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

• System of two coupled periodically driven Duffing oscillators is studied in general case.
• Implicit equations for amplitude profiles of non-linear resonances are derived.
• Singular points of the resulting 3D curve are computed.
• Changes of dynamics near the singular points of the curve are studied.

We study dynamics of two coupled periodically driven oscillators in general case. Periodic steady-state solutions of the system of two equations are determined within the Krylov–Bogoliubov–Mitropolsky approach. The corresponding amplitude profiles, A(ω),B(ω)A(ω),B(ω), which are given by two implicit equations, F(A,B,ω)=0,G(A,B,ω)=0F(A,B,ω)=0,G(A,B,ω)=0, where ωω is frequency of the driving force, are computed. These two equations, each describing a surface, define a 3D3D curve-intersection of these surfaces. In the present paper we carry out preliminary investigation of metamorphoses of this curve, induced by changes of control parameters. The corresponding changes of dynamics near singular points of the curve are studied.