Jacobian Matrix and Determinant - Jacobian Matrix - Uses - Dynamical Systems

Dynamical Systems

Consider a dynamical system of the form x' = F(x), where x' is the (component-wise) time derivative of x, and F : Rn → Rn is continuous and differentiable. If F(x₀) = 0, then x₀ is a stationary point (also called a critical point, not to be confused with a fixed point). The behavior of the system near a stationary point is related to the eigenvalues of J_F(x₀), the Jacobian of F at the stationary point. Specifically, if the eigenvalues all have real part with a magnitude less than 1, then the system is stable in the operating point, if any eigenvalue has a real part with a magnitude greater than 1, then the point is unstable.