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# Stability of a fixed point in a system of ODE

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Stability of a fixed point in a system of ODE

Yasuyuki Nakamura
Graduate School of Information Science, Nagoya University
A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
nakamura@nagoya-u.jp
http://www.phys.cs.is.nagoya-u.ac.jp/~nakamura/

Let us consider the following system of ODE

...

System of ODE

For the simplisity, we consider the follwoing system of autonomous ODE with two variables.

(Please input and without independent variable , like for and for .)

Fixed point

Fixed points  are defined with the condition  . Let one of them to be . Note that there could be more than one fixed points.

(Note, when solutions are not expressed in explicit form, the solution are not listed above.)

Linearization

In order to analize a behaviour of solutions near fixed points, let us consider the system of ODE for . We linearize the original ODE under the condition .

When we linearize ODE near th fixed point (, ),  ODE for is calculated to be as follows.

 in matrix form,

Stability of fixed points

 Stability of a fixed point can be determined by eigen values of matrix  .   Eigen values of   are  , , therfore   the fixed point  (, ) is .   Direction field near the fixed point (, ) is displayed in the right figure.    Solution curve starting (, ) can also diplayed with animation.

Function

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