- #1
JuanC97
- 48
- 0
Homework Statement
I want to find conditions over A,B,C,D to observe a stable limit cycle in the following system:
[tex] \frac{dx}{dt} = x \; (A-B y) \hspace{1cm} \frac{dy}{dt} = -y \; (C-D x) [/tex]
Homework Equations
Setting dx/dt=0 and dy/dt=0 you can find that (C/D , A/B) is a fixpoint.
The Attempt at a Solution
When I evaluate the Jacobian matrix in that point...
[tex]
J(x,y) = \left[
\begin{matrix}
A-B y & -B x
\\
D y & D x - C
\end{matrix} \right]
\: \Rightarrow \: J\left(\frac{C}{D}, \frac{A}{B}\right) =
\left[ \begin{matrix}
0 & -\frac{BC}{D}
\\
\frac{DA}{B} & 0
\end{matrix} \right]
[/tex]
Since λ1 = λ2 = 0, a Hopf bifurcation could occur (If and only if Tr(J)=0 ∧ Det(J)>0). As a consequence... it's necesary to have A,B,C,D such that AC > 0, however, by definition A,B,C,D are positive.
Does it mean that I will always have a bifurcation? If so, why am I getting stable and unstable orbits when I change the parameters?. Also, if this approach is not useful to determine the requirements to have a stable limit cycle, then, what could be the best approach? (Sorry if these questions sound kinda basic - I'm trying to learn about this topic by myself)