Linear stability of a planar equilibrium

For a hyperbolic equilibrium of a smooth planar system, the eigenvalues of the Jacobian matrix determine local stability. If its determinant and trace are positive, both eigenvalues have positive real part and the equilibrium is a repeller.

A planar equilibrium point whose Jacobian matrix has nonreal conjugate eigenvalues with positive real part. Its nearby linearized trajectories rotate while their amplitude grows exponentially; the smooth nonlinear system has the same local qualitative behavior when the equilibrium is hyperbolic.

A node equilibrium with a repeated nonzero real eigenvalue. If the planar Jacobian matrix has a one-dimensional eigenspace, its linearized solutions contain a term $t{e}^{\lambda t}$ and the node is also called improper. A negative repeated eigenvalue still gives asymptotic stability; a repeated value does not by itself indicate a bifurcation.

A planar equilibrium with real nonzero eigenvalues of the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Repeated real eigenvalues may give a degenerate or improper node, depending on the eigenspace.

A planar node whose Jacobian matrix has a repeated nonzero real eigenvalue and only one independent eigenvector is an improper node. In suitable real coordinates its linearization of a dynamical system has the displayed form, with solutionsIt is a stable node for $\lambda <0$ and an unstable node for $\lambda >0$. The factor $t$ makes generic trajectories tangent to the unique eigendirection as they approach the equilibrium in the attracting time direction. For a nonlinear autonomous system with this hyperbolic equilibrium, the sign of $\lambda$ still determines local stability; the local phase portrait has the same topological type as its linearization.

A planar equilibrium whose linearization has a complex-conjugate pair of nonreal eigenvalues with nonzero real part. Negative real part gives a stable focus; positive real part gives an unstable focus.

A planar hyperbolic equilibrium is a saddle when the Jacobian determinant is negative. It has one stable and one unstable eigendirection and corresponding one-dimensional invariant manifolds.

A center equilibrium of a planar dynamical system is surrounded by closed trajectories. It is stable but not asymptotically stable.

Let $F$ be a continuously differentiable planar vector field with an isolated equilibrium point $p$, and let the twice continuously differentiable first integral $H$ satisfy $F\cdot \nabla H=0$ and $\nabla H(p)=0$. If the Hessian matrix $D^{2}H(p)$ is positive or negative definite, then $p$ is a centre equilibrium.

For a positive-definite matrix, sufficiently short rays from $p$ have strictly increasing $H$ with radial derivative bounded below by a positive constant times distance. Every sufficiently small positive level difference therefore meets each ray exactly once, and the implicit function theorem makes these intersections a simple closed curve. These regular level sets shrink to $p$; use $-H$ in the negative-definite case. The vector field is tangent and nonzero on each such compact curve. Its speed is bounded away from zero, so a traversal takes finite time and each curve is a periodic orbit. An arbitrary strict local extremum without these regularity and nondegeneracy conditions does not by itself supply this argument.

A planar equilibrium is a stable node when its two Jacobian eigenvalues are real and negative. Every nearby trajectory approaches it, tangent asymptotically to an eigendirection.

A planar equilibrium is an unstable node when its two Jacobian eigenvalues are real and positive. Every nearby nonstationary trajectory moves away from it in forward time.

A planar equilibrium is a stable spiral when its Jacobian has a complex-conjugate pair of eigenvalues with negative real part. Nearby nonstationary trajectories spiral toward it.