Conditions for the existence of isolated backbone curves

Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example—a single-mode nonlinear structure with a nonlinear tuned mass damper—is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.