📈 Nonlinear Oscillations — Duffing Oscillator

The βx³ term makes the restoring force nonlinear: unlike a simple spring where force is proportional to displacement, here the force grows with the cube of x. Combined with a negative linear term α, this creates a double-well potential where the net restoring force can push the mass toward either of two stable positions instead of a single equilibrium.

When α is negative and β is positive, the potential energy function V(x) = ½αx² + ¼βx⁴ has a local maximum at x = 0 and two local minima on either side. These minima are the two wells; without forcing, the system would settle into whichever well it started closer to, but periodic forcing can kick it back and forth between them.

A Poincaré section takes a continuous trajectory and records its state only at fixed intervals, here once every forcing period T = 2π/ω. For a purely periodic orbit this produces a single repeating point, for a period-2 cycle it produces two alternating points, and for chaotic motion it produces a scattered, fractal-like cloud of points called a strange attractor.

The bifurcation diagram is built by fixing all parameters except γ, running the simulation to a steady state for each γ value, and plotting the resulting Poincaré x-coordinates as vertical slices. Where the diagram shows a single point per γ, the phase portrait shows a simple periodic loop; where it fans out into a dense band, the phase portrait shows the tangled, non-repeating trajectory characteristic of chaos.

As γ grows, the energy pumped in each period increases and eventually the trajectory has enough energy to escape one well and cross into the other. Near this transition the system becomes extremely sensitive to initial conditions: two nearly identical starting states diverge exponentially, which is the defining signature of deterministic chaos, visible here as period-doubling cascades leading into broadband, aperiodic motion.