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 JF(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.
Read more about this topic: Jacobian Matrix And Determinant, Jacobian Matrix, Uses
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