Degenerate Bifurcation Scenarios in the Dynamics of Coupled Oscillators With Symmetric Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. Both the potential of the oscillator and of the coupling spring are adopted to be even-power polynomials with nonnegative coefficients. By defining the coupling parameter ε, the dynamics of this system at the limit ε → 0 and for finite ε is investigated. Bifurcation scenario of the nonlinear normal modes is revealed. The degeneracy in the dynamics is manifested by a ‘bifurcation from infinity’ where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by mechanism of successive cusp catastrophes with growth of the coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincare sections). For particular case of pure cubic nonlinearity of the oscillator and the coupling spring good agreement between quantitative analytical predictions and numerical results is observed.