Exact and approximate solutions for the anti-symmetric quadratic truly nonlinear oscillator

• The exact solution of the anti-symmetric quadratic truly nonlinear oscillator was expressed as a piecewise function.
• The Fourier coefficients of the exact solution were computed numerically and we showed these decrease rapidly.
• Using just a few of Fourier coefficients provides an accurate analytical representation of the exact periodic solution.
• Analytical approximate solutions are built up containing only two harmonics as well as a rational harmonic representation.
• The two-harmonic representation is more accurate than the rational harmonic representation.

The exact solution of the anti-symmetric quadratic truly nonlinear oscillator is derived from the first integral of the nonlinear differential equation which governs the behavior of this oscillator. This exact solution is expressed as a piecewise function including Jacobi elliptic cosine functions. The Fourier series expansion of the exact solution is also analyzed and its coefficients are computed numerically. We also show that these Fourier coefficients decrease rapidly and, consequently, using just a few of them provides an accurate analytical representation of the exact periodic solution. Some approximate solutions containing only two harmonics as well as a rational harmonic representation are obtained and compared with the exact solution.