Fixed Points, Stability, and Phase Portraits

Reading a flow without solving it

Most nonlinear differential equations have no closed-form solution — you cannot write x(t) as a tidy formula. Poincaré's revolution was to stop asking for formulas and instead ask geometric questions: where does the flow stand still? Which way does it move nearby? What happens in the long run? The answers form a phase portrait, and they need no exact solution.

The skeleton of any phase portrait is its fixed points together with their stability, plus the special trajectories that connect them. Master those and you understand the system's behaviour qualitatively — which is often all a physicist needs.

Fixed points and one-dimensional flows

A fixed point \mathbf{x}^* is a state where the flow vanishes, \mathbf{f}(\mathbf{x}^*)=0: place the system exactly there and it never moves. These are the equilibria. Start with the simplest case, a single variable on a line.

\dot{x} = f(x), \qquad f(x^*) = 0

In one dimension the fixed points are the zeros of f. Between them the sign of f says which way the state slides — right where f > 0, left where f < 0.

Stability is decided by a nudge. Perturb the state to x = x^* + \eta and linearize: since f(x^*)=0, the small displacement obeys \dot\eta = f'(x^*)\,\eta. If the slope f'(x^*) is negative the nudge decays and the point is stable; if positive it grows and the point is unstable. This is linear stability analysis in its purest form.

\dot{\eta} = f'(x^*)\,\eta \quad\Longrightarrow\quad \eta(t) = \eta_0\, e^{\,f'(x^*)\,t}

The perturbation grows or decays exponentially at the rate f'(x*). Negative slope ⇒ stable equilibrium; positive slope ⇒ unstable.

Linear stability in higher dimensions

The same idea works in any number of dimensions — we just replace the derivative by the Jacobian matrix of partial derivatives, evaluated at the fixed point. Its eigenvalues \lambda_k govern how each direction of perturbation grows or shrinks.

\dot{\boldsymbol{\eta}} = J\,\boldsymbol{\eta}, \qquad J_{ij} = \left.\frac{\partial f_i}{\partial x_j}\right|_{\mathbf{x}^*}, \qquad \boldsymbol{\eta}(t) = \sum_k c_k\, \mathbf{v}_k\, e^{\lambda_k t}

Near a fixed point the flow is governed by the Jacobian J. The fixed point is stable exactly when every eigenvalue has negative real part.

Classifying two-dimensional fixed points

In two dimensions you rarely need to compute eigenvalues by hand: two numbers suffice, the trace \tau = \operatorname{tr}J and the determinant \Delta = \det J, because \tau = \lambda_+ + \lambda_- and \Delta = \lambda_+\lambda_-.

\lambda_\pm = \tfrac{1}{2}\left(\tau \pm \sqrt{\tau^2 - 4\Delta}\,\right), \qquad \tau = \operatorname{tr}J,\ \ \Delta = \det J

The trace–determinant plane classifies every 2D fixed point at a glance.

The rules: if \Delta < 0 the eigenvalues are real with opposite signs — a saddle (always unstable). If \Delta > 0, the point is a node or spiral: stable when \tau < 0, unstable when \tau > 0; the sign of \tau^2 - 4\Delta decides node (real, >0) versus spiral (complex, <0). The borderline \tau = 0,\ \Delta > 0 is a center — closed loops, as in an undamped oscillator.

Worked example: the damped pendulum

Let us apply the whole method to a real system: a damped pendulum, governed by \ddot\theta + \gamma\dot\theta + \omega_0^2\sin\theta = 0. Keeping the full \sin\theta (no small-angle cheat) makes it nonlinear, so we analyse it geometrically.

Write it as a first-order system. Let v = \dot\theta. The second-order equation becomes the pair \dot\theta = v and \dot v = -\gamma v - \omega_0^2\sin\theta — a flow in the (\theta, v) phase space.
Find the fixed points. Set \dot\theta = \dot v = 0: we need v = 0 and \sin\theta = 0, giving \theta = 0 (hanging straight down) and \theta = \pi (balanced straight up).
Build the Jacobian. Differentiating, J = \begin{pmatrix} 0 & 1 \\ -\omega_0^2\cos\theta & -\gamma \end{pmatrix}, so the trace is \tau = -\gamma and the determinant is \Delta = \omega_0^2\cos\theta.
Classify the bottom (\theta = 0). Here \Delta = \omega_0^2 > 0 and \tau = -\gamma < 0: a stable point — a spiral (damped oscillation) if \gamma < 2\omega_0, a node if overdamped. The pendulum settles here, exactly as expected.
Classify the top (\theta = \pi). Here \cos\pi = -1 so \Delta = -\omega_0^2 < 0: always a saddle, hence unstable. The slightest push sends it away — matching the everyday truth that balancing a pendulum bolt upright is hopeless.

Limit cycles, and why the plane forbids chaos

Some systems settle not to a point but to an isolated closed loop — a limit cycle, a self-sustained oscillation with its own preferred amplitude (a heartbeat, a laser, the van der Pol circuit). Unlike the ring of neutral orbits around a center, a limit cycle is isolated and attracts nearby trajectories, so its amplitude is robust against disturbance.

Now a deep structural fact, the Poincaré–Bendixson theorem: in a two-dimensional continuous autonomous flow, a bounded trajectory can only approach a fixed point or a limit cycle — nothing more exotic. Trajectories cannot cross, and the plane is simply too cramped to let a bounded path wander forever without repeating. So chaos is impossible in the plane.