A Degenerate bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to a nonlinear one that possesses essential stiffness nonlinearity. By defining the small coupling parameter ε, we study the dynamics of this system at the limit ε → 0. The degeneracy in the dynamics is manifested by a 'bifurcation from infinity' where a bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) bifurcation point is generated close to the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The degenerate bifurcation structure can be directly attributed to the high degeneracy of the uncoupled system in the limit ε → 0, whose linearized structure possesses a double zero, and a conjugate pair of purely imaginary eigenvalues. First we construct local analytical approximations to the NNMs in the neighborhoods of the bifurcation points and at other energy ranges of the system. Then, we 'connect' the local approximations by global approximants, and identify global branches of NNMs where unstable and stable mode and inverse mode localization between the linear and nonlinear oscillators take place for decreasing energy.