say that the system is autonomous . So for example x = x + y + t y = t sin( xy ) is not an autonomous system, while x = x + y y = sin( xy ) is autonomous. Phase Plane linear We say that a system is linear , if the functions F j ( t , y 1 , y 2 , . . . , y n ) are linear functions of the dependent variables i.e. dy 1 dt = a 11 ( t ) y 1 + a 12 ( t ) y 2 + + a 1 n ( t ) y n + f 1 ( t ) dy 2 dt = a 21 ( t ) y 1 + a 22 ( t ) y 2 + + a 2 n ( t ) y n + f 2 ( t ) . . . . . . dy n dt = a n 1 ( t ) y 1 + a n 2 ( t ) y 2 + + a nn ( t ) y n + f n ( t ) We say that a linear system is homogeneous if all of the f j ( t ) are 0 Phase Plane Of course, if the system is linear and autonomous the a ij and the f i are all constants, so the system would look like dy 1 dt = a 11 y 1 + a 12 y 2 + + a 1 n y n + f 1 dy 2 dt = a 21 y 1 + a 22 y 2 + + a 2 n y n + f 2 . . . . . . dy n dt = a n 1 y 1 + a n 2 y 2 + + a nn y n + f n Phase Plane Phase Plane We will be mostly focussing on autonomous systems with two dependent variables....
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PPbeam - Phase Plane Autonomous first order systems with...

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