Analyzing bifurcation point of a 2D dynamical system

Question

I have the following nonlinear dynamical system:

\begin{align} \frac{dX}{dt} &= Y(\gamma - 2X) \\\ \frac{dY}{dt} &= - \gamma X + X^2- Y^2 + 1 \end{align}

And I'm trying to understand what type of bifurcation is happening at the point $\gamma = 2$. I'm fairly new to dynamical analysis and not sure what would be the best way to approach this specific problem.

So far what I tried is to look at the jacobian: $$ \mathbf{J} = \begin{bmatrix} -2Y & \gamma \\[1ex] % <-- 1ex more space between rows of matrix -\gamma + 2X & -2Y \end{bmatrix} $$ And when $\gamma =2$ the equilibrium point of the system is $(X = 1, Y = 0)$, which gives the following: $$ \mathbf{J}= \begin{bmatrix} 0 & 2 \\[1ex] % <-- 1ex more space between rows of matrix 0 & 0 \end{bmatrix} $$ The problem is that I'm not sure how to continue from here, since the eigenvalue is zero and I don't see a way to characterize the bifurcation.

Answer 1

We're looking for stationary points: $$ \left\{\begin{aligned} & y (\gamma - 2x) = 0\\ & -\gamma x + x^2 - y^2 + 1 = (x - 0.5 \gamma)^2 - y^2 + (1-0.25 \gamma^2) = 0 \end{aligned}\right. . $$

There are two possibilities: $y = 0$ or $x = 0.5\gamma$.

In the first case we have: $$ \left\{\begin{aligned} & \varnothing, \quad \gamma < 2\\ & (1,0), \quad \gamma = 2\\ & (0.5\gamma \pm \sqrt{0.25\gamma - 1}, 0), \quad \gamma > 2 \end{aligned}\right. . $$

In the second case we have: $$ \left\{\begin{aligned} & (0.5\gamma, \pm \sqrt{1 - 0.25\gamma}), \quad \gamma < 2\\ & (1,0), \quad \gamma = 2\\ & \varnothing, \quad \gamma > 2 \end{aligned}\right. . $$

Jacobian: $$ J(x,y) = \begin{pmatrix} -2y & \gamma - 2x \\ 2x - \gamma & -2y \end{pmatrix}. $$

Hence:

1. point $(0.5\gamma, \sqrt{1 - 0.25\gamma})$, $\gamma < 2$, is a stable star (the Jacobian has two negative equal eigenvalues);
2. point $(0.5\gamma, -\sqrt{1 - 0.25\gamma})$, $\gamma < 2$, is an unstable star (the Jacobian has two positive equal eigenvalues);
3. points $(0.5\gamma \pm \sqrt{0.25\gamma - 1}, 0)$, $\gamma > 2$, are unstable focuses (the Jacobian has complex eigenvalues with positive real part);
4. point $(1,0)$, $\gamma = 2$, is a degenerate equilibrium point (the Jacobian has zero eigenvalue, $J(1,0) = 0$, $\gamma = 2$).

When $\gamma = 2$, we have the following equation: $$ \left\{\begin{aligned} & \dot{\xi} = - 2 \eta \xi\\ & \dot{\eta} = \xi^2 - \eta^2 \end{aligned}\right. , $$ where $\xi = x - 1$ and $\eta = y$. This point doesn't have a special name, I think. Degenerate points are topologically classified by their sectors. This point has two elliptic sectors (they contain homoclinic loops). You can find more in the literature listed here.

The scenario of the bifurcation is simple: nodes merge with each other into one degenerate point $(1,0)$, and then this point splits into two focuses.

The bifurcation is already atypical, it is not Morse--Smale, because the point $(1,0)$, $\gamma = 2$, has infinitely many homoclinic orbits.