Numerical solutions for the canonical escape equation (Q1304197)

This paper is concerned with the numerical solution of initial value problems for a damped nonlinear oscillator described by the second-order equation \( x''+ \beta x' + x - x^2 = F \sin (w t)\) where \(\beta,F,w\) are constant parameters. Due to the special form of this equation the author has been able to derive a one-step second-order explicit numerical method from a linear combination of two suitable first-order methods. A stability analysis of the numerical scheme is carried out showing that, under some restriction on the size of the step, the numerical scheme retains the stability of the fixed point of the underlying equation for \( F=0\). Finally some numerical experiments are presented to confirm the above stability analysis.