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Periodic Orbits (periodic + orbit)

Distribution by Scientific Domains

Physics and Astronomy	58%

Selected Abstracts

Periodic Orbits near equilibria

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2010
Luis Barreira
Lyapunov, Weinstein, and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory, we establish a similar result for a differential system without assuming the existence of a first integral. Our result can also be interpreted as a kind of special Hopf bifurcation. © 2010 Wiley Periodicals, Inc. [source]

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 1 2006
N. Voglis
ABSTRACT In the presence of a strong m= 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L1 or L2. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine,Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar,spiral pattern. [source]

Boxy/peanut ,bulges': comparing the structure of galaxies with the underlying families of periodic orbits

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2006
P. A. Patsis
ABSTRACT The vertical profiles of disc galaxies are built by the material trapped around stable periodic orbits, which form their ,skeletons'. Therefore, knowledge of the stability of the main families of periodic orbits in appropriate 3D models enables one to predict possible morphologies for edge-on disc galaxies. In a pilot survey we compare the orbital structures that lead to the appearance of ,peanut'- and ,X'-like features with the edge-on profiles of three disc galaxies (IC 2531, NGC 4013 and UGC 2048). The subtraction from the images of a model representing the axisymmetric component of the galaxies reveals the contribution of the non-axisymmetric terms. We find a direct correspondence between the orbital profiles of 3D bars in models and the observed main morphological features of the residuals. We also apply a simple unsharp masking technique in order to study the sharpest features of the images. Our basic conclusion is that the morphology of the boxy ,bulges' of these galaxies can be explained by considering disc material trapped around stable 3D periodic orbits. In most models, these building-block periodic orbits are bifurcated from the planar central family of a non-axisymmetric component, usually a bar, at low-order vertical resonances. In such a case, the boxy ,bulges' are parts of bars seen edge-on. For the three galaxies we study, the families associated with the ,peanut' or ,X'-shape morphology are probably bifurcations at the vertical 2/1 or 4/1 resonance. [source]

The stellar dynamics of spiral arms in barred spiral galaxies

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY: LETTERS (ELECTRONIC), Issue 1 2006
P. A. Patsis
ABSTRACT A dynamical mechanism is proposed that explains the spiral structure observed frequently as a continuation of the bars in barred spiral galaxies. It is argued that the part of the spirals attached to the bar is due to chaotic orbits. These are chaotic orbits that exhibit for long time intervals a 4:1-resonance orbital behaviour. They are of the same type of orbit as is responsible for the boxiness of the outer isophotes of the bar in cases like NGC 4314, as indicated by Patsis, Athanassoula & Quillen. The spirals formed this way are faint with respect to the bar, open as they wind out, and do not extend over an angle larger than ,/2. A possible continuation of the spiral structure towards larger angles can be due to orbits trapped around stable periodic orbits at the corotation region. We present a family of stable, banana-like periodic orbits, precessing as EJ increases, that can play this role. [source]

STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR

NATURAL RESOURCE MODELING, Issue 2 2009
JEAN M. TCHUENCHE
Abstract The study of three-species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super-predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator-top-predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long-term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations. [source]

Dynamic optimization and Skiba sets in economic examples

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 5-6 2001
Wolf-Jürgen Beyn
Abstract We discuss two optimization problems from economics. The first is a model of optimal investment and the second is a model of resource management. In both cases the time horizon is infinite and the optimal control variables are continuous. Typically, in these optimal control problems multiple steady states and periodic orbits occur. This leads to multiple solutions of the state,costate system each of which relates to a locally optimal strategy but has its own limiting behaviour (stationary or periodic). Initial states that allow different optimal solutions with the same value of the objective function are called Skiba points. The set of Skiba points is of interest, because it provides thresholds for a global change of optimal strategies. We provide a systematic numerical method for calculating locally optimal solutions and Skiba points via boundary value problems. In parametric or higher dimensional systems Skiba curves (or manifolds) appear and we show how to follow them by a continuation process. We apply our method to the models above where Skiba sets consist of points or curves and where optimal solutions have different stationary or periodic asymptotic behaviour. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

ANNALEN DER PHYSIK, Issue 9 2007
M.I. Krivoruchenko
Abstract In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ,. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p -forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior. [source]

Ordered and chaotic spiral arms

ASTRONOMISCHE NACHRICHTEN, Issue 9-10 2008
P.A. Patsis
Abstract The stellar flow at the arms of spiral galaxies is qualitatively different among different morphological types. The stars that reinforce the spiral arms can be either participating in an ordered or in a chaotic flow. Ordered flows are associated with normal (non-barred) spiral galaxies. Typically they are described with precessing ellipses corresponding to stable periodic orbits at successive energies (Jacobi constants). On the contrary, the spiral arms in barred-spiral systems may be supported by stars in chaotic motion. The trajectories of these stars are associated with the invariant manifolds of the unstable Lagrangian points (L1,2). Response and orbital models indicate that this kind of spirals either stop at an azimuth smaller than , /2, or present large gaps at about this angle. Chaotic spirals appear in strong bars having (L1,2) close to the ends of the bar. The arms of barred-spiral systems with corotation away from the end of the bar can be either as in the case of normal spirals, or supported by banana-like orbits surrounding the stable Lagrangian points (L4,5). We find also models combining ordered and chaotic flows. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Periodic Orbits near equilibria

On the Floer homology of cotangent bundles

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2006
Alberto Abbondandolo
This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L, estimate for the solutions of the Floer equation, which allows us to deal with a larger,and more natural,class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc. [source]