ESPM-EEP%202010%20Lecture%2027

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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. An equilibrium
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ˆ 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 ˆ x 1 , ˆ x 2 ( ) matrix J have the following properties: i) Both are real and negative ⇒ ˆ x is a stable node ii) Both are real and positive ⇒ ˆ x is an unstable node iii) Both are real, with one negative and one positive ⇒ ˆ x is an unstable node iv) They are complex conjugative with negative real parts ⇒ ˆ x is an stable center (spiral) v) They are complex conjugative with positive real parts ⇒ ˆ x is an stable center (spiral)...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online