Lindstedt Duffing Secular Removal

Worked Example: Strained Time

Consider the inputs ε = 0.1, Amplitude = 1, and Time = 20.

1. Secular Growth: The naive perturbation solution incorporates a growing secular term: - (3 * ε * A³ * t * sin(t)) / 8.
   Evaluating at t=20: -(3 * 0.1 * 1 * 20 * 0.912945) / 8 = -0.6847089380457209.
2. Naive Perturbation: A cos(t) + (ε*A³/32)(cos(3t) - cos(t)) + secular term.
   Evaluates to approximately -0.28087842323929313 at t=20.
3. Strained Frequency: The first-order frequency correction is ω = 1 + 3εA²/8.
   Evaluating: 1 + 3(0.1)(1)/8 = 1.0375.
4. Strained Perturbation: The strained time is τ = ωt = 1.0375 * 20 = 20.75.
   A cos(τ) + (ε*A³/32)(cos(3τ) - cos(τ)) evaluates to -0.3200874056682934.
5. Exact Frequency: Calculated from the exact energy integral, giving ω = 1.0367169070746336.
   The exact energy period is approximately 6.060656736957466.

Interactive Experiment

Open the Lindstedt Duffing Secular Removal module.

1. Set the Nonlinearity epsilon to 0.1, Amplitude to 1, and Time to 20.
2. Observe the Secular correction value. It grows linearly in magnitude, causing the Naive perturbation to diverge from bounded periodic motion.
3. Check the Strained-time perturbation value, which remains bounded across large time intervals.
4. Increase Time to 40. The Secular correction term grows even larger, highlighting the breakdown of the naive solution, while the Strained solution stays bounded within the expected amplitude range.

Limitations

The validity of the Lindstedt-Poincaré first-order approximation is asymptotic for small εA². As the nonlinearity or amplitude increases substantially, the first-order frequency correction deviates from the exact energy frequency, and higher-order terms become necessary. Additionally, neither the naive nor the strained solution represents an exact trajectory, and strained time is not guaranteed to be strictly closer to the exact solution at every selected time point.

References

The mathematical derivation of Poincaré-Lindstedt's method applied to the Duffing oscillator is explored in introductory perturbation courses.

• Lars-Erik Persson - Poincaré-Lindstedt's Method: Discusses the formulation of strained coordinates to remove secular terms in nonlinear differential equations.