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

Metamorphoses of resonance curves in systems of coupled oscillators

“…More exactly, singular points of amplitude profiles are computed. We demonstrate that qualitative changes of dynamics, referred to as metamorphoses, induced by changes of control parameters, occur in neighbourhoods of singular points of amplitude profiles, see also Kyzioł andOkniński (2011, 2013) and references therein.…”

“…Eq. (1.1), initiated in Kyzioł (2015). More exactly, we have investigated: We have studied the amplitude (resonance) equations for steady states, obtained via the KBM approach, within the theory of singular points of 2D and 3D algebraic curves (Wall, 2004;Hartmann, 2003).…”

“…Equations (4.5) and these conditions are used to compute singular points. We have shown in our previous papers that qualitative changes of dynamics, induced by changes of control parameters, occur in neighbourhoods of singular points of amplitude profiles (Kyzioł andOkniński, 2011, 2013;Kyzioł, 2015).…”

“…and functions g(u, u,ż, z, τ ), k(u, u,ż, z, τ ) are defined in (Kyzioł, 2015). Equations (4.1) have been prepared in such a way that for ε = 0 the solutions are u(τ ) = B cos(Ωτ + ψ), z(τ ) = A cos(Ωτ + ϕ).…”

“…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).…”