To classify the equilibrium point of the system $\frac{dY}{dt} = AY$ with $A = \begin{bmatrix} 2 & 2 \\ -4 & 6 \end{bmatrix}$, we need to:

1. Compute the eigenvalues of the matrix $A$ to determine the nature of the equilibrium point.

Step 1: Calculate the eigenvalues of $A$

The eigenvalues $\lambda$ of $A$ are found by solving the characteristic equation: $\text{det}(A - \lambda I) = 0$ where $I$ is the identity matrix. The equation becomes: $\text{det} \begin{bmatrix} 2 - \lambda & 2 \\ -4 & 6 - \lambda \end{bmatrix} = 0$

The determinant is: $(2 - \lambda)(6 - \lambda) - (-4)(2) = 0$ $(2 - \lambda)(6 - \lambda) + 8 = 0$ $12 - 2\lambda - 6\lambda + \lambda^2 + 8 = 0$ $\lambda^2 - 8\lambda + 20 = 0$

Step 2: Solve the quadratic equation

$\lambda^2 - 8\lambda + 20 = 0$

Using the quadratic formula: $\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -8$, and $c = 20$, we find: $\lambda = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(20)}}{2(1)}$ $\lambda = \frac{8 \pm \sqrt{64 - 80}}{2}$ $\lambda = \frac{8 \pm \sqrt{-16}}{2}$ $\lambda = \frac{8 \pm 4i}{2}$ $\lambda = 4 \pm 2i$

Step 3: Classify the equilibrium

The eigenvalues $\lambda = 4 \pm 2i$ are complex with a positive real part ($\text{Re}(\lambda) = 4$). This indicates that the equilibrium point at the origin is a spiral source.

The direction (clockwise or counterclockwise) is determined by the sign of the imaginary part ($\text{Im}(\lambda) = 2$):

• Since $\text{Im}(\lambda) > 0$, the rotation is counterclockwise.

Final Answer:

The equilibrium point is a spiral source, counterclockwise.

Let me know if you'd like further clarification!

Related Questions:

1. How do we interpret eigenvalues geometrically in 2D systems?
2. Why does a positive real part indicate instability (source behavior)?
3. What changes if the eigenvalues have a negative real part instead?
4. Can a center equilibrium point occur with complex eigenvalues?
5. How does the sign of the imaginary part determine rotation direction?

Tip: For 2D systems, always consider both real and imaginary parts of eigenvalues for classification.