Van der Pol-Duffing oscillator: Global view of metamorphoses of the amplitude profiles

Abstract

Dynamics of the Duffing-Van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude A on forcing frequency Ω as an implicit function, F (A, Ω) = 0, referred to as resonance curve or amplitude profile.In singular points of the amplitude curve the conditions ∂F ∂A = 0, ∂F ∂Ω = 0 are fulfilled, i.e. in such points neither of the functions A = f (Ω), Ω = g (A), continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed.Several examples of singular points and the corresponding metamorphoses of dynamics are presented.

Localizing Bifurcations in Non-Linear Dynamical Systems via Analytical and Numerical Methods

“…We consider a damped periodically driven pendulum, applying our results obtained in [23]. Then we study a driven pendulum with van der Pol's type damping, generalizing our previous studies [30,31]. We show how changes of differential properties of asymptotic solutions are related to changes of existence and stability of dynamical states.…”

“…Equation (32) and Equation ( 3) of [31] are compatible (but, of course, not equivalent) for ν = 1 and λ = − 1 6 , and we obtain Equation ( 18) for ν = 0, µ = h, and G = f . The amplitude equation was computed by the Krylov-Bogoliubov-Mitropolsky method as:…”

“…This surface has a richer structure than the bifurcation set for the Duffing-van der Pol equation (i.e., M 1 ), cf. Figure 1 in [31]. We note that the surface d 3 = 0 (sienna) consists of several patches intersecting one another along some lines and intersected by the plane G = 0 (blue and green).…”

“…Firstly, we consider a damped periodically driven pendulum, applying our results obtained in [23]. Secondly, we study a driven pendulum with van der Pol's type damping, generalizing our previous studies [30,31]. Since we compute non-linear resonances (2) by the Krylov-Bogoliubov-Mitropolsky (KBM) method, which applies to polynomial non-linearities only, we have to expand the typical pendulum term sin(y), appearing in the function f y, dy dτ , τ; c in (1), and consider contributions from terms of the expansion sin(y) ∑ n k=1 c k y 2k+1 for growing n. In Section 3, we compute bifurcation sets D n (c) = 0, where D n is a non-linear function of parameters c and n + 1 is the number of terms in the expansion of sin(y).…”

“…It follows from Figure 6 that for some µ = µ 0 there are self-intersections of the surface d 3 (µ, ν, G) = 0. Therefore, a point (ν 0 , G 0 ) on the intersection curve can be computed from the following equations [31]:…”