A new integrable system on the sphere

We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.Comment: LaTeX, 15 page

Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$Comment: 7 pages, no figure

Two-dimensional superintegrable metrics with one linear and one cubic integral

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and the first examples of superintegrable metrics of nonconstant curvature on a closed surfaceComment: 35 page

Extended phase diagram of the Lorenz model

The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz model equations for thermal convection in Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been investigated as a function of r, the normalized Rayleigh number, at fixed Prandtl number \sigma. The present work extends the analysis to the entire (r,\sigma) parameter plane. An onion like periodic pattern is found which is due to the alternating stability of symmetric and non-symmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and \sigma. In addition to the limit which was previously analyzed by Sparrow, we identify two more distinct asymptotic regimes in which either \sigma/r or \sigma^2/r is constant. In both limits the dynamics is approximately described by Airy functions whence the periodicity in parameter space can be calculated analytically. Furthermore, some observations about sequences of bifurcations and coexistence of attractors, periodic as well as chaotic, are reported.Comment: 36 pages, 20 figure

On a family of integrable systems on S2S^2 with a cubic integral of motion

We discuss a family of integrable systems on the sphere $S^2$ with an additional integral of third order in momenta. This family contains the Coryachev-Chaplygin top, the Goryachev system, the system recently discovered by Dullin and Matveev and two new integrable systems. On the non-physical sphere with zero radius all these systems are isomorphic to each other.Comment: LaTeX, 8 page

Maslov Indices and Monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived.Comment: 6 page

Vanishing Twist near Focus-Focus Points

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page

A Poincar\'e section for the general heavy rigid body

A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.Comment: 25 pages, 10 figure

Stability of stationary solutions for nonintegrable peakon equations

The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family. In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable

Non-Invasive Optical Motion Tracking Allows Monitoring of Respiratory Dynamics in Dystrophin-Deficient Mice

Duchenne muscular dystrophy (DMD) is the most common x-chromosomal inherited dystrophinopathy which leads to progressive muscle weakness and a premature death due to cardiorespiratory dysfunction. The mdx mouse lacks functional dystrophin protein and has a comparatively human-like diaphragm phenotype. To date, diaphragm function can only be inadequately mapped in preclinical studies and a simple reliable translatable method of tracking the severity of the disease still lacks. We aimed to establish a sensitive, reliable, harmless and easy way to assess the effects of respiratory muscle weakness and subsequent irregularity in breathing pattern. Optical respiratory dynamics tracking (ORDT) was developed utilising a camera to track the movement of paper markers placed on the thoracic-abdominal region of the mouse. ORDT successfully distinguished diseased mdx phenotype from healthy controls by measuring significantly higher expiration constants (k) in mdx mice compared to wildtype (wt), which were also observed in the established X-ray based lung function (XLF). In contrast to XLF, with ORDT we were able to distinguish distinct fast and slow expiratory phases. In mdx mice, a larger part of the expiratory marker displacement was achieved in this initial fast phase as compared to wt mice. This phenomenon could not be observed in the XLF measurements. We further validated the simplicity and reliability of our approach by demonstrating that it can be performed using free-hand smartphone acquisition. We conclude that ORDT has a great preclinical potential to monitor DMD and other neuromuscular diseases based on changes in the breathing patterns with the future possibility to track therapy response