- #1

hunt_mat

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I am looking at the following system of ODEs:

[tex]

\begin{eqnarray*}

\dot{\omega}_{3}+\alpha\omega_{3} & = & \frac{\beta_{1}+\beta_{3}}{\rho_{0}}J_{3} \\

\dot{J_{3}}+2(\alpha_{2}-\alpha_{1})\beta_{2} & = & 0 \\

\dot{\beta}_{1}+\omega_{3}\beta_{2} & = & 0 \\

\dot{\beta}_{2}-\frac{\bar{\alpha}}{2}J_{3}+\frac{1}{2}\omega_{3}(\beta_{3}-\beta_{1}) & = & 0 \\

\dot{\beta}_{1}+\dot{\beta}_{3} & = & 0

\end{eqnarray*}

[/tex]

With suitable scaling I can reduce the system down to the following:

[tex]

\begin{eqnarray*}

\dot{\omega} & = & -\omega+CJ \\

\dot{J} & = & -\beta_{2} \\

\dot{\beta}_{1} & = & -\omega\beta_{2} \\

\dot{\beta}_{2} & = & \hat{\alpha}J-\frac{1}{2}\omega(C-2\beta_{1})

\end{eqnarray*}

[/tex]

Where [itex]\beta_{1}(t)+\beta_{3}(t)=C[/itex]. As I said in the title, I am interested in the stability of this system, so the first thing I so is look for the equilibrium points. I find there are two such points, one rather trivial one which is easy to analyse is [itex](\omega,J,\beta_{1},\beta_{2})=(0,0,C/2,0)[/itex], and another one which is: [itex](\omega,J,\beta_{1},\beta_{2})=(CJ_{0},J_{0},(C-2\hat{\alpha})/2),0)[/itex].

So my question is this: How do I determine [itex]J_{0}[/itex]? Do I consider it a parameter and look at different cases?