The question provided is as follows:

Q2:

1. (a) Determine the type of critical point and its stability.
2. (b) Find the critical point for the system given by:

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -3 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}.$

Solution Outline

Let's address each part step-by-step.

Part (a): Determine the Type and Stability of the Critical Point

1. Identify the critical point: For linear systems of the form $\mathbf{x}' = A \mathbf{x}$, the origin $(0, 0)$ is typically a critical point.

2. Find the Eigenvalues of the Matrix: To classify the critical point and determine stability, we need to find the eigenvalues of the matrix $A = \begin{pmatrix} 0 & 1 \\ -3 & -2 \end{pmatrix}.$

   The eigenvalues $\lambda$ are found by solving the characteristic polynomial: $\det(A - \lambda I) = 0.$ This becomes: $\det \begin{pmatrix} -\lambda & 1 \\ -3 & -2 - \lambda \end{pmatrix} = 0.$

   Expanding the determinant: $(-\lambda)(-2 - \lambda) - (1)(-3) = \lambda^2 + 2\lambda + 3 = 0.$

   Solving this quadratic equation: $\lambda = \frac{-2 \pm \sqrt{4 - 4 \cdot 3}}{2} = \frac{-2 \pm \sqrt{-8}}{2} = -1 \pm i\sqrt{2}.$

3. Classify the Eigenvalues: Since the eigenvalues are complex, $\lambda = -1 \pm i\sqrt{2}$, they indicate a spiral type of critical point. The negative real part $-1$ means that the system is asymptotically stable (the origin is an attracting spiral).

Part (b): Find the Critical Point

Since this is a linear homogeneous system, the only critical point is at the origin, $(0, 0)$.

Summary

1. Type of Critical Point: Spiral (because of complex eigenvalues).
2. Stability: Asymptotically stable (because the real part of the eigenvalues is negative).

Would you like further explanation on any step, or details on eigenvalue calculation?

Here are some related questions:

1. What does it mean for a system to be asymptotically stable?
2. How do complex eigenvalues affect the behavior of a dynamical system?
3. What types of critical points exist in dynamical systems?
4. How do we determine stability for non-linear systems?
5. What is the role of the Jacobian matrix in stability analysis?

Tip: Complex eigenvalues with a negative real part indicate an attracting spiral, which means trajectories will spiral towards the critical point.