91Ó°ÊÓ

Equilibrium points are where the system remains in a steady state. For a system of differential equations, these points occur when the rate of change of all variables is zero. In simple terms, the system stops evolving at these points.
To find equilibrium points, one must set the derivatives

• \( x' = x(2-y) \)
• \( y' = y(-3+x) \)

to zero and solve the resulting equations. This helps identify combinations of \( x \) and \( y \) where neither changes over time.
For example, if we solve these conditions together, we find the equilibrium points for our specific system:

• \((0, 0)\)
• \((0, 2)\)
• \((3, 0)\)
• \((3, 2)\)

Each point is an equilibrium state—meaning at these coordinates, the system won't change unless perturbed.

Once equilibrium points are found, understanding their stability gives insight into how the system behaves near these points. Stability analysis helps determine if a small disturbance will cause the system to return to equilibrium or move away.
Stability uses eigenvalues derived from the Jacobian matrix (which we will cover in the next section). The eigenvalues indicate how solutions evolve around equilibrium points.
Analyzing the stability involves:

• If all eigenvalues are negative, the point is stable—systems return to equilibrium after a disturbance.
• If at least one eigenvalue is positive, the point is unstable—systems move away from equilibrium.
• If eigenvalues are imaginary without a real part, the equilibrium point might be a center, leading to periodic motion (neutral stability).

This method of analysis provides qualitative understanding of the behaviors around equilibrium points.

The Jacobian matrix is a fundamental tool in analyzing systems of differential equations, especially for stability. The matrix is essentially a snapshot of how variables in a system influence each other at an equilibrium.
For a two-variable system like \( x \) and \( y \), the Jacobian matrix is calculated by taking partial derivatives:

• The element \( J_{11} \) is the partial derivative of \( x' \) with respect to \( x \).
• The element \( J_{12} \) is the partial derivative of \( x' \) with respect to \( y \).
• The element \( J_{21} \) is the partial derivative of \( y' \) with respect to \( x \).
• The element \( J_{22} \) is the partial derivative of \( y' \) with respect to \( y \).

For our exercise, we obtained:\[J = \begin{pmatrix} 2-y & -x \ y & -3+x \end{pmatrix}\]This matrix is evaluated at each equilibrium point to reveal the system's local behavior and allows us to determine the nature (stable, unstable, or neutral) based on the eigenvalues. Understanding this process is key to interpreting how dynamic systems behave around stable and unstable points.