Hyperbolic equilibrium point

If T \colon \mathbb{R}^{n} \to \mathbb{R}^{n} is a C1 map and p is a fixed point then p is said to be a hyperbolic fixed point when the Jacobian matrix \operatorname{D} T (p) has no eigenvalues on the complex unit circle. One example of a map whose only fixed point is hyperbolic is Arnold's cat map: :\begin{bmatrix} x_{n+1}\\ y_{n+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 2\end{bmatrix} \begin{bmatrix} x_n\\ y_n\end{bmatrix} Since the eigenvalues are given by :\lambda_1=\frac{3+\sqrt{5}}{2} :\lambda_2=\frac{3-\sqrt{5}}{2} We know that the Lyapunov exponents are: :\lambda_1=\frac{\ln(3+\sqrt{5})}{2}>1 :\lambda_2=\frac{\ln(3-\sqrt{5})}{2} Therefore, it is a saddle point. == Flows ==

Let F \colon \mathbb{R}^{n} \to \mathbb{R}^{n} be a C1 vector field with a critical point p, i.e., F(p) = 0, and let J denote the Jacobian matrix of F at p. If the matrix J has no eigenvalues with zero real parts then p is called hyperbolic. Hyperbolic fixed points may also be called hyperbolic critical points or elementary critical points. The Hartman–Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic equilibrium point is topologically equivalent to the orbit structure of the linearized dynamical system. Example Consider the nonlinear system : \begin{align} \frac{dx}{dt} & = y, \\[5pt] \frac{dy}{dt} & = -x-x^3-\alpha y,~ \alpha \ne 0 \end{align} (0, 0) is the only equilibrium point. The Jacobian matrix of the linearization at the equilibrium point is :J(0,0) = \left[ \begin{array}{rr} 0 & 1 \\ -1 & -\alpha \end{array} \right]. The eigenvalues of this matrix are \frac{-\alpha \pm \sqrt{\alpha^2-4}}{2}. For all values of α ≠ 0, the eigenvalues have non-zero real part. Thus, this equilibrium point is a hyperbolic equilibrium point. The linearized system will behave similar to the non-linear system near (0, 0). When α = 0, the system has a nonhyperbolic equilibrium at (0, 0). == Comments ==

In the case of an infinite dimensional system—for example systems involving a time delay—the notion of the "hyperbolic part of the spectrum" refers to the above property. == See also ==