Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point

2012 ◽  
Vol 22 (3) ◽  
pp. 033115 ◽  
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima
Keyword(s):  
Bifurcation Point ◽  
Van Der Pol ◽  
Rotating Waves ◽  
Codimension Two ◽  
Van Der Pol Oscillators ◽  
Codimension Two Bifurcation

NEW PHENOMENA IN THE NEIGHBORHOOD OF THE CODIMENSION-TWO BIFURCATION IN THE TWIN-WELL DUFFING OSCILLATOR

2000 ◽  
Vol 10 (06) ◽  
pp. 1367-1381 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
A. ZUBRZYCKI ◽  
E. TYRKIEL
Keyword(s):  
Bifurcation Point ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Codimension Two ◽  
Periodic Attractors ◽  
The Cross ◽  
Complex Phenomena ◽  
New Phenomena ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

Fuzzy Codimension Two Bifurcations

2011 ◽  
Author(s):  
Ling Hong ◽  
Jian-Qiao Sun
Keyword(s):  
Symmetry Breaking ◽  
Mapping Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Generalized Cell Mapping ◽  
Codimension Two ◽  
Deterministic Systems ◽  
Breaking Parameter ◽  
Fuzzy Noise ◽  
Codimension Two Bifurcation

By means of fuzzy generalized cell mapping method, a Duffing-Van der Pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period-one attractors grow and merge as the intensity of fuzzy noise is increased. By introducing a small symmetry-breaking parameter to break the symmetry, the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations. Considering both the intensity of fuzzy noise and the symmetry-breaking parameter together as controls, a codimension two bifurcation of fuzzy attractors is defined, and two examples of additive and multiplicative fuzzy noise are given. Such a codimension two bifurcation is fuzzy noise-induced effects which cannot be seen in the deterministic systems.

SHILNIKOV’S SADDLE-NODE BIFURCATION

1996 ◽  
Vol 06 (06) ◽  
pp. 1153-1160 ◽  
Author(s):  
PAUL GLENDINNING ◽  
COLIN SPARROW
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Infinite Number ◽  
Bifurcation Point ◽  
Stationary Point ◽  
Saddle Node Bifurcation ◽  
Codimension Two ◽  
Full Shift ◽  
Parameter Values ◽  
Codimension Two Bifurcation

In 1969, Shilnikov described a bifurcation for families of ordinary differential equations involving n≥2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols. We investigate a simple (piecewise-smooth) example with an infinite number of homoclinic loops. We also present a smooth example which shows how Shilnikov’s mechanism is related to the Lorenz bifurcation by considering the unfolding of a previously unstudied codimension two bifurcation point.

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

2014 ◽  
Vol 59 (9) ◽  
pp. 932-938
Author(s):  
V.A. Danylenko ◽  
◽  
S.I. Skurativskyi ◽  
I.A. Skurativska ◽  
◽  
...  
Keyword(s):  
Van Der Pol ◽  
Wave Solutions ◽  
Van Der Pol Oscillators

scholarly journals Modal synchronization of coupled bistable van der Pol oscillators

2021 ◽  
Vol 143 ◽  
pp. 110555
Author(s):  
I.B. Shiroky ◽  
O.V. Gendelman
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Oscillators

scholarly journals Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

2017 ◽  
Vol 4 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Mohit Sinha ◽  
Florian Dorfler ◽  
Brian B. Johnson ◽  
Sairaj V. Dhople
Keyword(s):  
Nonlinear Dynamics ◽  
Droop Control ◽  
Van Der Pol ◽  
Control Laws ◽  
Van Der Pol Oscillators

scholarly journals Dynamics of hierarchical weighted networks of van der Pol oscillators

2020 ◽  
Vol 30 (12) ◽  
pp. 123146
Author(s):  
Daniel Monsivais-Velazquez ◽  
Kunal Bhattacharya ◽  
Rafael A. Barrio ◽  
Philip K. Maini ◽  
Kimmo K. Kaski
Keyword(s):  
Van Der Pol ◽  
Weighted Networks ◽  
Van Der Pol Oscillators
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