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Equivariant Hopf bifurcation for functional differential equations of mixed type
2011
Applied Mathematics Letters
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In this paper we employ the Lyapunov-Schmidt procedure to set up equivariant Hopf bifurcation theory of functional differential equations of mixed type. ...
In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction. ...
Near a = a j,k for each j ∈ {1, 2} and k ∈ N, system (16) undergoes a Hopf bifurcation, the bifurcation direction is determined by the sign of a j,k g ′′′ (0). ...
doi:10.1016/j.aml.2010.12.017
fatcat:jw2m7ayiwzfqhalr27jbg4t2yu
S1-degree and global Hopf bifurcation theory of functional differential equations
1992
Journal of Differential Equations
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The recently developed S'-degree and bifrucation theory are applied to provide a purely topological argument of a global Hopf bifurcation theory for functional differential equations of mixed type. ...
In the special case where the equation is of retarded type, the established result represents an analog of Alexander and Yorke's global Hopf bifurcation theorem which has been obtained by Chow, Fiedler ...
Huaxing Xia for providing additional references. ...
doi:10.1016/0022-0396(92)90094-4
fatcat:3e4nkoqmd5hq5dxmj2anvkcrgu
PATTERNS OF OSCILLATION IN A RING OF IDENTICAL CELLS WITH DELAYED COUPLING
2007
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
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We show the existence of codimension two bifurcation points involving both standard and D 3 -equivariant, Hopf and pitchfork bifurcation points. ...
Further, these secondary bifurcations give rise to 10 different types of periodic solutions. ...
., 1997; Krawcewicz and Wu, 1999; Wu, 1998 ] extended the theory of equivariant Hopf bifurcation to systems with time delays (functional differential equations). ...
doi:10.1142/s0218127407018907
fatcat:bobsh3jxbzby5g3bb7toqdwmc4
Symmetry and bifurcation of periodic solutions in Neumann boundary value problems
2008
Portugaliae Mathematica
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For completeness, we include a description of the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steadystate/Hopf and Hopf/Hopf. ...
connecting at least one solution of standing wave type. ...
Rocha for a fruitful discussion of a preliminary version of this work. ...
doi:10.4171/pm/1818
fatcat:sbfbyvl2nnc3ddjbi3jb36e3yy
A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square
1995
Mathematics of Computation
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We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. ...
This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. ...
We also acknowledge support from a European Community Laboratory Twinning Grant during completion of this work. Bibliography ...
doi:10.1090/s0025-5718-1995-1284661-x
fatcat:scbnvscbf5bpbhuordjru3kk7i
Symmetry Breaking in Dynamical Systems
[chapter]
1996
Nonlinear Dynamical Systems and Chaos
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Then this cycle will reduce to a homoclinic orbit if we project the equation onto the orbit ~pace. Therefore techniques to study homoclinic bifurcations become available. ...
Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. ...
This does not remain true if we per-
turb the type of boundary condition, like Dirichlet to mixed or Neumann
to mixed, by adding small terms. ...
doi:10.1007/978-3-0348-7518-9_6
fatcat:tqphikirxneevi4uq2yz3ubtx4
Periodically Forced Hopf Bifurcation
2011
SIAM Journal on Applied Dynamical Systems
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It is worth remarking that the dynamics and bifurcation diagrams shown in [7] were given in the parameter space whose coordinates are functions of the forcing amplitude ε, the Hopf bifurcation parameter ...
Our results are presented in terms of bifurcation diagrams of amplitude of periodic solution versus ω for fixed ε and λ. ...
We thank Don Aronson, Edgar Knobloch, Claire Postlethwaite, and LieJune Shiau for helpful conversations. ...
doi:10.1137/10078637x
fatcat:u4qik43vqnditbylgzco3quhhq
Normal form for Hopf bifurcation of partial differential equations on the square
1995
Nonlinearity
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Recommended by R S MacKay Abstrad We derive and analyse anormal form governing dynamics of Hopf bifurwtions of paiial differential evolution equations on a square domain. ...
The nor& form reduces to that invesfigated by Swift [23] for bifurcation of modes with odd parity but is new for modes with even parity where the centre eigenspace carries a reducible action of 0 4 x St ...
z-) where f+(z+ z-) are k times differentiable functions with zero linear part, and equivariant under the appropriate action of DaxS' [24] . ...
doi:10.1088/0951-7715/8/5/004
fatcat:rjumtaf2graijbnk62iwbn366u
Recent advances in symmetric and network dynamics
2015
Chaos
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Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries ...
of periodic states in equivariant dynamics. ...
ACKNOWLEDGMENTS The research of M.G. was supported in part by NSF Grant No. DMS-0931642 to the Mathematical Biosciences Institute. ...
doi:10.1063/1.4918595
pmid:26428565
fatcat:qzpbaxnmune5nbl5i2nnmojspe
Mode Interactions With Symmetry
1995
Dynamics and stability of systems (Print)
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We. prove several results concerning problems invariant under the action of an arbitrary compact Lie group IF. These include the existence of mixed-mode solutions and secondary Hopf bifurcations. ...
Mixed-mode patterns are first considered in Busse and Riahi [6] where the study is made for two consecutive modes, i. e., neighbouring degrees I and I* of spherical harmonics. ...
These values of x and y determine the points at which a Hopf bifurcation occurs along the mixed-mode branch. ...
doi:10.1080/02681119508806192
fatcat:bo5tr32y7nb7pfci52slx7s7ga
Page 373 of Mathematical Reviews Vol. , Issue 96a
[page]
1996
Mathematical Reviews
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They apply their results to Hopf bifur-
cation for functional parabolic partial differential equations. ...
[Wu, Jian Hong] (3-YORK-MS; North York, ON)
Hopf bifurcation for parametrized equivariant coincidence problems
and parabolic equations with delays.
Funkcial. Ekvac. 37 (1994), no. 3, 415-446. ...
Hopf bifurcation on a sphere
2010
Nonlinearity
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The equivariant Hopf theorem guarantees the existence of periodic solutions with each of these symmetries in O(3)×S 1 equivariant differential equations. ...
We compute conditions for each of these solution branches to be stable and by restricting the O(3) × S 1 equivariant differential equations to four-dimensional invariant subspaces we are able to find additional ...
The author also wishes to thank Paul Matthews and Stephen Cox for their guidance and support throughout this work. This work formed part of the authors PhD thesis which was funded by the EPSRC. ...
doi:10.1088/0951-7715/23/12/011
fatcat:45u6ucjw6fahde3ufljqynr5pq
Pulsating and rotating spirals in a delayed feedback diffractive nonlinear optical system
[article]
2019
arXiv
pre-print
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Based on the explicitly constructed normal form of the Hopf bifurcation for the one-dimensional delayed scalar diffusion equation, we make predictions about the existence and stability of two-dimensional ...
Starting from a delayed scalar diffusion equation in a thin annulus with oblique derivative boundary conditions, we shrink the annulus and derive the limiting equation on a circle. ...
The normal form of the SO(2)-equivariant Hopf bifurcation is just the standard Hopf bifurcation and consists of 2 equations; so there are only two coefficients that define the qualitative properties of ...
arXiv:1909.02796v1
fatcat:7c6ll5zmxzhanmcmofr32eclm4
Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator
2014
Symmetry
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We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. ...
Some of the results apply to rate equations on more general networks. ...
Acknowledgments I am grateful to Marty Golubitsky for introducing me to rate equations and for many helpful conversations.
Conflicts of Interest The author declares no conflict of interest. ...
doi:10.3390/sym6010023
fatcat:kcyendrgkfdrzdil7k5s5756da
Page 2957 of Mathematical Reviews Vol. , Issue 96e
[page]
1996
Mathematical Reviews
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Finally, it is shown that Hopf bifurcations are expected to be encountered along the branch of mixed-mode solutions un- der quite general assumptions. ...
(RS-VORO; Voronezh) Smooth marginal analysis of bifurcation of extremals. Geometry in partial differential equations, 345-375, World Sci. Publishing, River Edge, NJ, 1994. ...
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