The origin point of the unstable solution area of a forced softening Duffing oscillator

Each Duffing equation has an unstable solution area with a boundary, which is also a line of bifurcation. Generally, in a system that can be modeled by the Duffing equation, bifurcations can occur at frequencies lower than the origin point frequency of the unstable solution area for a softening system and at higher frequencies for a hardening system. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account. To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. For each system, for three combinations of linear and nonlinear stiffness coefficients with the same nonlinearity factor, bistability areas and unstable solution areas were determined for a series of damping coefficient values. For each case, curves determined for different damping values were grouped to obtain the origin point curve of the unstable solution, ultimately developing the target formulas.