Predator-prey, van der Pol, and limit cycles
Most real systems are nonlinear, and they can do things linear ones never can: settle onto a single rhythm from any start, or swing over the top.
Click the phase portrait to pick starting populations and press Play. Every start gives a closed loop, but the size of the loop depends on where you click.
Most real-world differential equations are nonlinear. Unlike linear systems, nonlinear systems can exhibit dramatically different behavior: closed orbits that are not ellipses, limit cycles that attract or repel nearby trajectories, and sensitive dependence on parameters. In this section, we explore three classic nonlinear systems that reveal the richness of nonlinear dynamics.
A central result is the Poincare-Bendixson theorem: in two-dimensional autonomous systems, the only possible long-term behaviors are convergence to an equilibrium, convergence to a limit cycle, or escape to infinity. There is no chaos in the plane -- that requires at least three dimensions.
The Lotka-Volterra equations model the interaction between prey (x) and predators (y). Prey grow exponentially in the absence of predators, while predators decline without prey. Their interaction creates closed orbits in the phase plane -- perpetual oscillations where prey and predator populations rise and fall in a delayed cycle.
Key insight: This is a conservative system -- it has a conserved quantity (the Lotka-Volterra integral), so trajectories form closed orbits. The equilibrium point at (gamma/delta, alpha/beta) is a center, not a spiral. Energy is neither added nor dissipated, so oscillations persist forever without damping or growing. Click different starting points to see orbits of different amplitudes.
Press Play and watch trajectories from inside and outside spiral onto the same loop, the limit cycle. Raise mu to turn smooth swings into sudden jumps.
The Van der Pol oscillator is a dissipative system with a remarkable property: it has a stable limit cycle. Trajectories starting inside the limit cycle spiral outward, and trajectories starting outside spiral inward -- all converging to the same periodic orbit. The parameter mu controls the strength of the nonlinear damping.
Key insight: For small mu, oscillations are nearly sinusoidal. As mu increases, the system exhibits relaxation oscillations -- long periods of slow change punctuated by rapid jumps. The limit cycle exists for all mu > 0 and is guaranteed by the Poincare-Bendixson theorem: the origin is an unstable equilibrium, and trajectories are bounded, so they must converge to a periodic orbit.
Click the phase portrait to pick a starting state, then press Play. Small pushes swing it back and forth; big ones send it over the top. The separatrix is the dividing line.
The simple pendulum is one of the most important nonlinear systems. Its phase portrait reveals three qualitatively different types of motion: libration (back-and-forth swinging), rotation (going over the top), and the separatrix -- the critical energy level that divides the two regimes.
Key insight: The pendulum is a conservative system (Hamiltonian), so its phase portrait consists of level curves of the energy function E = (1/2)omega^2 - (g/L)cos(theta). The saddle points at theta = +-pi correspond to the unstable upright equilibrium. The separatrix connects these saddle points and represents the trajectory where the pendulum just barely reaches the top with zero velocity.
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