MS08.3: Non-Smooth Dynamics

Extending a previous analysis of the Hopf bifurcation at infinity in symmetric piecewise linear differential systems with three zones, which is responsible for the appearance of a large amplitude limit cycle, in this work we analyze the degeneration of such a bifurcation at infinity, providing a theoretical result that guarantees the existence of a saddle-node bifurcation curve of periodic orbits. Combining this new result with other known bifurcation results, we analyze an extended Bonhöffer-van der Pol piecewise linear electronic oscillator. We get a deeper insight on the dynamical richness of this oscillator, showing rigorously the simultaneous existence of six limit cycles in some region of its bifurcation set.}

An extension of a collocation based periodic orbit continuation framework developed for piecewise-smooth delay differential equations is presented, to incorporate neutral delayed terms in the problem formulation. Numerical difficulties arising from the consequent disappearance of the natural smoothing property of delay differential equations are discussed in detail, along with mesh improvement approaches which may help mitigate this issue. The capabilities of the extended continuation framework are demonstrated on a low-degree-of-freedom traffic dynamics model, where driver behaviour is modelled with the help of piecewise-smooth controllers, which also take reaction delays into account.

A wing flap with free play interacting with the flow is is studied using the latest ideas in non-smooth dynamical systems theory in this paper. Piecewise linear spring is used to model the restoring moment due to the freeplay in rotational hinge. The non-smoothness in such a piecewise-smooth system (PWS ) will create enormous interesting dynamical behaviours. High contact stiffness assumption is used to further simplify the model as impacting hybrid system and the restitution coefficient is to characterize the hardness of the impact. Special bifurcations like BEB (boundary equilibrium bifurcation), fold, grazing bifurcation are found in the simulation. We observed a Hopf-like bifurcation, period one LCOs (limit cycle oscillations) born after the BEB. Regarding such phenomenon, only results of planar system are given in the literature and we first come up with a semi-analytical method to determine the existence and stability of the in such higher dimensional system ($n>2$). The numerical continuation is used to track and unfold the bifurcations of the LCOs and the transition to the chaotic attractor is investigated. The conclusion is that the free-play can introduce different attractors in the system, and the softer impact will suppress the single impact LCO.

This work explores discontinuities in dynamical systems, emphasizing the pivotal concept of the switching surface, where continuity is lost. This phenomenon holds particular significance in mechanical systems, contributing to stick-slip vibrations, impact, or a combination of these phenomena. Achieving uniqueness in the presence of such discontinuities poses challenges. Rigorous ideas from piecewise-smooth dynamics and singular perturbation theory provide valuable perspectives on the problem. Filippov employed the convex combination of neighbouring vector fields, defining Filippov sliding dynamics (linear switching). Jeffrey extended this framework and incorporated a nonlinear term into the Filippov convex model. This term, referred to as the hidden term, vanishes outside the switching surface, marking the introduction of hidden dynamics (nonlinear switching). We employ hidden dynamics to investigate a self-excited smooth discontinuous (SD) oscillator with geometric nonlinearity at the switching surface. This exploration highlights a gap in Filippov’s theory regarding dry friction oscillators, where the static friction coefficient is often larger than the kinetic coefficient. By modelling the belt friction in the SD oscillator as Coulomb friction, we investigate the consequences of the discontinuity in the friction model. The sliding regions are calculated from the theory and verified using numerical simulations. Bifurcations within the system are discussed, and a comparative analysis of the system’s dynamics under Filippov and hidden dynamics theories is presented. Additionally, we analyze the system’s response to harmonic excitation, providing valuable insights into its behaviour under external forces.