Resonances and Twist in Volume-Preserving Mappings

The phase space of an integrable, volume-preserving map with one action and $d$ angles is foliated by a one-parameter family of $d$-dimensional invariant tori. Perturbations of such a system may lead to chaotic dynamics and transport. We show that near a rank-one, resonant torus these mappings can be reduced to volume-preserving "standard maps." These have twist only when the image of the frequency map crosses the resonance curve transversely. We show that these maps can be approximated---using averaging theory---by the usual area-preserving twist or nontwist standard maps. The twist condition appropriate for the volume-preserving setting is shown to be distinct from the nondegeneracy condition used in (volume-preserving) KAM theory.Comment: Many typos fixed and notation simplified. New $n^{th}$ order averaging theorem and volume-preserving variant. Numerical comparison with averaging adde

A NORMALLY ELLIPTIC HAMILTONIAN BIFURCATION

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained. having 'integrable' approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

A NORMALLY ELLIPTIC HAMILTONIAN BIFURCATION

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained. having 'integrable' approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

A NORMALLY ELLIPTIC HAMILTONIAN BIFURCATION

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained. having 'integrable' approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

Bifurcation curves of subharmonic solutions

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -- i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory.Comment: 29 pages, 2 figure

Burden of genetic risk variants in multiple sclerosis families in the Netherlands

Background: Approximately 20% of multiple sclerosis patients have a family history of multiple sclerosis. Studies of multiple sclerosis aggregation in families are inconclusive. Objective: To investigate the genetic burden based on currently discovered genetic variants for multiple sclerosis risk in patients from Dutch multiple sclerosis multiplex families versus sporadic multiple sclerosis cases, and to study its influence on clinical phenotype and disease prediction. Methods: Our study population consisted of 283 sporadic multiple sclerosis cases, 169 probands from multiplex families and 2028 controls. A weighted genetic risk score based on 102 non-human leukocyte antigen loci and HLA-DRB1*1501 was calculated. Results: The weighted genetic risk score based on all loci was significantly higher in familial than in sporadic cases. The HLA-DRB1*1501 contributed significantly to the difference in genetic burden between the groups. A high weighted genetic risk score was significantly associated with a low age of disease onset in all multiple sclerosis patients, but not in the familial cases separately. The genetic risk score was significantly but modestly better in discriminating familial versus sporadic multiple sclerosis from controls. Conclusion: Familial multiple sclerosis patients are more loaded with the common genetic variants than sporadic cases. The difference is mainly driven by HLA-DRB1*1501. The predictive capacity of genetic loci is poor and unlikely to be useful in clinical settings.</p

Effect of Ordering on Spinodal Decomposition of Liquid-Crystal/Polymer Mixtures

Partially phase-separated liquid-crystal/polymer dispersions display highly fibrillar domain morphologies that are dramatically different from the typical structures found in isotropic mixtures. To explain this, we numerically explore the coupling between phase ordering and phase separation kinetics in model two-dimensional fluid mixtures phase separating into a nematic phase, rich in liquid crystal, coexisting with an isotropic phase, rich in polymer. We find that phase ordering can lead to fibrillar networks of the minority polymer-rich phase