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
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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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