Nonlinear Oscillations §

• $\beta > 0$: hardening spring (stiffness increases with amplitude; period decreases with amplitude).
• $\beta < 0$: softening spring (stiffness decreases; period increases with amplitude).
• $\delta = 0, \gamma = 0$: undriven, undamped conservative case.
• $\delta > 0, \gamma > 0$: damped and driven; can exhibit bifurcations and chaos.

The Duffing equation arises from expanding a nonlinear potential $V(x) = \tfrac{1}{2}\alpha x^2 + \tfrac{1}{4}\beta x^4$ to fourth order, so that $-V'(x) = -\alpha x - \beta x^3$.

The damping term $\delta\dot{x}$ and the driving term $\gamma\cos(\omega t)$ are the nonlinear generalizations of the linear counterparts treated in Damped oscillations and Driven oscillations.

For small nonlinearity ($|\beta|A^2 \ll \alpha$), a perturbation expansion gives:

where $T_0 = 2\pi/\sqrt{\alpha}$ is the linear period.

Pendulum example. The exact period of a pendulum of length $L$ released at angle $\theta_0$ is:

For small angles: $T \approx T_0(1 + \theta_0^2/16 + 11\theta_0^4/3072 + \cdots)$. At $\theta_0 = 90°$, the period is about 18% longer than the small-angle value.

A stable limit cycle means the system oscillates with a fixed amplitude regardless of initial conditions (within the basin of attraction). Examples: the van der Pol oscillator (modeling heartbeats, relaxation oscillations), the Hopf bifurcation in fluid dynamics.

The Poincaré–Bendixson theorem guarantees that in a 2-D system, any bounded trajectory that does not approach a fixed point must approach a limit cycle. This is why 2-D nonlinear oscillators are so tractable.

In contrast, the Harmonic oscillator has a family of closed orbits (one for each energy), not a single limit cycle.

The parameter $\mu \geq 0$ controls the strength of the nonlinear damping:

• $\mu = 0$: reduces to the linear harmonic oscillator.
• $0 < \mu \ll 1$: weakly nonlinear regime; the limit cycle is nearly sinusoidal and the period is close to $2\pi$.
• $\mu \gg 1$: relaxation oscillation regime; the trajectory spends most of its time near $x = \pm 2$ with rapid transitions, producing a sawtooth-like waveform. The period scales as $T \approx \mu(3 - 2\ln 2)$.

The van der Pol oscillator is a self-excited system: energy is injected when $|x| < 1$ (negative damping) and dissipated when $|x| > 1$ (positive damping), producing a stable limit cycle.

Biological relevance. The van der Pol equation and its generalizations model:

• Cardiac pacemaker cells (sinoatrial node firing).
• Neural spike generation (integrate-and-fire models).
• Circadian rhythm oscillators.

The connection to the Duffing equation: both are second-order nonlinear ODEs, but the Duffing nonlinearity is in the restoring force (position-dependent stiffness) while the van der Pol nonlinearity is in the damping (velocity-dependent friction). The Duffing oscillator is conservative (or externally driven); the van der Pol oscillator is self-sustained.

The Duffing oscillator with $\delta > 0$ and varying $\gamma$ exhibits a period-doubling route to chaos. The Lyapunov exponent becomes positive in the chaotic regime, meaning nearby trajectories diverge exponentially.

Poincaré map. To study the steady-state of a periodically driven system, one samples the trajectory once per driving period (at $t = 2\pi n/\omega$). This Poincaré section converts the continuous flow into a discrete map: a fixed point of the map corresponds to a periodic orbit, a 2-cycle to a period-doubled orbit, and an infinite set of points to chaos. This is the standard tool for constructing bifurcation diagrams.

Practical relevance:

• Metronomes coupled on a common base synchronize via nonlinear interaction.
• Cardiac cells modelled by van der Pol–type equations can fibrillate (chaotic) under perturbation.
• Nonlinear energy sinks in structural engineering use intentional nonlinearity to absorb vibration over a broad frequency band.

Small-angle period: $T_0 = 2\pi\sqrt{L/g} = 2\pi\sqrt{1/9.81} \approx 2.006$ s.

First correction: $T \approx T_0(1 + \theta_0^2/16)$. With $\theta_0 = \pi/3$:

$\theta_0^2/16 = (\pi/3)^2/16 = \pi^2/144 \approx 0.0685$.

$T \approx 2.006 \times 1.0685 \approx 2.143$ s.

The exact value (from the elliptic integral) is $T = 4\sqrt{1/9.81}\,K(\sin 30°) = 4 \times 0.3193 \times K(0.5) \approx 1.277 \times 1.686 \approx 2.154$ s. The first-order correction is within 0.5%.