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## 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 λ

_{1}= λ

_{2}= 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)