Question

Determine the critical points of the system below. (b) Explain

(with a brief discussion) the stability/instability behaviour of
these critical points. dx dt = −10x − 5; dy dt = x 2 − y + 3

Answer 1

The system's critical point is (-0.5, 2.75) by solving the equations dx/dt = -10x - 5 and dy/dt = x² - y + 3. The stability of this point cannot be determined without further information, but the x component shows characteristics of an unstable equilibrium.

Step-by-step explanation:

To determine the critical points of the given system, we need to set the derivatives to zero:
dx/dt = −10x − 5 = 0
dy/dt = x² − y + 3 = 0
Solving for dx/dt gives us x = −0.5. Substituting this into the dy/dt equation, we get the quadratic equation (−0.5)² − y + 3 = 0, which simplifies to y = 2.75. Therefore, the critical point is (−0.5, 2.75).

Examining the stability/instability at this critical point involves analyzing the system's behavior around the point. For the x component, any perturbation will lead the system to move away from x = −0.5 due to the negative coefficient of x in the dx/dt equation, indicating an unstable equilibrium. For the y component, we need to examine the curvature of the potential energy or consider the Jacobian matrix at the critical point; however, details are missing to conclude its stability rigorously. In general, in potential energy curves, a concave-up curve at the equilibrium position indicates stability, while a concave-down curve indicates instability.