Nonlinear Phase-Plane Analysis, Equilibria & Limit Cycles

For an autonomous second-order system

\dot x_1=f_1(x_1,x_2),\qquad \dot x_2=f_2(x_1,x_2),

each state is a point in the phase plane and $(f_1,f_2)$ is the local velocity vector. A trajectory is the curve traced by the state. Time determines how quickly it is traversed, while the portrait reveals equilibria, separatrices, attraction, repulsion, and recurring motion.

Local arrows classify behavior near an equilibrium. Separatrices partition basins; a limit cycle is an isolated periodic orbit, not a neutrally stable family of linear oscillations.

Equilibria and nullclines #

An equilibrium $x^*$ satisfies

f_1(x^*)=0,\qquad f_2(x^*)=0.

The $x_1$ -nullcline is where $\dot x_1=0$ ; the $x_2$ -nullcline is where $\dot x_2=0$ . Their intersections are equilibrium candidates. Nullclines are not trajectories in general: on one nullcline the vector field is tangent to one coordinate direction, and the state can cross it.

They divide the plane into regions with known signs of $\dot x_1$ and $\dot x_2$ . This sign map often reveals rotation direction, trapping regions, or possible basin boundaries before numerical integration.

Local classification by the Jacobian #

Linearize at $x^*$ :

\delta\dot x=J(x^*)\delta x,\qquad J=\frac{\partial f}{\partial x}.

For a hyperbolic equilibrium—no eigenvalue with zero real part—the eigenvalues classify local behavior:

two negative real parts: asymptotically stable node or focus;
two positive real parts: unstable node or focus;
opposite signs: saddle;
complex pair: rotation with decay or growth set by the real part.

At a saddle, stable and unstable invariant manifolds approach and leave along eigenvector directions locally. They can form separatrices that divide initial conditions with different long-term outcomes.

If an eigenvalue has zero real part, linearization may be inconclusive. A nonlinear center can become a stable or unstable focus after higher-order terms are included; use energy, Lyapunov, center-manifold, or normal-form reasoning.

Worked pendulum portrait #

For a damped pendulum,

\dot\theta=\omega,

\dot\omega=-\frac{g}{\ell}\sin\theta-\frac{b}{m\ell^2}\omega.

Equilibria occur at $(\theta,omega)=(k\pi,0)$ . Downward positions $\theta=2k\pi$ are locally asymptotically stable when $b>0$ ; upright positions $\theta=(2k+1)\pi$ are saddles. The angular coordinate is periodic, so a cylinder is the natural state space even when the portrait is drawn as repeated strips in the plane.

For $b=0$ , mechanical energy is conserved and trajectories lie on energy contours: small oscillations form closed curves, the separatrix passes through upright saddles, and higher energy gives continuous rotation. With damping, energy decreases and trajectories spiral or flow toward a downward equilibrium.

This example shows why a local transfer function around the downward equilibrium cannot describe swing-up, full rotation, the upright saddle, or basin geometry.

Constructing trajectories without solving time explicitly #

Where $f_1\ne0$ ,

\frac{dx_2}{dx_1}=\frac{f_2(x_1,x_2)}{f_1(x_1,x_2)}.

This first-order equation describes trajectory shape, but it is singular on the $x_1$ -nullcline and still may lack a closed form. Numerical integration is normally more reliable. Use adaptive integration with event checks, and verify that apparent cycles or equilibria are not step-size artifacts.

Phase arrows must follow increasing time. Reversing both differential equations preserves geometric curves but reverses stability and causality.

Limit cycles #

A limit cycle is an isolated closed trajectory. Nearby trajectories approach it if stable and recede if unstable. Its amplitude and period emerge from nonlinear dynamics; scaling a trajectory generally does not produce another solution.

This differs from an undamped linear oscillator, whose phase plane contains a continuum of closed ellipses selected by initial energy. None is isolated or attracting, so those orbits are centers, not limit cycles.

Limit cycles can be intended—oscillators, walking gaits, switching converters—or harmful—hunting valves, relay chatter, actuator wear. Their existence can result from saturation, relay action, dead zones, friction, quantization, or nonlinear energy balance.

Poincaré–Bendixson reasoning #

For a smooth planar autonomous system, if a forward trajectory remains in a compact region containing only finitely many equilibria, its nonempty limit set is constrained: absent convergence to an equilibrium or connecting orbit, a periodic orbit can arise. A trapping region whose boundary vectors point inward can therefore support an existence argument.

This result is special to two-dimensional continuous autonomous systems. Higher-dimensional systems can have quasiperiodic or chaotic attractors, and hybrid/discontinuous systems require their own conditions.

Poincaré map and orbital stability #

Choose a transverse section through a periodic orbit. The Poincaré return map takes one crossing to the next. A fixed point of the map represents the cycle; the derivative (or Floquet multipliers in higher dimension) determines transverse stability. One multiplier associated with motion along an autonomous orbit is neutral; stability concerns perturbations across the orbit.

This reduces a continuous periodic-orbit question to a discrete fixed-point question and is more precise than judging convergence from a finite simulation.

Numerical phase-plane workflow #

Find every equilibrium in the region, including boundary and periodic-coordinate copies.
Plot nullclines and sign regions.
Compute Jacobian eigenvalues/eigenvectors at each equilibrium.
Integrate forward from a grid and from both sides of suspected separatrices.
Check conservation or dissipation laws and solver convergence.
Detect recurring crossings and estimate a Poincaré map for suspected cycles.
Vary parameters and initial conditions to map bifurcations and basins.
Reintroduce actuator limits, switching, sampling, and noise.

A phase portrait is not decoration. It is a compact map of which futures are possible from which states.