Coupled nonlinear oscillators: metamorphoses of amplitude profiles. The case of the approximate effective equation

Exact nonlinear fourth-order equation for two coupled nonlinear oscillators: metamorphoses of resonance curves

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-Bogoliubov-Mitropolsky 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 qualitativel…

驱动范德波尔振荡器中共振的诞生与蜕变

Birth and metamorphoses of resonances in the driven van der Pol oscillator

本文利用幅频响应隐函数的微分性质，研究了驱动范德波尔振荡器中1:1和1:3共振的诞生与蜕变。

非对称Duffing振子：$1:2$共振的变形及其与主共振的相互作用

Asymmetric Duffing oscillator: metamorphoses of $1:2$ resonance and its interaction with the primary resonance

本文研究非对称Duffing振子中$1:2$共振的变形及其与主共振的相互作用，通过计算幅频隐函数并利用隐函数微分性质揭示其复杂动力学行为。

Coupled nonlinear oscillators: metamorphoses of amplitude profiles for the approximate effective equation - the case of 1:3 resonance

We study dynamics of two coupled periodically driven oscillators. An important example of such a system is a dynamic vibration absorber which consists of a small mass attached to the primary vibrating system of a large mass. Periodic solutions of the approximate effective equation (derived in our earlier papers) are determined within the Krylov-Bogoliubov-Mitropolsky approach to compute the amplitude profiles $A(Ω)$. In the present paper we investigate metamorphoses of the function $A(Ω)$ induced by changes of the control parameters in the case of 1:3 resonances.

Effective equation for two coupled oscillators: towards a global view of metamorphoses of the amplitude profiles

Dynamics of nonlinear coupled driven oscillators is investigated. Recently, we have demonstrated that the amplitude profiles -- dependence of the amplitude $A$ on frequency $Ω$ of the driving force, computed by asymptotic methods in implicit form as $F\left( A,Ω\right) =0$, permit prediction of metamorphoses of dynamics which occur at singular points of the implicit curve $F\left( A,Ω\right) =0$. In the present study we strive at a global view of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude …

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

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, $F\left( A,Ω\right) =0$, referred to as resonance curve or amplitude profile. In singular points of the amplitude curve the conditions $\frac{\partial F}{\partial A}=0$, $\frac{\partial F}{\partial Ω}=0$ are fulfilled, i.e. in such points neither of the functions $A=f\left( Ω\right) $, $Ω=g\left( A\right) $, contin…