Nonlinear dynamics studies systems whose equations of motion are nonlinear in the state variables. Nonlinearity breaks the superposition principle, so behavior can include amplitude-dependent periods, multiple equilibria, limit cycles, bifurcations, and chaos.
What makes a system nonlinear
Typical nonlinear terms include trigonometric functions and higher powers:
θ¨ + (g/ℓ) sin θ = 0
x¨ + x + α x^3 = 0
Even small nonlinear terms can accumulate and shape long-term motion.
Phase space and fixed points
Many systems can be written as
x˙ = f(x, t)
In phase space one studies equilibria, their stability, and invariant sets such as periodic orbits.
Linearization (and its limits)
Near an equilibrium x*, one often approximates
δx˙ ≈ J(x*) δx
This describes local behavior, but global trajectories can differ because nonlinear terms dominate away from x*.
Bifurcations
As parameters change, equilibria and periodic orbits can be created, destroyed, or change stability (saddle-node, pitchfork, Hopf, period-doubling).
Limit cycles and self-oscillation
Some nonlinear systems have stable periodic orbits (limit cycles). The van der Pol oscillator is a classic example where effective damping depends on amplitude.
Common pitfalls
- Linear intuition may fail outside a small neighborhood of equilibrium.
- Small parameter changes can cause qualitative transitions.