Bifurcation diagram for transition to chaos with nonzero initial θ

Construct a bifurcation diagram that characterizes the transition to chaos in the conservative two-body fallen pendula system—two masses connected by a rope that slides without friction over a rod of radius R with a screw-like surface, described by the dimensionless equations of motion for the variables φ, θ, and ℓ—specifically for cases with nonzero initial value of θ, where the prevalence of rapidly terminating trajectories impedes conventional bifurcation analysis.

Background

The paper investigates a conservative mechanical system consisting of two masses connected by a rope that can slide without friction over a rod with a screw-like surface, shifting the oscillation planes of the two loads. The system often exhibits transient chaos with finite lifetimes, as oscillations end when the rope segments l1, l2, or the rod-contact segment l3 reach zero.

Standard tools such as Poincaré sections and largest Lyapunov exponents require long-lived trajectories, but for many initial conditions, the motion ends quickly. The authors report that when θ is nonzero, constructing a conventional bifurcation diagram of the transition to chaos was not feasible due to the quickly ending trajectories, and they resorted instead to plotting lifetime (escape time) and using FTLE maps.