Harmonic balance method and global analysis of dynamical systems
One of the main problems in the global analysis of oscillatory systems is the finding of all its periodic motions for given correlations of the parameters and priori considerations give reason to believe that for systems of differential equations with polynomial nonlinearity the use of the harmonic balance method (HBM) for this purpose seems to be very attractive. Indeed, the HBM enables us to reduce the finding of stationary motions of such systems to the solving of systems of polynomial equations, the number of solutions of which, presumably, can be set using the theory of Newton polyhedra. And, then, with the help of the interval approaches or methods of continuation, which are currently being developed within the framework of tropical geometry, you can determine the whole set of solutions of polynomial equations and thus, the entire range of motions of the dynamical system. In this paper, this hypothesis is being tested for the differential equation with cubic nonlinearity and harmonic exciting force. We consider two versions of HBM, - trigonometric one and complex exponential form. On their basis for the differential equation with cubic nonlinearity the construction of polynomial equations is fulfilled and in accordance with the theorem of Bernstein, attempt to estimate the number of solutions of the obtained system has been undertaken. Then, with use of interval bisection method solutions of the system of polynomial equations in a given part of phase space are determined, comparative evaluation of the complexity of the considered versions of HBM is conducted, advantages and disadvantages of the described approach are marked.