Autonomous planar system

An autonomous planar system is an ordinary differential equationwhose vector field does not depend explicitly on time.

The phase plane is the two-dimensional state space of an autonomous planar system. Its oriented trajectories, equilibria, nullclines, and invariant sets display the system's qualitative dynamics.

An equilibrium point, or fixed point, of $\dot{z}=f(z)$ is a state $z_{\ast }$ with $f(z_{\ast })=0$.

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 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.

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.