Fixed points and the geometry of motion

Write a smooth autonomous system as $\dot{\mathbf z}=\mathbf f(\mathbf z)$. Autonomous means that time does not appear explicitly; identical states have identical future directions. For a mechanical oscillator the state is position together with velocity. Two objects passing the same position with opposite velocities are different phase-space points. A fixed point requires every component of the vector field to vanish simultaneously, so a zero velocity by itself is merely a possible turning point.

Nullclines mark where one component vanishes. Their intersections are candidate fixed points, while arrows between nullclines reveal which component increases or decreases. Draw a few arrows before drawing trajectories. For a continuously differentiable field, local uniqueness prevents two distinct trajectories from crossing at an ordinary state with different directions. A portrait can therefore rule out an imagined motion even when integrating the equations is difficult; it is a constrained representation rather than freehand decoration.

Near a fixed point, the Jacobian matrix supplies the leading linear field. In two dimensions let $\tau$ be its trace and $\Delta$ its determinant. Negative determinant gives real eigenvalues of opposite sign and therefore a saddle. Positive determinant gives eigenvalues with matching real-part signs. The discriminant separates real eigenvalues from a complex-conjugate pair: real stable directions form a node, whereas a negative discriminant with negative trace produces an inward spiral.

A saddle attracts only along its stable manifold and repels along an unstable manifold; approaching along one specially chosen line does not make it stable. Repeated real eigenvalues require inspecting eigenvectors because a defective matrix shears as well as contracts. Purely imaginary eigenvalues describe a center for the linear system, but nonlinear terms can turn that apparent center into weak attraction or repulsion. Classification is strongest for hyperbolic equilibria, whose eigenvalues all have nonzero real parts.

A useful portrait answers an experiment: after a small displacement, does a device return to rest, ring down while returning, or depart? Node versus spiral predicts whether sufficiently small displacements oscillate locally. Stable and unstable manifolds can separate initial conditions that end in different outcomes. Those boundaries are especially valuable when a system has multiple attractors, since a stable operating point may coexist with a dangerous distant state.

Keep the scales and assumptions visible. An arrow field supplies instantaneous velocity, not equal-time spacing unless speeds are represented consistently. Local classification says nothing by itself about escape, distant cycles, or the size of an attraction basin. For a damped pendulum the mechanical energy decreases, giving a second piece of evidence beyond the Jacobian. Combining local eigenvalues with an energy argument is stronger than extrapolating a small region of a drawing to all angles.