Metastable states of a ferromagnet on random thin graphs

We calculate the mean number of metastable states of an Ising ferromagnet on random thin graphs of fixed connectivity c. We find, as for mean field spin glasses that this mean increases exponentially with the number of sites, and is the same as that calculated for the +/- J spin glass on the same graphs. An annealed calculation of the number of metastable states of energy E is carried out. For small c, an analytic result is obtained. The result is compared with the one obtained for spin glasses in order to discuss the role played by loops on thin graphs and hence the effect of real frustration on the distribution of metastable states.Comment: 15 pages, 3 figure

Multiple scattering of ultrasound in weakly inhomogeneous media: application to human soft tissues

Waves scattered by a weakly inhomogeneous random medium contain a predominant single scattering contribution as well as a multiple scattering contribution which is usually neglected, especially for imaging purposes. A method based on random matrix theory is proposed to separate the single and multiple scattering contributions. The experimental set up uses an array of sources/receivers placed in front of the medium. The impulse responses between every couple of transducers are measured and form a matrix. Single-scattering contributions are shown to exhibit a deterministic coherence along the antidiagonals of the array response matrix, whatever the distribution of inhomogeneities. This property is taken advantage of to discriminate single from multiple-scattered waves. This allows one to evaluate the absorption losses and the scattering losses separately, by comparing the multiple scattering intensity with a radiative transfer model. Moreover, the relative contribution of multiple scattering in the backscattered wave can be estimated, which serves as a validity test for the Born approximation. Experimental results are presented with ultrasonic waves in the MHz range, on a synthetic sample (agar-gelatine gel) as well as on breast tissues. Interestingly, the multiple scattering contribution is found to be far from negligible in the breast around 4.3 MHz.Comment: 35 pages, 11 figures, final version, contains the appendix of the original articl

Dense heteroclinic tangencies near a Bykov cycle

This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes --- we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere

The Dark Z' Portal: Direct, Indirect and Collider Searches

We perform a detailed study of the dark Z' portal using a generic parametrization of the Z'-quarks couplings, both for light (8-15)GeV and heavy (130-1000)GeV dark matter scenarios. We present a comprehensive study of the collider phenomenology including jet clustering, hadronization, and detector artifacts, which allows us to derive accurate bounds from the search for new resonances in dijet events and from mono-jet events in the LHC 7TeV, LHC 8TeV, and Tevatron 1.96 TeV data. We also compute the dark matter relic abundance, the relevant scattering cross sections and pair-annihilation spectrum, and compare our results with the current PLANCK, Fermi-LAT and XENON100/LUX bounds. Lastly, we highlight the importance of complementary searches for dark matter, and outline the excluded versus still viable parameter space regions of the dark Z' portal.Comment: 29 pages, 21 figure

On Takens' Last Problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields $C^2$-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. We illustrate our results with an explicit example where historic behaviour arises $C^2$-close of a $\textbf{SO(2)}$-equivariant vector field

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions