Generic bifurcation

1971 ◽  
pp. 128-129
Author(s):  
K. Meyer
Keyword(s):  
Generic Bifurcation

Generic bifurcation and symmetry with an application to the Von Kármán equations

1978 ◽  
Vol 81 (3-4) ◽  
pp. 211-235 ◽  
Author(s):  
André L. Vanderbauwhede
Keyword(s):  
Rectangular Plate ◽  
Bifurcation Theory ◽  
Normal Load ◽  
Additional Symmetry ◽  
Von Karman ◽  
Symmetry Properties ◽  
Von Kármán Equations ◽  
Generic Bifurcation ◽  
Von Kármán

SynopsisA generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

scholarly journals Generic bifurcation of steady-state solutions

1984 ◽  
Vol 52 (3) ◽  
pp. 432-438 ◽  
Author(s):  
J Smoller ◽  
A Wasserman
Keyword(s):  
Steady State ◽  
Generic Bifurcation ◽  
Steady State Solutions

On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes

2007 ◽  
Author(s):  
C. H. Pak ◽  
Y. S. Choi
Keyword(s):  
Normal Mode ◽  
Natural Frequencies ◽  
Normal Modes ◽  
New Normal ◽  
Perturbed System ◽  
Saddle Node ◽  
Non Linear ◽  
The Difference ◽  
Generic Bifurcation ◽  
The Stability

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

Piecewise-Linear Map for Studying Border Collision Phenomena in DC/AC Converters

2020 ◽  
Vol 30 (07) ◽  
pp. 2030015 ◽  
Author(s):  
Viktor Avrutin ◽  
Zhanybai T. Zhusubaliyev
Keyword(s):  
Control Strategies ◽  
Piecewise Linear ◽  
Piecewise Smooth ◽  
Piecewise Linear Approximation ◽  
Specific Class ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Bifurcation Phenomena ◽  
Generic Bifurcation ◽  
Ac Converters

Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border points in 1D). Obviously, the multitude of control strategies applied in the modern power electronics leads to different maps belonging to this class of models. However, in this paper we show that several models can be studied using the same piecewise-linear approximation, so that the bifurcation phenomena which can be observed in this approximation are generic for many models. Based on the results obtained before for piecewise-smooth models with different kinds of nonlinearities resulting from the corresponding control strategies, in the present paper we discuss the generic bifurcation patterns in the underlying piecewise-linear map.

The middle Pleistocene transition as a generic bifurcation on a slow manifold

2015 ◽  
Vol 45 (9-10) ◽  
pp. 2683-2695 ◽  
Author(s):  
Peter Ashwin ◽  
Peter Ditlevsen
Keyword(s):  
Slow Manifold ◽  
Middle Pleistocene ◽  
Middle Pleistocene Transition ◽  
Generic Bifurcation

On the sensitivity of non-generic bifurcation of non-linear normal modes

2007 ◽  
Vol 42 (7) ◽  
pp. 973-980 ◽  
Author(s):  
C.H. Pak ◽  
Y.S. Choi
Keyword(s):  
Normal Modes ◽  
Non Linear ◽  
Generic Bifurcation

LIMIT CYCLE BIFURCATION FROM CENTER IN SYMMETRIC PIECEWISE-LINEAR SYSTEMS

1999 ◽  
Vol 09 (05) ◽  
pp. 895-907 ◽  
Author(s):  
EMILIO FREIRE ◽  
ENRIQUE PONCE ◽  
JAVIER ROS
Keyword(s):  
Linear Systems ◽  
Limit Cycle ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Stability Change ◽  
Stable Periodic ◽  
Limit Cycle Bifurcation ◽  
Generic Bifurcation ◽  
The Stability ◽  
Limit Cycle Amplitude

The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopf's case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given. The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmann's analysis are also included.

scholarly journals On damping in the vicinity of critical points

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120426 ◽  
Author(s):  
L. N. Virgin ◽  
R. Wiebe
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Damping Ratio ◽  
Transcritical Bifurcation ◽  
Nonlinear Dynamical ◽  
Free Decay ◽  
Generic Bifurcation ◽  
Primary Focus ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Bifurcation Behaviour

The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated. Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added. Oscillators with saddle–node, pitchfork and transcritical bifurcations are shown analytically to exhibit several interesting characteristics in the free decay response near a bifurcation. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. A transcritical bifurcation was selected because it may be used to represent generic bifurcation behaviour. It is shown that the damping ratio can be used to predict a change in the stability with respect to changing system parameters.

Sign in / Sign up

Export Citation Format

Share Document