Question

The solutions of a nonlinear system near a critical point can be approximated by the solutions of this system, in the case that the Jacobian matrix at the critical point, $underlinehspace0.5cm$, has eigenvalues with...

a) Real parts greater than zero
b) Real parts less than zero
c) Imaginary parts
d) Magnitude equal to one

Answer 1

The solutions of a nonlinear system near a critical point can be approximated by the solutions of this system if the Jacobian matrix at the critical point has eigenvalues with real parts less than zero.

Step-by-step explanation:

The solutions of a nonlinear system near a critical point can be approximated by the solutions of this system if the Jacobian matrix at the critical point has eigenvalues with real parts less than zero.

When the eigenvalues have real parts less than zero, it indicates that the critical point is stable. This means that solutions near the critical point will converge towards it over time.

On the other hand, if the eigenvalues have real parts greater than zero, it indicates that the critical point is unstable. In this case, solutions near the critical point will diverge away from it.