Driven nonlinear spring: chaos, bistability, and period-doubling
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The Duffing equation ẍ + γẋ + x + εx³ = F cos(ωt) describes a nonlinear oscillator with a hardening spring (cubic term ε=0.5). Unlike a linear oscillator, the resonance curve bends and can exhibit bistability — two stable amplitudes at the same drive frequency — and period-doubling routes to chaos at large forcing.