Corner bifurcations in non-smoothly forced impact oscillators

Abstract

Simultaneous impacts of multiple objects in mechanical systems are considered. It will be shown that in some cases the dynamics in a neighbourhood of such events can be explained by piecewise linear two-dimensional maps. These maps give rise to a variety of complex dynamics, which will be analysed. A special emphasis will be put on period-adding cascades.

Discontinuity-induced bifurcations in TCP/RED communication algorithms

“…In particular, it was shown [5] that corner bifurcations in impact oscillators may be associated with bifurcation diagrams characterized by period-adding cascades, devil staircases, and phenomena similar to bifurcations described in this paper. The observed phenomena were explained in terms of border-collisions of fixed points of two-dimensional piecewise linear maps with a gap.…”

“…Period-adding cascades with orbits of increasing period alternate with bands of more complex behavior. This behavior was observed in a two-dimensional discontinuous map described in [5].…”

“…A stable period-2 region is observed for p max ∈ [0.1, 0.2], when the average queue size is either 7.5 or 8.2 packets. As p max increases, the system exhibits a devil staircase (predicted in [5] for maps with a gap), before locking to higher-periodic solutions for larger p max implying that more packers are dropped (marked) as indicated by large variations of the average queue size. …”

“…Such maps have been recently studied in [5]- [7]. It was shown that they can exhibit a novel type of so-called discontinuityinduced bifurcations [5] that may organize intricate bifurcation diagrams characterized by period-adding cascades and sudden transitions to higher-periodic orbits and chaotic attractors.…”

Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems

“…Particularly, cumbersome dynamics can be observed in the case of impact oscillators with moving boundaries. For example, in [11], it is suggested that a novel bifurcation phenomenon termed as corner-impact can occur in discontinuously forced impact oscillators.…”