Exact Nonlinear Fourth-order Equation for Two Coupled Oscillators: Metamorphoses of Resonance Curves

Abstract

We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov-BogoliubovMitropolsky approach -we compute the amplitude profiles, which from mathematical point of view are algebraic curves.In the present paper we investigate metamorphoses of amplitude profiles induced by changes of control parameters near singular points of these curves. It follows that dynamics changes qualitatively in the neighbourhood of a singular point.

Metamorphoses of resonance curves for two coupled oscillators: The case of small non-linearities in the main mass frame

“…(19) and (16) (2)) then the function L 2 becomes independent on Z. In this case it is possible to separate variables in Eqs.…”

“…The main result of this work is generalization of our earlier results on metamorphoses of amplitude profiles and bifurcations of dynamics. More exactly, we have designed method, based on the theory of singular points of 2D curves, permitting computation of parameter values at which qualitative changes (metamorphoses) of 2D amplitude curves occur [16][17][18][19]. We have also shown that metamorphoses of amplitude profiles are visible in bifurcation diagrams as qualitative changes of dynamics (bifurcations).…”

Metamorphoses of resonance curves in systems of coupled oscillators

“…it is possible to separate off the variable y to obtain the following equation for relative motion (Kyzioł and Okniński, 2013)…”

“…In our earlier papers, we have designed a method based on the theory of singular points of 2D curves, permitting computation of parameter values at which qualitative changes (metamorphoses) of 2D amplitude curves occur, see Kyzioł and Okniński (2013) and references therein. We have also shown that metamorphoses of amplitude profiles are visible in bifurcation diagrams as qualitative changes of dynamics (bifurcations).…”

“…Recently, our approach has been generalized to the case of 3D resonance curves and applied to compute bifurcations in dynamical system (1.1) with small nonlinearities in the main mass frame (Kyzioł, 2015). It is thus possible to treat system (1.1) as a small perturbation of model with linear functions R(x), V (ẋ) analyzed in Kyzioł and Okniński (2013) (let us recall that in this case internal motion can be separated off, leading to a simpler equation for the corresponding amplitude profile) and use the results obtained by Kyzioł and Okniński (2013). We show that the method is a powerful tool to predict bifurcations of nonlinear resonances present in such dynamical systems.…”