Equivalence of energy methods in stability theory

We will prove the equivalence of three methods, the so called energy methods, for establishing the stability of an equilibrium point for a dynamical system. We will illustrate by examples that this result simplifies enormously the amount of computations especially when the stability can not be decided with one of the three methods.Comment: 10 pages, no figures, minnor correction

Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods

We study the stability of the equilibrium points of a skew product system. We analyze the possibility to construct a Lyapunov function using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case

A short proof of chaos in an atmospheric system

We will prove the presence of chaotic motion in the Lorenz five-component atmospheric system model using the Melnikov function method developed by Holmes and Marsden for Hamiltonian systems on Lie Groups.Comment: PACS: 02.20.Sv; 02.30.Hg; 02.40.-k; 92.60.-e. 5 page

Asymptotic Stability for a Class of Metriplectic Systems

Using the framework of metriplectic systems on $\R^n$ we will describe a constructive geometric method to add a dissipation term to a Hamilton-Poisson system such that any solution starting in a neighborhood of a nonlinear stable equilibrium converges towards a certain invariant set. The dissipation term depends only on the Hamiltonian function and the Casimir functions

On the generalized Hamiltonian structure of 3D dynamical systems

The Poisson structures for 3D systems possessing one constant of motion can always be constructed from the solution of a linear PDE. When two constants of the motion are available the problem reduces to a quadrature and the structure functions include an arbitrary function of them

A computational model unifies apparently contradictory findings concerning phantom pain

Amputation often leads to painful phantom sensations, whose pathogenesis is still unclear. Supported by experimental findings, an explanatory model has been proposed that identifies maladaptive reorganization of the primary somatosensory cortex (S1) as a cause of phantom pain. However, it was recently found that BOLD activity during voluntary movements of the phantom positively correlates with phantom pain rating, giving rise to a model of persistent representation. In the present study, we develop a physiologically realistic, computational model to resolve the conflicting findings. Simulations yielded that both the amount of reorganization and the level of cortical activity during phantom movements were enhanced in a scenario with strong phantom pain as compared to a scenario with weak phantom pain. These results suggest that phantom pain, maladaptive reorganization, and persistent representation may all be caused by the same underlying mechanism, which is driven by an abnormally enhanced spontaneous activity of deafferented nociceptive channels

On the symmetry breaking phenomenon

We investigate the problem of symmetry breaking in the framework of dynamical systems with symmetry on a smooth manifold. Two cases will be analyzed: general and Hamiltonian dynamical systems. We give sufficient conditions for symmetry breaking in both cases

Becoming British Application

2017 has been a politically challenging year, with great impacts for each individual living in the UK as well as, across Europe. Due to Brexit, many emigrants are now looking to get their British citizenships to secure stability for what the future brings. Therefore, solutions to support each individual on their journey are important. Although there are various books, websites and app-based games one could argue that the information they share is limited as well as lacks completeness. Each resource appears to focus on one particular stage and it does not guide the user through completing the full journey. There is no doubt that information consumption has changed and will continue to change drastically as technology is capable of providing the user with more than classrooms ever did; bespoke learning. This application will include: - revision material for the 5 chapter within the Life in the UK book, and tests at each chapter - 10 tests similar to the ""Life in the UK"" test - support and links on where to book your language test - support, links and information on submitting your final document

Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty

We analyze the quantum dynamics of the non-relativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence does not arise from ambiguities in the ordering of operators but from the fact of having several symplectic structures defined with respect to the same set of coordinates. It is also shown that the uncertainty relations between the fundamental observables depend on the particular quantum theory chosen. It is important to emphasize that these (somehow paradoxical) results emerge from the combination of two paradigms: Dirac's quantization rule and the usual Copenhagen interpretation of quantum mechanics.Comment: 8 pages, LaTex file, no figures. Accepted for publication in Phys. Rev.