scholarly journals The Slow Passage through a Hopf Bifurcation: Delay, Memory Effects, and Resonance

1989 ◽  
Vol 49 (1) ◽  
pp. 55-71 ◽  
Author(s):  
S. M. Baer ◽  
T. Erneux ◽  
J Rinzel
Keyword(s):  
Hopf Bifurcation ◽  
Memory Effects ◽  
Bifurcation Delay ◽  
Slow Passage

The slow passage through a steady bifurcation: Delay and memory effects

1987 ◽  
Vol 48 (5-6) ◽  
pp. 1059-1070 ◽  
Author(s):  
Paul Mandel ◽  
Thomas Erneux
Keyword(s):  
Memory Effects ◽  
Bifurcation Delay ◽  
Slow Passage

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Hopf-bifurcation-delay-induced bursting patterns in a modified circuit system

2016 ◽  
Vol 36 ◽  
pp. 517-527 ◽  
Author(s):  
Xiujing Han ◽  
Fubing Xia ◽  
Peng Ji ◽  
Qinsheng Bi ◽  
Jürgen Kurths
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Delay

Control of bifurcation-delay of slow passage effect by delayed self-feedback

2017 ◽  
Vol 27 (1) ◽  
pp. 013104 ◽  
Author(s):  
D. Premraj ◽  
K. Suresh ◽  
Tanmoy Banerjee ◽  
K. Thamilmaran
Keyword(s):  
Passage Effect ◽  
Bifurcation Delay ◽  
Slow Passage
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