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I have the following dynamical system $$ \dot{x} = \mu - mx - xy^2 $$ $$ \dot{y} = -\mu y + xy^2 $$ and I need to "analyse" any bifurcations that occur as $m$ and $\mu$ are varied. I've worked out that the critical points are $(\frac{\mu}{m}, 0)$ for $m \neq 0$, $(\frac{2\mu}{1 \pm \sqrt{1 - 4m}}, \frac{1 \pm \sqrt{1 - 4m}}{2})$ for $m \leq \frac{1}{4}$, and the entire $y$ axis for $\mu = 0$. To find the bifurcations, I worked out when the critical points were non-hyperbolic by finding the determinant and trace of the Jacobian matrix at each critical point, and setting them to zero. It seems that there is a saddle-node bifurcation occurring at $m = \frac{1}{4}$ for the points $(\frac{2\mu}{1 \pm \sqrt{1 - 4m}}, \frac{1 \pm \sqrt{1 - 4m}}{2})$, and a Hopf bifurcation occurring at $$\mu = \frac{1 + \sqrt{1 - 4m}}{2} = y$$ for the point $(\frac{2\mu}{1 + \sqrt{1 - 4m}}, \frac{1 + \sqrt{1 - 4m}}{2})$.

The problem is, I don't know how to prove that these bifurcations are in fact saddle-nodes and Hopfs, respectively. In a one-dimensional system, I know that one has to calculate certain partial derivatives of the original equation (with respect to the variable and parameter) to confirm that a bifurcation is a saddle-node. But since this is a two-dimensional system, I'm not sure what do to. Similarly, the fact that there are two parameters seems to complicate the process of confirming that the bifurcation is a Hopf. I have managed to get the system into normal form for each bifurcation. I did this by first shifting $x$, $y$, and $m$ so that the bifurcations in each case occurred at $(\bar{x}, \bar{y}, \bar{m}) = (0, 0, 0)$. Then I substituted $\bar{x}$, $\bar{y}$, and $\bar{m}$ (the shifted variables) into the original equations to get a new system of the form $$\dot{\bar{\mathbf{x}}} = A\bar{\mathbf{x}} + \text{nonlinear terms},$$ and I transformed this into normal form in the usual way (i.e. by defining $\mathbf{u} = P^{-1}\bar{\mathbf{x}}$, where $P$ is the matrix of eigenvectors of $A$, to get $\dot{\mathbf{u}} = \Lambda\mathbf{u}$).

Presumably, I have to use this new normal form system in order to classify the bifurcations that occur in each case, but I'm not sure how to do this.

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1 Answer 1

If you have transformed the system to the normal form of a bifurcation, then you are done. This is the proof.

In general you find the critical point $x_0$ in the state space and $\mu_0$ in parameters, you translate these to $0$ of the respective space and then you try to transform the system to the normal form of the bifurcation. Once this is done, you look at certain coefficients to see the type of the bifurcation, for example, in Hopf bifurcation it is called first Lyapunov coefficient.

Hopf bifurcation requires at least a 2-dimensional state space, so it shouldn't be a problem. On the other hand saddle-node bifurcation can happen in 1 dimensional state space, so you have to find a center manifold. In short, you have to decompose the system in 2 1-dimensional systems with one's linear terms independent of the other. So you have to find a manifold (if you are lucky this will be a plane) in state space on which nothing interesting happens, the eigenvalue never vanishes and an other manifold on which the saddle-node bifurcation happens. I was very brief with that, but do check the center manifold theorem.

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