Quasiperiodic Time Dependence Research Articles

Aggregation of slightly buoyant microplastics in 3D vortex flows

Abstract. Although the movement and aggregation of microplastics at the ocean surface have been well studied, less is known about the subsurface. Within the Maxey–Riley framework governing the movement of small, rigid spheres with high drag in fluid, the aggregation of buoyant particles is encouraged in vorticity-dominated regions. We explore this process in an idealized model that is qualitatively reminiscent of a 3D eddy with an azimuthal and overturning circulation. In the axially symmetric state, buoyant spherical particles that do not accumulate at the top boundary are attracted to a loop consisting of periodic orbits. Such a loop exists when drag on the particle is sufficiently strong. For small, slightly buoyant particles, this loop is located close to the periodic fluid parcel trajectory. If the symmetric flow is perturbed by a symmetry-breaking disturbance, additional attractors for small, rigid, slightly buoyant particles may arise near periodic orbits of fluid parcels within the resonance zones created by the disturbance. Disturbances with periodic or quasiperiodic time dependence may produce even more attractors, with a shape and location that recurs periodically. However, not all such loops attract, and rigid particles released in the vicinity of one loop may instead be attracted to a nearby attractor. Examples are presented along with mappings of the respective basins of attraction.

On the Stability of the Nonlinear Center under Quasi-periodic Perturbations

The problem of stability of the zero solution of a system with a “center”-type critical point at the origin of coordinates is considered. For the first time, such a problem for autonomous systems was investigated by A.M. Lyapunov. We continued Lyapunov’s investigations for systems with a periodic dependence on time. In this paper, systems with a quasi-periodic time dependence are considered. It is assumed that the basic frequencies of quasi-periodic functions satisfy the standard Diophantine-type condition. The problem under consideration can be interpreted as the problem of stability of the state of equilibrium of the oscillator $$\ddot {x} + {{x}^{{2n - 1}}}$$ = 0, n is an integer number, n ≥ 2, under “small” quasi-periodic perturbations.

Prandtl–Batchelor theorem for flows with quasiperiodic time dependence

The classical Prandtl–Batchelor theorem (Prandtl, Proc. Intl Mathematical Congress, Heidelberg, 1904, pp. 484–491; Batchelor, J. Fluid Mech., vol. 1 (02), 1956, pp. 177–190) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasiperiodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasiperiodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contours form a family of closed curves in a quasiperiodic viscous flow, then at the limit of zero viscosity, vorticity is constant in the area enclosed by those curves at all times.

Koopman Mode Decomposition for Periodic/Quasi-periodic Time Dependence

Abstract: In this paper we propose an extension of Koopman operator framework for non-autonomous systems with periodic and quasi-periodic time dependence. Using a time parametrized family of Koopman operators and the associated time dependent eigenvalues and eigenfunctions, and concepts from Floquet theory, we extend the notion of the Koopman Mode Decomposition. We illustrate our framework on several examples.

Magnetization Dynamics Under a Quasiperiodic Magnetic Field

In the present work, we study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a time dependent magnetic field using the Landau-Lifshitz-Gilbert equation. In particular, we study the case when the magnetic field consists in two terms. One is perpendicular to the anisotropy direction and has quasiperiodic time dependence, while the other term is constant and parallel to the anisotropy direction. We numerically characterize the dynamical behavior of the system by monitoring the Lyapunov exponents, and by calculating Poincare sections and Fourier spectra. In addition, we calculate analytically the corresponding Melnikov function which gives us the bifurcations of the homoclinic orbits. We find a rather complicated landscape of sometimes closely intermingled chaotic and non-chaotic areas in parameters space. Finally, we show that the system exhibits strange nonchaotic attractors.

New graphical techniques for studying acoustic ray stability

Alternatives to the standard Poincaré section are proposed to cater for some conditions arising in the study of chaotic ray propagation where the usual method of dimension reduction by the Poincaré section is inadequate because the driving is not periodic. There are three alternatives proposed which all use the same surface of intersection, but which differ in their use of the values of the dependent variables at the intersections of the rays with the surface. The new reduction techniques are used to examine ray behaviour in a harmonically perturbed Munk profile which supports ray chaos. It is found that all three techniques provide a graphical means of distinguishing between regular and irregular motions, and that the space of the mapping associated with one of them is partitioned into nonintersecting regular and chaotic regions as with the Poincaré section. A further model with quasiperiodic time dependence of the Hamiltonian is examined, and it turns out that the quasiperiodic nature of the motion is revealed as Lissajous curves by one technique.

Almost periodic orbits and stability for quantum time-dependent Hamiltonians

We study almost periodic orbits of quantum systems and prove that for periodic time-dependent Hamiltonians an orbit is almost periodic if, and only if, it is precompact. In the case of quasiperiodic time-dependence we present an example of a precompact orbit that is not almost periodic. Finally we discuss some simple conditions assuring dynamical stability for nonautonomous quantum system.

Centrifugally driven convection in the rotating cylindrical annulus with modulated boundaries

Abstract. The effect of sinusoidally modulated conical end boundaries on convection in a rotating cylindrical annulus is investigated theoretically and experimentally. A quasiperiodic time dependence of convection in the form of thermal Rossby waves is found and semi-quantitative agreement between theory and measurements can be established. The results are relevant to convection in the Earth's outer core close to the tangent cylinder touching the inner core at its equator.

On the logistic delay differential equations with piecewise constant argument and quasi-periodic time dependence

In this paper, we discuss the quasi-periodic logistic delay differential equations. As a corollary, we give a more sharp result than that in [G. Seifert, J. Differential Equations 164 (2000) 451–458] for the periodic logistic delay differential equations.

Chaotic Mixing and Transport Barriers in an Idealized Stratospheric Polar Vortex

Chaotic mixing processes and transport barriers around the wintertime stratospheric polar vortex are investigated with an idealized barotropic model, previously used by Ishioka and Yoden. A barotropically unstable jet is forced in order to obtain a fluctuating polar vortex. A flow with quasiperiodic time dependence and an aperiodic flow with similar behavior are investigated using several Lagrangian methods. A typical chaotic mixing process is observed in the quasiperiodic flow, resulting in effective mixing inside and outside of the polar vortex. The mixing regions are on the critical latitudes of several planetary waves that grow through barotropic instability. Poincaré sections give accurate locations of chaotic mixing regions, and transport barriers are identified as the edges of invariant torus regimes. In addition to the transport barriers associated with strong potential vorticity gradients, another type of transport barrier exists, which is not related to the steep potential vorticity gradient. Chaotic mixing is dominant also in the aperiodic flow. Comparing with the quasiperiodic flow, an aperiodic flow with the same wave energy has a higher average Lyapunov exponent. This arises because the area involved in chaotic zones increases. The evolution of the correlation function is also more typical of a chaotic zone. Isolated regions are found near the center of the polar vortex, which can be explained by the invariant tori in the Poincaré sections of the quasiperiodic flow. Implications of the results for the observed “4-day wave” in the upper stratosphere are discussed.

Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

Boundedness of Solutions for Quasiperiodic Potentials

In this paper conservative systems are studied describing the motion of a particle on the line in the field of a potential force with additional quasiperiodic time dependence. It is shown that superquadratic growth of the potential at infinity results in the near-integrability of the Hamiltonian system in question (for a large class of potentials), despite the fact that no smallness assumptions are made on the quasiperiodic dependence of the potential on time. As a consequence all the solutions of such systems are bounded for all time. Some specific examples are given, together with a counterexample which shows that, without the quasiperiodicity assumption, the boundedness breaks down.

Symmetry breaking in a nonlinear αω-dynamo

Abstract An idealized nonlinear αω-dynamo is investigated. Emphasis is placed upon the different spatial symmetries, and the asymmetries that arise after secondary bifurcations. On varying the main control parameter D (the dynamo number), many transitions are found involving solutions without an equatorial symmetry, and solutions with quasiperiodic time dependence, but no chaos. Instead of a cascade to smaller spatial scales when D is highly supercritical it is found that additional asymmetries are introduced at tertiary bifurcations. Our complete bifurcation diagrams allow us to follow in detail how stability is passed from one solution to another as D varies. In these diagrams there are typically multiple stable solutions at any value of D, which suggests that similar stars can have different magnetic patterns.

Multiphoton ionization of hydrogen by an intense 248-nm linearly polarized field.

We study multiphoton ionization of hydrogen by a linearly polarized 248-nm field (the photons have an energy of 5.0 eV) up to an intensity of ${10}^{15}$ W/${\mathrm{cm}}^{2}$. We compare two very different computational methods for the calculation of the total ionization rate: on the one hand, the full numerical solution of the time-dependent Schr\"odinger equation; on the other hand, results obtained within the Floquet formalism, which assumes a quasiperiodic time dependence. We obtain very good agreement between the two sets of calculations throughout the whole intensity range. The calculated rates also agree well with third-order perturbation theory up to ${10}^{13}$ W/${\mathrm{cm}}^{2}$, but are orders of magnitude lower at ${10}^{15}$ W/${\mathrm{cm}}^{2}$. The shortfall at high intensities is attributed to the change from a nonresonant third-order process to a resonant fourth-order process caused by ac Stark-induced shifts in the energy-level structure of hydrogen. In fact, the ionization rate between 2 and 6\ifmmode\times\else\texttimes\fi{}${10}^{14}$ W/${\mathrm{cm}}^{2}$ is almost constant, giving rise to a plateau feature in the rate versus intensity curve.

A two-dimensional adaptive pseudo-spectral method

We develop a two-dimensional adaptive pseudo-spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions. The method is based on introducing two-dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated. The method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel. Spatio-temporal patterns are computed as a parameter μ, related to the activation energy, is increased above a critical value μc. The spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder, together with a standing wave pattern in the direction of the azimuthal angle ψ For small values of μ − μc the solutions exhibit a nearly sinusoidal dependence in both time and ψ As μ is increased further relaxation oscillations in both time and ψ occur. Beyond a critical value of μ stable time-periodic solutions are no longer found and the solution exhibits a quasi-periodic time dependence.

Numerical simulation of thermal convection in a two-dimensional finite box

The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Bénard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

A quasi-periodic time dependence of interfacial tension in an oil-water-surfactant system

A strong fluctuation of interfacial tension (IFT), that is quasi-periodic in time, has been observed for the water-dodecane interface, with 8-phenylhexadecane sodium sulfonate (“Texas no. 1”) surfactant. This fluctuation was observed with sessile but not pendant drops, and in a massive cell. It ceased when vibrations were introduced. The periodic effect was probably caused by the growth of the interfacial layer, from a monolayer of surfactant, into a layer of liquid crystal (LC) material of appreciable thickness, which can be considered to be a degenerate (or underdeveloped) middle phase. Growth would occur by capture of upward-drifting liquid crystal particles from the aqueous phase. The IFT on the aqueous side of the middle phase is probably relatively high, and controls the overall drop shape. The IFT on the oil side of the middle phase is low, and falls with time, to a value so low that, when the layer attains sufficient thickness, droplets can be thrown off by small random vibrations. With a vibration-insulated cell, vibrations of sufficient amplitude come along only at widely separated intervals. When the excess liquid crystal material is thrown off, the layer thickness decreases to a small enough value that the oil-LC layer boundary prevents the water-LC layer boundary from achieving its maximum IFT. A brief discussion is given of the relation between the phenomenon we have observed and the disjoining pressure.