Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

Abstract

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

“…In this paper, we consider the limit case for the novel SD oscillator introduced in [18,19] as shown in Fig. 1.…”

“…A non-linear mechanical model with a lump mass and a pair of springs pinned to rigid supports; the oblique springs provide irrational non-linearity behavior with smooth and discontinuous characteristics [18,19]. branch x¼0 bifurcate to two unstable branches x ¼ 7 α and two stable branches x ¼ 7 1.…”

“…The rig-coupled SD system is based upon an SD oscillator [3] which derived from an arch model. This simple system has multiple well dynamics, high degeneracy singularities, chaotic behaviors, codimension three bifurcation phenomena and multi-stability or multiple snapthrough buckling [19]. The current study is focussed on the chaotic criterion for PWLD system with multiple well potentials in the limiting discontinuous case.…”

Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

“…Then this fashion went away and terminology from catastrophe returned to singularities, discontinuous bifurcations and so on. But the terms "catastrophe, catastrophe point" are used now too [22][23][24][25]. The Catastrophe theory is used in works of Saratov (in Russia) scientific school [26].…”

Chaotic threshold for a class of impulsive differential system

“…Moreover, the well-known Melnikov theory for smooth systems was extended to a class of two-zonal planar hybrid piecewise smooth systems which has a discontinuum of periodic orbits [15]. Furthermore, for other types of discontinuities, bifurcations and chaotic threshold have been among the most active fields of research due to its diverse applications in science and engineering [16][17][18][19][20].…”