Existence and stability of forced oscillations in nonlinear systems with one degree of freedom (Q1905473)

The model equation is \(\ddot x+\mu\varphi(x,\dot x)+f(x)= p(\omega t)\), \(p(\omega t)=p(\omega t+2\pi)\), where \(\mu\) is small, \(\varphi(x,\dot x)\), \(f(x)\) and \(p(\omega t)\) are differentiable; \(\mu\varphi\) means the nonconservative force, \(f(x)\) the position and \(p(\omega t)\) the external force. Using qualitative analysis, the paper investigates existence and stability of harmonic and subharmonic oscillations for certain types of positions and external forces. The relations of these properties to the cases \(\mu=0\) and \(\mu\neq 0\) are examined. Several kinds of position forces (for example, the cases \(f_x(x)\geq 0\); \(f_x(x)< 0\); \(f(x)x<0\) for \(x\neq 0\); the case of a system with three equilibrium points) are taken into consideration. Finally, some cases are discussed, in which approximate analytical methods concerning stability criteria may be erroneous.