Moser Tori Research Articles

Lagrangian transport in the time-periodic two-dimensional lid-driven square cavity

The Lagrangian transport in the laminar incompressible flow in a two-dimensional square cavity driven by a harmonic tangential oscillation of the lid is investigated numerically for a wide range of Reynolds and Strouhal numbers. The topology of fluid trajectories is analyzed by stroboscopic projections revealing the co-existence of chaotic trajectories and regular Kolmogorov–Arnold–Moser (KAM) tori. The pathline structure strongly depends on the Reynolds number and the oscillation frequency of the lid. Typically, most pathlines are chaotic when the oscillation frequency is small, with few KAM tori being strongly stretched along instantaneous streamlines of the flow. As the frequency is increased, the fluid motion becomes more regular and the size of the KAM tori grows until, at high frequencies, they resemble streamlines of a mean flow.

Attractors for the motion of a finite-size particle in a cuboidal lid-driven cavity

The motion of a finite-size particle in the cuboidal lid-driven cavity flow is investigated experimentally for Reynolds numbers $100$ and $200$ for which the flow is steady. These steady three-dimensional flows exhibit chaotic and regular streamlines, where the latter are confined to Kolmogorov–Arnold–Moser (KAM) tori. The interaction between the moving wall and the particle creates global particle attractors. For neutrally buoyant particles, these attractors are periodic or quasi-periodic, strongly attracting and located in or near KAM tori of the flow. As the density mismatch between particle and fluid increases, buoyancy and inertia become important, and the attractors evolve from those for neutrally buoyant particles, changing their shape, position and attraction rates.

Stellar instability from parametric resonance

In this paper we examine the stability of stellar configurations in which the interior solution is described by a closed Friedmann-Lemaître-Robertson-Walker (FLRW) geometry sourced with a charged pressureless fluid and radiation. An interacting vacuum component and a conformally coupled massive scalar field are also included. Given a simple factor for the energy transfer between the pressureless fluid and the vacuum component we obtain bounded interior oscillatory solutions. We show that in proper domains of the parameter space the interior dynamics is highly unstable so that the break of the Kolmogorov–Arnold–Moser (KAM) tori leads to a disruptive ejection of mass. For such configurations the interior solution asymptotically matches an exterior Reissner–Nordström–de Sitter spacetime.

Invariant tori for multi-dimensional integrable Hamiltonians coupled to a single thermostat

This paper demonstrates sufficient conditions for the existence of Kolmogorov-Arnol’d-Moser (KAM) tori in a singly thermostated, integrable Hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2—which include the Nosé thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends theorem 3.2 of Legoll et al (2009 Nonlinearity 22 1673–94) to prove that a ‘typical’ singly thermostated, integrable, real-analytic Hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable Hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion.

Experimental Study on Coherent Structures by Particles Suspended in Half-Zone Thermocapillary Liquid Bridges: Review

Coherent structures by the particles suspended in the half-zone thermocapillary liquid bridges via experimental approaches are introduced. General knowledge on the particle accumulation structures (PAS) is described, and then the spatial–temporal behaviours of the particles forming the PAS are illustrated with the results of the two- and three-dimensional particle tracking. Variations of the coherent structures as functions of the intensity of the thermocapillary effect and the particle size are introduced by focusing on the PAS of the azimuthal wave number m=3. Correlation between the particle behaviour and the ordered flow structures known as the Kolmogorov–Arnold—Moser tori is discussed. Recent works on the PAS of m=1 are briefly introduced.

Attractors for the motion of a finite-size particle in a two-sided lid-driven cavity

Abstract

A generic mechanism for finite-size coherent particle structures

Particles transported by a fluid flow can accumulate in preferential regions. Usually, the clustering is caused by particle inertia, leading to inertial coherent particle structures. In the absence of inertia, particles can also cluster massively in a steady flow, solely owing to their size when they are repelled from a boundary by lubrication and lift forces. These forces can transfer the particles from the chaotic region to Kolmogorov–Arnold–Moser tori of the unperturbed incompressible flow, where they may focus on three-dimensional limit cycles. Numerical simulations for two different flow systems, taking into account the boundary-repulsion effect for finite-size particles, are in very good agreement with corresponding experiments. The results provide evidence that finite-size coherent particle structures are generic for a class of incompressible flows.

On the co-existence of maximal and whiskered tori in the planetary three-body problem

In this paper, we discuss about the possibility of the coexistence of stable and unstable quasi-periodic Kolmogorov–Arnold–Moser (kam) tori in a region of the phase space of the three-body problem. The argument of proof goes along kam theory and, especially, the production of two non-smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for “properly degenerate” systems, by a theorem of Nekhoroshev and Miščenko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in Arnold [“Small denominators and problems of stability of motion in classical and celestial mechanics,” 18, 85–191 (1963)].

A KAM Tori Algorithm for Earth Satellite Orbits

This paper offers a new approach for constructing Kolmogorov - Arnold - Moser (KAM) tori for orbits in the full potential for a non-spherical planet. The Hamilton - Jacobi equation is solved numerically by a Newton-Rhapson iteration, achieving convergence to machine precision, and still retaining literal variable dependence. Similar iteration methods allow correcting the orbital frequencies, and permit the calculation of the state transition matrix for the full problem. Some initial numerical examples are offered.

Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

Cellular flow in a partially filled rotating drum: regular and chaotic advection

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

État de l’art en imagerie médicale

Particles transported by a fluid flow can accumulate in preferential regions. Usually, the clustering is caused by particle inertia, leading to inertial coherent particle structures. In the absence of inertia, particles can also cluster massively in a steady flow, solely owing to their size when they are repelled from a boundary by lubrication and lift forces. These forces can transfer the particles from the chaotic region to Kolmogorov–Arnold–Moser tori of the unperturbed incompressible flow, where they may focus on three-dimensional limit cycles. Numerical simulations for two different flow systems, taking into account the boundary-repulsion effect for finite-size particles, are in very good agreement with corresponding experiments. The results provide evidence that finite-size coherent particle structures are generic for a class of incompressible flows.

Lagrangian chaos in confined two-dimensional oscillatory convection

AbstractThe chaotic advection of passive tracers in a two-dimensional confined convection flow is addressed numerically near the onset of the oscillatory regime. We investigate here a differentially heated cavity with aspect ratio 2 and Prandtl number 0.71 for Rayleigh numbers around the first Hopf bifurcation. A scattering approach reveals different zones depending on whether the statistics of return times exhibit exponential or algebraic decay. Melnikov functions are computed and predict the appearance of the main mixing regions via the break-up of the homoclinic and heteroclinic orbits. The non-hyperbolic regions are characterised by a larger number of Kolmogorov–Arnold–Moser (KAM) tori. Based on the numerical extraction of many unstable periodic orbits (UPOs) and their stable/unstable manifolds, we suggest a coarse-graining procedure to estimate numerically the spatial fraction of chaos inside the cavity as a function of the Rayleigh number. Mixing is almost complete before the first transition to quasi-periodicity takes place. The algebraic mixing rate is estimated for tracers released from a localised source near the hot wall.

Stagnant Motion in Chaotic Region of Mushroom Billiard System with Dielectric Medium

Generic hamiltonian systems often exhibit slow dynamics. The origin of the slow behavior is thought to be from the stagnant motion in the neighborhood of the outermost Kolmogorov–Arnold–Moser (KAM) tori. We investigate the onset mechanism of such slow dynamics by using a mushroom billiard with a dielectric medium. Bifurcation processes are precisely studied in the vicinity of the outermost KAM tori when the refraction index of the dielectric medium is changed smoothly, and it is shown that the stagnant motion is generated near singular points in the chaotic sea. There are two types of stagnant aspects coexisting in the generic case, for which the distributions of the escape time of stagnant motion are theoretically determined.

On the atmosphere of a moving body

We explore whether a rigid body moving freely with no circulation around it in a two-dimensional ideal fluid can carry a fluid “atmosphere” with it in its motion. Somewhat surprisingly, the answer appears to be “yes.” When the body is elongated and the motion is dominated by rotation, we demonstrate numerically that, indeed, regions of fluid follow the body in its motion. We see this as an example of the stability of Kolmogorov–Arnold–Moser tori. These observations of an atmosphere around a moving body with no circulation around it appear to be new.

Relative volume of Kolmogorov–Arnold–Moser tori and uniform distribution, stickiness and nonstickiness in Hamiltonian systems

We discuss several open problems in the theory of Hamiltonian systems. They are all related to the Hamiltonian systems with divided phase space, where Kolmogorov–Arnold–Moser tori coexist with ergodic components of positive measure.

Non-linear resonance in bouncing braneworld universes and initial conditions forinflation

We examine the phase space dynamics of closed Friedmann–Robertson–Walker universeswith a massive inflaton field, where the Friedmann equations contain additional termsarising from high energy corrections to cosmological scenarios. The model is basedupon a Randall–Sundrum type of action, with an extra timelike dimension, andthe corrections implement non-singular bounces in the early evolution of theuniverse. In narrow windows of the parameter space of the models non-linearresonance phenomena of Kolmogorov–Arnold–Moser tori are shown to occur, leadingto the destruction of tori that trap the inflaton. As a consequence the escapeinto inflation takes place. These resonance windows are labeled with an integern≥2,where n is related to the ratio of the frequencies in the scale factor to those in the scalar field degrees offreedom. We examine the constraints imposed by non-linear resonance in the physicaldomain of parameters of the model so that inflation may be realized. The larger the ordern of the resonance, the stronger the gravitational interaction in the braneworld universeinflated from initial conditions connected with the resonance considered. We also discussthe structural stability of the resonance pattern, the complex dynamics arising inthis pre-inflationary phase and some of its possible imprints in the physics ofinflation.

Nonlinear resonance of Kolmogorov–Arnold–Moser tori in bouncing universes

The dynamics of closed Friedmann–Robertson–Walker (FRW) universes with a massiveinflaton field is examined where Friedmann equations are corrected by the introduction of apotential term that implements non-singular bounces in the early evolution of theuniverse. This potential term arises from quantum gravity/high-energy corrections tocosmological scenarios near the singularity and is semiclassical in nature, being effectiveonly when the scale factor is very small. For certain windows in the parameterspace labelled by the scalar field mass and the conserved Hamiltonian, nonlinearresonance phenomena take place. Nonlinear resonance may induce the destruction ofKolmogorov–Arnold–Moser (KAM) tori that trap the inflaton, leading to a rapid growth ofthe scale factor and the scalar field, with disruption of metastable states and consequentescape of the universe into inflation. We make a numerical/analytical approach tothe nonlinear resonance phenomena, characterizing a particular resonance by itscharacteristic periodic orbits and by the structure of the associated diffusion pattern.The diffusion occurs when the orbit escapes through a Cantorus in the borderof primary KAM islands that encloses the characteristic periodic orbits of theresonance. The windows of parametric resonance, characterized by an integern ≥ 2 (associated with the ratio of the frequencies in the scale factor/scalar field degrees offreedom) are the ones that strongly favour inflation in the system. We discuss how genericthis behaviour is for inflationary models, and its possible consequences for structureformation.

Experimental observation of resonance overlap responsible for Hamiltonian chaos

A test electron beam is propagated in a specially designed traveling wave tube. The beam intensity is low enough to ensure that beam-plasma instabilities are ruled out. By recording the beam energy distribution at the output of the tube, we report the experimental observation of the resonant domain of a single wave and of the overlap of the resonance domains of two waves associated to the destruction of Kolmogorov–Arnold–Moser tori constituting barriers in phase space. This overlap mechanism is responsible for the transition to large scale chaos common to a large class of Hamiltonian systems.

Production of enhanced beam halos via collective modes and colored noise

We investigate how collective modes and colored noise conspire to produce a beam halo with much larger amplitude than could be generated by either phenomenon separately. The collective modes are lowest-order radial eigenmodes calculated self-consistently for a configuration corresponding to a direct-current, cylindrically symmetric, warm-fluid Kapchinskij-Vladimirskij equilibrium. The colored noise arises from unavoidable machine errors and influences the internal space-charge force. Its presence quickly launches statistically rare particles to ever-growing amplitudes by continually kicking them back into phase with the collective-mode oscillations. The halo amplitude is essentially the same for purely radial orbits as for orbits that are initially purely azimuthal; orbital angular momentum has no statistically significant impact. Factors that do have an impact include the amplitudes of the collective modes and the strength and autocorrelation time of the colored noise. The underlying dynamics ensues because the noise breaks the Kolmogorov-Arnol'd-Moser tori that otherwise would confine the beam. These tori are fragile; even very weak noise will eventually break them, though the time scale for their disintegration depends on the noise strength. Both collective modes and noise are therefore centrally important to the dynamics of halo formation in real beams.