11,794 research outputs found
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
Singularity theory study of overdetermination in models for L-H transitions
Two dynamical models that have been proposed to describe transitions between
low and high confinement states (L-H transitions) in confined plasmas are
analysed using singularity theory and stability theory. It is shown that the
stationary-state bifurcation sets have qualitative properties identical to
standard normal forms for the pitchfork and transcritical bifurcations. The
analysis yields the codimension of the highest-order singularities, from which
we find that the unperturbed systems are overdetermined bifurcation problems
and derive appropriate universal unfoldings. Questions of mutual equivalence
and the character of the state transitions are addressed.Comment: Latex (Revtex) source + 13 small postscript figures. Revised versio
The case of the trapped singularities
A case study in bifurcation and stability analysis is presented, in which
reduced dynamical system modelling yields substantial new global and predictive
information about the behaviour of a complex system. The first smooth pathway,
free of pathological and persistent degenerate singularities, is surveyed
through the parameter space of a nonlinear dynamical model for a
gradient-driven, turbulence-shear flow energetics in magnetized fusion plasmas.
Along the route various obstacles and features are identified and treated
appropriately. An organizing centre of low codimension is shown to be robust,
several trapped singularities are found and released, and domains of
hysteresis, threefold stable equilibria, and limit cycles are mapped.
Characterization of this rich dynamical landscape achieves unification of
previous disparate models for plasma confinement transitions, supplies valuable
intelligence on the big issue of shear flow suppression of turbulence, and
suggests targeted experimental design, control and optimization strategies.Comment: 21 pages, 12 figures, 34 postscript figure file
Dynamics of a hyperbolic system that applies at the onset of the oscillatory instability
A real hyperbolic system is considered that applies near the onset of the oscillatory instability in large spatial domains. The validity of that system requires that some intermediate scales (large compared with the basic wavelength of the unstable modes but small compared with the size of the system) remain inhibited; that condition is analysed in some detail. The dynamics associated with the hyperbolic system is fully analysed to conclude that it is very simple if the coefficient of the cross-nonlinearity is such that , while the system exhibits increasing complexity (including period-doubling sequences, quasiperiodic transitions, crises) as the bifurcation parameter grows if ; if then the system behaves subcritically. Our results are seen to compare well, both qualitatively and quantitatively, with the experimentally obtained ones for the oscillatory instability of straight rolls in pure Rayleigh - Bénard convection
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