ESPM-EEP%202010%20Lecture%2027

ESPM-EEP%202010%20Lecture%2027 - x to the system of...

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ESPM 104/EEP 115 Fall 2010: Lecture 27 1. Consider the 2-dimensional nonlinear system of differential equations dx 1 dt = f 1 x 1 , x 2 ( ) dx 2 dt = f 2 x 1 , x 2 ( ) which can be written in vector notion as d x dt = fx where f = f 1 f 2 . Suppose it has an equilibrium at ˆ x = x 1 x 2 . Now define the perturbed variables (size of the perturbation of x ( t ) from ˆ x to be z ( t ) = x ( t ) ˆ x = x 1 ( t ) x 1 x 2 ( t ) x 2 . The the original system linearized around it equilibrium ˆ x is dz 1 dt = f 1 x 1 ˆ x 1 , ˆ x 2 ( ) + f 1 x 2 ˆ x 1 , ˆ x 2 ( ) dz 2 dt = f 2 x 1 ˆ x 1 , ˆ x 2 ( ) + f 2 x 2 ˆ x 1 , ˆ x 2 ( ) which can be written in matrix form as d z dt = J ˆ x 1 , ˆ x 2 ( ) z where J ˆ x 1 , ˆ x 2 ( ) = f 1 x 1 ˆ x 1 , ˆ x 2 ( ) f 1 x 2 ˆ x 1 , ˆ x 2 ( ) f 2 x 1 ˆ x 1 , ˆ x 2 ( ) f 2 x 2 ˆ x 1 , ˆ x 2 ( ) is called the Jacobian matrix.
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2. An equilibrium
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Unformatted text preview: x to the system of nonlinear ordinary differential equations d x dt = fx is of the following type if the eigenvalues of the Jacobian J x 1 , x 2 ( ) matrix J have the following properties: i) Both are real and negative x is a stable node ii) Both are real and positive x is an unstable node iii) Both are real, with one negative and one positive x is an unstable node iv) They are complex conjugative with negative real parts x is an stable center (spiral) v) They are complex conjugative with positive real parts x is an stable center (spiral)...
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at University of California, Berkeley.

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ESPM-EEP%202010%20Lecture%2027 - x to the system of...

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