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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 

`/`(`*`(dx), `*`(dt)) = f[1](x, y, z, () .. ()) 

`/`(`*`(dy), `*`(dt)) = f[2](x, y, z, () .. ()) 

`/`(`*`(dz), `*`(dt)) = f[3](x, y, z, () .. ()) 

... 

 

System of ODE 

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

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

Embedded component`≡`(f[1](x, y)) 

Embedded component`≡`(f[2](x, y)) 

 

Fixed point 

Fixed points x, y  are defined with the condition  f[1](x, y) = 0, f[2](x, y) = 0. Let one of them to be x[0], y[0]. Note that there could be more than one fixed points.  

Calc fixed point 

Embedded component 

(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 `+`(`≡`(X, x), `-`(x[0])), `+`(`≡`(Y, y), `-`(y[0])). We linearize the original ODE under the condition X, `≪`(Y, 1).  

When we linearize ODE near Embedded componentth fixed point (Embedded component, Embedded component),  ODE for X, Y is calculated to be as follows. 

Linearization 

Embedded component 

Embedded component 

in matrix form,  

Embedded component`*`(`≡`(binomial(X, Y), A), `*`(binomial(X, Y))) 

 

Stability of fixed points 

Stability of a fixed point can be determined by eigen values of matrix  A.  

Eigen values of  A are  Embedded component, Embedded component, therfore  

the fixed point  (Embedded component, Embedded component) is Embedded component.  

Direction field near the fixed point (Embedded component, Embedded component) is displayed in the right figure. 

Calculation 

Solution curve starting (Embedded component, Embedded component) can also diplayed with animation.  

Solution curveEmbedded component 

Embedded component 

Function 

 

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