Editing Jacobian matrix and determinant (section)

=== Dynamical systems ===

Consider a [[dynamical system]] of the form <math>\dot{\mathbf{x}} = F(\mathbf{x})</math>, where <math>\dot{\mathbf{x}}</math> is the (component-wise) derivative of <math>\mathbf{x}</math> with respect to the [[evolution parameter]] <math>t</math> (time), and <math>F \colon \mathbb{R}^{n} \to \mathbb{R}^{n}</math> is differentiable. If <math>F(\mathbf{x}_{0}) = 0</math>, then <math>\mathbf{x}_{0}</math> is a [[stationary point]] (also called a [[steady state]]). By the [[Hartman–Grobman theorem]], the behavior of the system near a stationary point is related to the [[eigenvalue]]s of <math>\mathbf{J}_{F} \left( \mathbf{x}_{0} \right)</math>, the Jacobian of <math>F</math> at the stationary point.<ref>{{cite book |first1=D. K. |last1=Arrowsmith |first2=C. M. |last2=Place |title=Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour |chapter=The Linearization Theorem |publisher=Chapman & Hall |location=London |year=1992 |isbn=0-412-39080-9 |pages=77–81 |chapter-url=https://books.google.com/books?id=8qCcP7KNaZ0C&pg=PA77 }} </ref> Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.<ref>{{cite book |first1=Morris |last1=Hirsch |first2=Stephen |last2=Smale |title=Differential Equations, Dynamical Systems and Linear Algebra |year=1974 |isbn=0-12-349550-4 }}</ref>