SINGULAR HOPF BIFURCATION PROBLEMS AND ROTATING–SLIDING SPIRAL FLOWS

1994 ◽  
pp. 345-360
Author(s):  
GEORGE H. KNIGHTLY ◽  
D. SATHER
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Problems

Two new approaches to computing Hopf bifurcation problems

1994 ◽  
Vol 4 (12) ◽  
pp. 2203-2215
Author(s):  
Wu Baisheng ◽  
Tassilo Küpper
Keyword(s):  
Hopf Bifurcation ◽  
New Approaches ◽  
Bifurcation Problems

Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology

2009 ◽  
Vol 2 (3) ◽  
pp. 493-515 ◽  
Author(s):  
Thomas Sturm ◽  
Andreas Weber ◽  
Essam O. Abdel-Rahman ◽  
M’hammed El Kahoui
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Problems ◽  
Algebraic Biology

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

2020 ◽  
Vol 30 (02) ◽  
pp. 2050028 ◽  
Author(s):  
Chuncheng Wang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Transmission Line ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Line Equation ◽  
Bifurcation Problems ◽  
Functional Differential

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.

scholarly journals Flow Around a Slender Circular Cylinder: A Case Study on Distributed Hopf Bifurcation

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
J. A. P. Aranha ◽  
K. P. Burr ◽  
I. C. Barbeiro ◽  
I. Korkischko ◽  
J. R. Meneghini
Keyword(s):  
Hopf Bifurcation ◽  
Circular Cylinder ◽  
Chaotic Behavior ◽  
Landau Equation ◽  
Lift Coefficient ◽  
Harmonic Oscillation ◽  
Length Scale ◽  
Ginzburg Landau ◽  
Bifurcation Problems

This paper presents a short overview of the flow around a slender circular cylinder, the purpose being to place it within the frame of the distributed Hopf bifurcation problems described by the Ginzburg-Landau equation (GLE). In particular, the chaotic behavior superposed to a well tuned harmonic oscillation observed in the range Re > 270, with Re being the Reynolds number, is related to the defect-chaos regime of the GLE. Apparently new results, related to a Kolmogorov like length scale and thermsof the response amplitude, are derived in this defect-chaos regime and further related to the experimentalrmsof the lift coefficient measured in the range Re > 270.

scholarly journals Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120467 ◽  
Author(s):  
C. M. Postlethwaite ◽  
G. Brown ◽  
M. Silber
Keyword(s):  
Hopf Bifurcation ◽  
Unstable Periodic Orbits ◽  
Feedback Controls ◽  
Bifurcation Theorem ◽  
Non Invasive ◽  
Subcritical Hopf Bifurcation ◽  
Bifurcation Problems ◽  
Spatio Temporal ◽  
The Stability ◽  
Equivariant Hopf Bifurcation

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

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