dynamical systems
Erosion of safe basin of forced-damped nonlinear oscillator
The safe basin of a dynamical system refers to the set of initial conditions from which the system evolves in a way that avoids undesirable outcomes (e.g. the solution exceeds certain critical threshold).
\[\begin{aligned} \dot{x} &= y, \\ \dot{y} &= - p(x) + \epsilon\left(-\beta_1 y - \beta_2 y|y| + \alpha \cos(\Omega t + \theta)\right), \end{aligned}\]