Limit cycles, energy, and Lyapunov functions
Nonlinear systems can sustain oscillations that linear systems cannot, and stability can sometimes be proved without solving anything.
Limit cycles
A limit cycle is an isolated closed trajectory that nearby trajectories approach (stable) or leave (unstable). Unlike the family of orbits around a linear centre, a stable limit cycle has a definite amplitude regardless of starting point. Heartbeats, predator-prey oscillations in some models, and electronic oscillators are limit cycles.
The van der Pol oscillator
\(x'' - \mu(1 - x^2)x' + x = 0\) has negative damping for small \(x\) (amplifying oscillations) and positive damping for large \(x\) (shrinking them). The balance produces a stable limit cycle. Van der Pol introduced it in the 1920s to model vacuum-tube circuits.
x'' - \mu(1-x^2)x' + x = 0Poincare-Bendixson theorem
In the plane, if a trajectory stays in a bounded region containing no equilibria, it must approach a closed orbit. This theorem guarantees limit cycles in many two-dimensional systems. It also shows why chaos cannot occur in planar autonomous systems; at least three dimensions are needed.
Conservative systems and energy
For \(x'' + V'(x) = 0\), the energy \(E = \frac{1}{2}(x')^2 + V(x)\) stays constant along solutions. Trajectories are level curves of \(E\). For the frictionless pendulum, small energies give back-and-forth swings and large energies give full rotations.
Lyapunov functions
To prove an equilibrium is stable without solving the system, find a function \(V\) that is positive except at the equilibrium (where it is zero) and does not increase along trajectories (\(V' \le 0\)). If \(V' < 0\) away from the equilibrium, it is asymptotically stable. Energy is often a good candidate.
Worked example
Consider \(x' = -x\), \(y' = -y^3\). Take \(V = x^2 + y^2\). Then \(V' = 2xx' + 2yy' = -2x^2 - 2y^4 < 0\) except at the origin. So the origin is asymptotically stable, even though linearisation gives a zero eigenvalue in \(y\) and is inconclusive.
Bifurcations in systems
As a parameter changes, a stable spiral can lose stability and give birth to a limit cycle, a Hopf bifurcation. This explains how steady states turn into sustained oscillations, such as flutter in aircraft wings or onset of oscillations in chemical reactions.
Chaos
In three or more dimensions, nonlinear systems can be chaotic: deterministic yet unpredictable in the long run because tiny differences in starting conditions grow exponentially. The Lorenz equations, a simplified weather model, produce the famous butterfly-shaped strange attractor.
Key takeaways
Limit cycles are isolated closed orbits with fixed amplitudes, like the van der Pol oscillator. Poincare-Bendixson guarantees them in bounded planar regions without equilibria and rules out planar chaos. Energy and Lyapunov functions prove stability without solving. Hopf bifurcations create oscillations, and chaos appears in three dimensions.
Flashcard
Isolated closed trajectory
Check your understanding
Which is true of a stable limit cycle?