Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems

We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries characterising the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proven here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Analysis of the Equilibrium and Kinetics of the Ankyrin Repeat Protein Myotrophin

We apply the Wako-Saito-Munoz-Eaton model to the study of Myotrophin, a small ankyrin repeat protein, whose folding equilibrium and kinetics have been recently characterized experimentally. The model, which is a native-centric with binary variables, provides a finer microscopic detail than the Ising model, that has been recently applied to some different repeat proteins, while being still amenable for an exact solution. In partial agreement with the experiments, our results reveal a weakly three-state equilibrium and a two-state-like kinetics of the wild type protein despite the presence of a non-trivial free-energy profile. These features appear to be related to a careful "design" of the free-energy landscape, so that mutations can alter this picture, stabilizing some intermediates and changing the position of the rate-limiting step. Also the experimental findings of two alternative pathways, an N-terminal and a C-terminal one, are qualitatively confirmed, even if the variations in the rates upon the experimental mutations cannot be quantitatively reproduced. Interestingly, folding and unfolding pathway appear to be different, even if closely related: a property that is not generally considered in the phenomenological interpretation of the experimental data.Comment: 27 pages, 7 figure

Results from RHIC with Implications for LHC

Results from the PHENIX experiment at RHIC in p-p and Au+Au collisions are reviewed from the perspective of measurements in p-p collisions at the CERN-ISR which serve as a basis for many of the techniques used. Issues such as J/Psi suppression and hydrodynamical flow in A+A collisions require data from LHC-Ions for an improved understanding. Suppression of high pT particles in Au+Au collisions, first observed at RHIC, also has unresolved mysteries such as the equality of the suppression of inclusive pi0 (from light quarks and gluons) and direct-single electrons (from the decay of heavy quarks) in the transverse momentum range 4< pT < 9 GeV/c. This disfavors a radiative explanation of suppression and leads to a fundamental question of whether the Higgs boson gives mass to fermions. Observation of an exponential distribution of direct photons in central Au+Au collisions for 1< pT <2 GeV/c where hard-processes are negligible and with no similar exponential distribution in p-p collisions indicates thermal photon emission from the medium at RHIC, making PHENIX at the moment ``the hottest experiment in Physics''.Comment: Invited lectures at the International School of Subnuclear Physics, 47th Course, "The most unexpected at LHC and the status of High Energy Frontier'', Erice, Sicily, Italy, August 29-September 7. 2009. 32 pages, 22 figure

Binary black hole merger in the extreme-mass-ratio limit: a multipolar analysis

Building up on previous work, we present a new calculation of the gravitational wave (GW) emission generated during the transition from quasi-circular inspiral to plunge, merger and ringdown by a binary system of nonspinning black holes, of masses $m_1$ and $m_2$, in the extreme mass ratio limit, $m_1 m_2\ll(m_1+m_2)^2$. The relative dynamics of the system is computed {\it without making any adiabatic approximation} by using an effective one body (EOB) description, namely by representing the binary by an effective particle of mass $\mu=m_1 m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M=m_1+m_2$ and submitted to an \O(\nu) 5PN-resummed analytical radiation reaction force, with $\nu=\mu/M$. The gravitational wave emission is calculated via a multipolar Regge-Wheeler-Zerilli type perturbative approach (valid in the limit $\nu\ll 1$). We consider three mass ratios, $\nu={10^{-2},10^{-3},10^{-4}}$,and we compute the multipolar waveform up to $\ell=8$. We estimate energy and angular momentum losses during the quasi-universal and quasi-geodesic part of the plunge phase and we analyze the structure of the ringdown. We calculate the gravitational recoil, or "kick", imparted to the merger remnant by the gravitational wave emission and we emphasize the importance of higher multipoles to get a final value of the recoil $v/(c\nu^2)=0.0446$. We finally show that there is an {\it excellent fractional agreement} ($\sim 10^{-3}$) (even during the plunge) between the 5PN EOB analytically-resummed radiation reaction flux and the numerically computed gravitational wave angular momentum flux. This is a further confirmation of the aptitude of the EOB formalism to accurately model extreme-mass-ratio inspirals, as needed for the future space-based LISA gravitational wave detector.Comment: 20 pages, 12 figures. Version published in Phys. Rev.

The three-body problem and the Hannay angle

The Hannay angle has been previously studied for a celestial circular restricted three-body system by means of an adiabatic approach. In the present work, three main results are obtained. Firstly, a formal connection between perturbation theory and the Hamiltonian adiabatic approach shows that both lead to the Hannay angle; it is thus emphasised that this effect is already contained in classical celestial mechanics, although not yet defined nor evaluated separately. Secondly, a more general expression of the Hannay angle, valid for an action-dependent potential is given; such a generalised expression takes into account that the restricted three-body problem is a time-dependent, two degrees of freedom problem even when restricted to the circular motion of the test body. Consequently, (some of) the eccentricity terms cannot be neglected {\it a priori}. Thirdly, we present a new numerical estimate for the Earth adiabatically driven by Jupiter. We also point out errors in a previous derivation of the Hannay angle for the circular restricted three-body problem, with an action-independent potential.Comment: 11 pages. Accepted by Nonlinearit

Brans-Dicke gravity and the capture of stars by black holes: some asymptotic results

In the context of star capture by a black hole, a new noticeable difference between Brans-Dicke theory and general relativity gravitational radiation is pointed out. This feature stems from the non-stationarity of the black hole state, barring Hawking's theorem.Comment: 4 pages. Submitted to Classical and Quantum Gravit