Find and classify the bifurcations that occur as $\mu$ varies for the system \begin{align}\frac{dx}{dt}&= y-2x \\ \frac{dy}{dt}&=\mu +x^2 -y\end{align}

What I have so far:

The equilibrium points of the system can be determined as $$E_1(x_1^*, y_1^*) = (1+ \sqrt{1-\mu}~, 2+2\sqrt{1-\mu}~) \\ E_2(x_2^*,y_2^*) =(1-\sqrt{1-\mu}~, 2-2\sqrt{1-\mu}~) $$ where $\mu \in (-\infty,1]$ since we are only interested in real equilibrium points.

Now using the Jacobian $$J= \begin{bmatrix}-2 & 1 \\2x&-1\end{bmatrix}$$

We find that the eigenvalues of $J_{E_1}$ are given by $$\lambda_{\pm}= -\frac{3}{2} \pm \frac{\sqrt{9+8\sqrt{1-\mu}}}{2}$$

Now, if $\mu < 1$ then we will always have $$\lambda_{+} >0,~ \lambda_- <0 \implies E_1~ \text{unstable saddle}$$

Now, if $\mu =1$ then $\lambda_+ =0 \implies E_1$ is non-hyperbolic.

The eigenvalues of $J_{E_2}$ are given by $$\lambda_{\pm} = -\frac{3}{2} \pm \frac{\sqrt{9-8\sqrt{1-\mu}}}{2}$$

Now if $\mu \le - \frac{17}{64}$ then $\Re(\lambda_\pm) <0 \implies E_2$ stable.

If $-\frac{17}{64}<\mu < 1$ then $\lambda_\pm <0 \implies E_2$ is stable.

If $\mu = 1$ then $\lambda_+ =0 \implies E_2$ is non-hyperbolic.

Is this correct so far? How can I use this information to classify the type of bifurcations that occur (and possibly draw a bifurcation diagram)?


1 Answer 1


So as $\mu \to 1-$ a stable node and a saddle collide and disappear. That's called a saddle-node bifurcation.


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