Identification of Upper and Lower Limits of the Unstable Region of the Undamped Duffing Oscillator with Softening Stiffness

The study concerns the Duffing oscillator (Duffing equation) with softening stiffness. Using numerical simulations, the upper and lower limits of the unstable region for damping equal to 0 were identified, and analytic formulas describing them were developed. The analysis shows that the developed formulas are effective for all combinations of stiffness coefficient values that were tested. The curve of the upper limit of the unstable region is a jump-down curve, and in the theory of nonlinear systems, this curve for damping equal to zero is identified as a backbone curve (the curve of natural frequency of a system). However, the classical backbone curve calculated via a formula that is commonly known and used differs visibly from that actually obtained via numerical simulations of the upper boundary of the unstable region at large amplitudes. It could therefore be concluded that the backbone curve is not equal to the upper boundary of the unstable solution region. Moreover, the paper shows that the use of the scale relative to a critical oscillation amplitude leads to the conclusion that for damping equal to 0, systems with different parameters have the same instability regions in dimensionless space.