Operator methods for calculating Lyapunov values in problems on local bifurcations of dynamical systems

Abstract

In the work we consider basic scenarios of local bifurcations in dynamical systems. We study the systems described by autonomous differential equations, discrete equations, as well as by non-autonomous periodic equations. We provide new formulae for calculating Lyapunov values. The formulae are obtained on the basis of a general operator approach for studying local bifurcations and they do not assume passing to normal forms and using the theorems on a central manifold. This method allows us to obtain new bifurcation formulae for studying main scenarios of local bifurcations. In the work we show how these bifurcation formulae lead one to new formulae for calculating Lyapunov values in problems on equilibria bifurcation, in Andronov–Hopf problems, in problems of doubling period, in problems on forced oscillations, etc. In the paper, the main attention is paid to obtain the first and the second Lyapunov value. The proposed approach allows us obtain Lyapunov values of higher order. As an application of the obtained formulae, in the paper we analyze basic scenarios of local bifurcations. We consider the problems on the direction of bifurcations, on stability of emerging solutions, on leading asymptotics for the solutions, etc. As an example, we calculate the Lyapunov values for Andronov–Hopf bifurcation in Langford system and for the problems on doubling period in Henon model.