1,975 research outputs found
Transition from rotating waves to modulated rotating waves on the sphere
We study non-resonant and resonant Hopf bifurcation of a rotating wave in
SO(3)-equivariant reaction-diffusion systems on a sphere. We obtained reduced
differential equations on so(3), the characterization of modulated rotating
waves obtained by Hopf bifurcation of a rotating wave, as well as results
regarding the resonant case. Our main tools are the equivariant center manifold
reduction and the theory of Lie groups and Lie algebras, especially for the
group SO(3) of all rigid rotations on a sphere
A reversible bifurcation analysis of the inverted pendulum
The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.
Symmetries in the Lorenz-96 model
The Lorenz-96 model is widely used as a test model for various applications,
such as data assimilation methods. This symmetric model has the forcing
and the dimension as parameters and is
equivariant. In this paper, we unravel its dynamics for
using equivariant bifurcation theory. Symmetry gives rise to invariant
subspaces, that play an important role in this model. We exploit them in order
to generalise results from a low dimension to all multiples of that dimension.
We discuss symmetry for periodic orbits as well.
Our analysis leads to proofs of the existence of pitchfork bifurcations for
in specific dimensions : In all even dimensions, the equilibrium
exhibits a supercritical pitchfork bifurcation. In dimensions
, , a second supercritical pitchfork bifurcation occurs
simultaneously for both equilibria originating from the previous one.
Furthermore, numerical observations reveal that in dimension , where
and is odd, there is a finite cascade of exactly
subsequent pitchfork bifurcations, whose bifurcation values are independent
of . This structure is discussed and interpreted in light of the symmetries
of the model.Comment: 31 pages, 9 figures and 3 table
Bifurcation from relative periodic solutions
Published versio
Reversible relative periodic orbits
AbstractWe study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits. Our results extend those recently obtained in the equivariant context by B. Sandstede et al. (1999, J. Nonlinear Sci.9, 439–478) and C. Wulff et al. (2001, Ergodic Theory Dynam. Systems21, 605–635)
Non-degenerate umbilics, the path formulation and gradient bifurcation problems
Parametrised contact-equivalence is successful for the understanding and classification of the qualitative local behaviour of bifurcation
diagrams and their perturbations. Path formulation is an alternative point of view. It makes explicit the singular behaviour due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter.
Here we show how path formulation can be used to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the non degenerate umbilics singularities are the generic cores in four situations: the general or gradient problems and the Z_2-equivariant (general or gradient) problems
where Z_2 acts on the second component of R^2 via
the reflection kappa(x,y)=(x,-y). The universal unfolding of the umbilic singularities have an interesting 'Russian doll' type of structure
of universal unfoldings in all those categories.
In our approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance some internal hierarchy). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new.
We discuss some application to the bifurcation of a cylindrical panel under different loads structure. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes
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