On a problem of a field method and its applications to the nonlinear vibrations (Q1412602)

Investigation of a dynamical system \[ dx_1/dt= X_1(t, x_1,\dots, x_n),\dots, dx_n/dt= X_n(t, x_1,\dots, x_n) \] is equivalent with the study of the first-order partial differential equation \(\partial U/\partial t+ \sum X_i\partial U/\partial x_i= 0\) for the first integrals \(U(t,x_1,\dots, x_n)= \text{const.}\) of the system. The author recalls some details of this classical idea and thoroughly deals with several applications to the perturbations of nonlinear oscillations. In particular approximate solutions of the system \(\dot x= p\), \(\dot p= -x+\mu(1+ x^4)p\) and \(\dot x= z\), \(\dot z=-x-\mu x^5\) with small parameter \(\mu\) are discussed.