Abstract
In this paper, we study the bifurcations of non-linear dynamical systems. We continue to develop the analytical approach, permitting the prediction of the bifurcation of dynamics. Our approach is based on implicit (approximate) amplitude-frequency response equations of the form FΩ,A;c̲ =0, where c̲ denotes the parameters. We demonstrate that tools of differential geometry make possible the discovery of the change of differential properties of solutions of the equation FΩ,A;c̲ =0. Such qualitative changes of the solutions of the amplitude-frequency response equation, referred to as metamorphoses, lead to qualitative changes of dynamics (bifurcations). We show that the analytical prediction of metamorphoses is of great help in numerical simulation.
Highlights
Non-linear dynamical systems find important applications in science and technology.It is necessary to analyse and solve non-linear equations governing dynamics of such systems with high accuracy
A novel contribution of our work consists of (i) the classification of a wide variety of metamorphoses of the dynamics of steady-state solutions of non-linear equations due to changes of the differential properties of the amplitude-frequency response curves; (ii) the definition and description of the construction of borderline sets, i.e., sets containing parameters for which jump phenomena are present; (iii) the computation of numerically exact bifurcation sets for pendulum systems using a standard asymptotic approach; and (iv) the demonstration of bifurcations corresponding to parameters belonging to the bifurcation set
For increasing values of n, the physical values of parameters c belonging to bifurcation sets Dn (c) = 0 converge quickly, providing a good approximation of the pendulum’s singular points
Summary
Non-linear dynamical systems find important applications in science and technology. It is necessary to analyse and solve non-linear equations governing dynamics of such systems with high accuracy. A novel contribution of our work consists of (i) the classification of a wide variety of metamorphoses of the dynamics of steady-state solutions of non-linear equations due to changes of the differential properties of the amplitude-frequency response curves; (ii) the definition and description of the construction of borderline sets, i.e., sets containing parameters for which jump phenomena are present; (iii) the computation of numerically exact bifurcation sets for pendulum systems using a standard asymptotic approach (this is a new result since pendulums are non-polynomial systems while standard asymptotic methods can be applied to polynomial systems only); and (iv) the demonstration of bifurcations corresponding to parameters belonging to the bifurcation set. Metamorphoses occur—a number of branches of the asymptotic solutions are changed, and this corresponds to the vertical tangency of the amplitude-frequency curve and the jump phenomenon for dynamics of the Duffing equation [18,24]. We draw the borderline sets determined by Equations (8) and (9), see Figure 2
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