Unstable Trajectories Research Articles

Unstable evolution of pointwise trajectory solutions to chaotic maps.

Simple chaotic maps are used to illustrate the inherent instability of trajectory solutions to the Frobenius-Perron equation. This is demonstrated by the difference in the behavior of delta-function solutions and of extended densities. Extended densities evolve asymptotically and irreversibly into invariant measures on stationary attractors. Pointwise trajectories chaotically roam over these attractors forever. Periodic Gaussian distributions on the unit interval are used to provide insight. Viewing evolving densities as ensembles of unstable pointwise trajectories gives densities a stochastic interpretation. (c) 1995 American Institute of Physics.

Problem of two fixed centers and a finite dipole: A unified treatment.

We investigate the classical dynamics of the problem of two centers and a finite dipole by means of a common Hamiltonian model. Conditions for trapped orbits are determined first by a qualitative analysis of the effective potential, revealing two types of bifurcation in the two-center problem as a control parameter passes through a critical value ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$. For equal masses there is a pitchfork bifurcation, for unequal masses a tangent bifurcation. Separating the common Hamiltonian in elliptic coordinates shows that the third invariants for the two-center problem and the finite dipole are isomorphic in scaled variables. Explicit trapping conditions are then found in terms of the coefficients of two quartics. A critical-point analysis for the finite dipole shows that a potential well exists for all values of scaled angular momentum below the same critical value ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$, at which the elliptic point runs off to infinity. In this case the existence of the third invariant does not confine any orbits not already trapped by energy conservation. A similar analysis of the effective potential for the point dipole shows that the only trapped orbits besides those impacting the origin are unstable zero-energy trajectories lying on a sphere.

Collision of dipolar vortices on a β plane

The interaction of two dipoles moving perpendicularly to the gradient of background vorticity is studied both numerically and experimentally. In the numerical computations the vorticity distribution is represented either by four point vortices (point-vortex model) or by thousands of them (vortex-in-cell method). The simplest model is used to study the dynamics and the advection of fluid particles in two kinds of interaction: coaxial couples of equal strength but different size, and equal parallel couples with a nonzero impact parameter (the distance between the dipoles’ axis). As a result of the interaction fluid masses are exchanged between the two dipoles and between each dipole and the ambient fluid. In the case of equal coaxial couples the amount of fluid exchanged depends on the gradient of ambient vorticity, with the largest mass exchange occurring always between the eastward traveling dipole and the ambient fluid. The collision of parallel couples with nonzero impact parameter leads to a large mass exchange, either because several interactions may occur or because when two independent couples arise, they have a nonuniform motion. Laboratory experiments in a rotating fluid (with a flat sloping bottom providing the β effect), confirm that an elastic interaction is a rare event. The unstable trajectory of the westward traveling dipole, as well as small perturbations unavoidable in the laboratory, invariably lead to collisions of nonaligned dipoles. The gross features of the vortex motion, as well as of the mass exchange, are well modeled using the point-vortex model, whereas the vortex-in-cell method reproduces many details of the vortex motion, the evolution of the vorticity field, and the exchange of mass.

Controlling chaos: Stabilization of unstable orbits using exponential control for maps and flows

We demonstrate that chaos can be controlled using multiplicative exponential feedback control. Unstable fixed points, unstable limit cycles and unstable chaotic trajectories can all be stabilized using such control which is effective both for maps and flows. The control is of particular significance for systems with several degrees of freedom, as knowledge of only one variable on the desired unstable orbit is sufficient to settle the system onto that orbit. We find in all cases that the transient time is a decreasing function of the stiffness of control. But increasing the stiffness beyond an optimum value can increase the transient time. We have also used such a mechanism to control spatiotemporal chaos is a well-known coupled map lattice model.

Phase space scars and quantization of non-integrable systems: the diamagnetic hydrogen atom

We present here a study of the quantum phase space localization (Wigner functions) in dependence of semiclassical quantization rules for the hydrogen atom in magnetic fields. We consider primarily two energy regions. In the first the corresponding classical system is near-integrable and in the second near fully chaotic. We study phase space localization (scars) close to stable and unstable trajectories, tori and the invariant manifold associated with the almost circular, unstable periodic trajectory. We find that these classical structures are an important element for the structure of the wavefunctions and the semiclassical quantization predictions are in good agreement with the quantal results.

CONTROLLING CHAOS IN CHUA’S CIRCUIT VIA SAMPLED INPUTS

We consider a class of autonomous, continuous time, chaotic dynamical systems the state equations of which can be represented in so-called Lur’e form. In particular we consider Chua’s circuit which is a paradigmatic chaotic system belonging to this class. It is shown that the dynamic behavior of such a system can be influenced in such a way as to obtain out of chaotic behavior a desired periodic orbit corresponding to an unstable periodic trajectory which exists in the system. This kind of control can be achieved via injection of a single continuous time signal representing the output of the system associated with an unstable periodic orbit embedded in the chaotic attractor. Further, we investigate the case when this signal is sampled, i.e. we supply to the system the control signal at discrete time moments only. We show via extensive numerical simulations that effective control can be achieved with a low number of samples only. Control proves to be very robust. Despite the presence of scaling of system variables and noise of a significant level introduced by signal quantization and offset, we were still able to control the chosen orbit (although with growing noise level the orbit becomes more and more distorted but maintains the same periodicity). We present a variety of simulation results to support this claim. First laboratory confirmations are also included. We claim that this method, proved to be functional for controlling chaos in Chua’s circuit, is also applicable to any chaotic system of the Lur’e type with a single nonlinearity.

Surface heat flux parameterization and the response of ocean general circulation models to high-latitude freshening

A global ocean general circulation model (OGCM) is forced using mixed boundary conditions (i.e. a restoring condition on the upper level temperature but using a fixed, specified surface salt flux). A freshwater flux anomaly is then applied over the western half of the sub-polar gyre in the northern North Atlantic. The response of the model is found to be dependent upon the details of the parameterization of the surface heat flux: In particular the “ coupling strength ” or Haney relaxation time is crucial. Responses range from a halocline catastrophe at short relaxation times (strong coupling) to a very modest perturbation at longer relaxation times (weaker coupling). An accurate parameterization is therefore required to properly model the evolution of the response. It is uncertain that the restorative condition is sufficiently realistic, especially in cases where a significantly different climatology is obtained. It is possible, for example, that the evolution could move from an unstable trajectory to a stable one if the parameters in the heat flux formulation are also allowed to evolve. This might help to explain why OGCMs under mixed boundary conditions are more sensitive than the observations suggest they should be. When a recovery does occur it does so on decadal time scales. It is therefore tempting to speculate that the positive feedback on the initial perturbation provided by the heat flux response plays a central role in the dynamics of North Atlantic variability, in a manner that is analogous to the wind-stress feedback in the El Nino, Southern Oscillation. DOI: 10.1034/j.1600-0870.1994.00008.x

Surface heat flux parameterization and the response of ocean general circulation models to high-latitude freshening

A global ocean general circulation model (OGCM) is forced using mixed boundary conditions (i.e. a restoring condition on the upper level temperature but using a fixed, specified surface salt flux). A freshwater flux anomaly is then applied over the western half of the sub-polar gyre in the northern North Atlantic. The response of the model is found to be dependent upon the details of the parameterization of the surface heat flux: In particular the “ coupling strength ” or Haney relaxation time is crucial. Responses range from a halocline catastrophe at short relaxation times (strong coupling) to a very modest perturbation at longer relaxation times (weaker coupling). An accurate parameterization is therefore required to properly model the evolution of the response. It is uncertain that the restorative condition is sufficiently realistic, especially in cases where a significantly different climatology is obtained. It is possible, for example, that the evolution could move from an unstable trajectory to a stable one if the parameters in the heat flux formulation are also allowed to evolve. This might help to explain why OGCMs under mixed boundary conditions are more sensitive than the observations suggest they should be. When a recovery does occur it does so on decadal time scales. It is therefore tempting to speculate that the positive feedback on the initial perturbation provided by the heat flux response plays a central role in the dynamics of North Atlantic variability, in a manner that is analogous to the wind-stress feedback in the El Nino, Southern Oscillation. DOI: 10.1034/j.1600-0870.1994.00008.x

Boundary effects and collisional activation in a quadrupole ion trap

AbstractThe effects of storing ions at different values of the stability parameters az and qzwere studied in a quadrupole ion trap, using helium or argon as buffer. A region was localized near the boundaries of the stability diagram in which the ions experience an increase in their kinetic energy. This is reflected in the occurrence both of fragmentation due to collisional activation and of a certain extent of ion loss due to unstable trajectories. The results of this excitation, referred to as ‘boundary effects,’ depend on the specific qzat which the ion storage is performed and on the buffer gas used, and point to a simpler means of obtaining tandem mass spectra with the ion trap without the need to apply resonant tickle voltages between the end ‐cap electrodes.

Computation of normal modes of three-dimensional resonators by semiclassical and variational methods: seismological implications

The problem investigated in this paper is that of calculating the eigenfrequencies of the bouncing-ball modes of oscillation of scalar waves in a three-dimensiol, arbitrarily shaped enclosure or resonator. With a view to future applications in seismology, we consider wave resonators limited above by a flat reflector—the earth's surface—and below by a curved one—the sediment/rock interface. The surface of the curved reflector is smooth but otherwise of arbitrary shape. The EBK (Einstein-Brillouin-Keller) method of semiclassical quantization is used to compute the eigenfrequencies. Conventional ray tracing is used to determine the trajectories of the rays in configuration space as well as in the Poincaré surfaces of section needed for the application of the method. For a prescribed set of mode numbers the EBK conditions produce a set of algebraic equations whose solution gives the required eigenfrequency and the corresponding eigentrajectory. Additionally, a system of four nonlinearly coupled ODE's is derived that simulates accurately the trajectories of the bouncing points of rays and can be used as a fast alternative to ray tracing. A variational method based upon the Rayleigh-Ritz formulation is used to obtain bounds on the eigenfrequencies and to determine the resonator's eigenfunctions. For separable and smooth nonseparable domains both approaches give consistent results. If the curvature of the reflector is convex towards the approaching ray, repeated reflections will result in highly unstable ray trajectories which ultimately becomne chaotic. In this case the EBK method fails. A brief discussion of the inherent tendency to chaotic behavior of the multiply reflecting ray system inside arbitrarily shaped enclosures is presented.

Nonlinear problems of shell theory and their solution methods (review)

In the present review we considered several problems of nonlinear shell theory, their deformation and stability. Speciailsts in many areas have devoted much attention recently to problems of the nonlinear behavior of evolutionary systems. New scientific trends related to the study of stability problems have been generated. They can be called the Hopf bifurcation theory; or, the theory of attractors, referring to the study of unstable trajectory behavior in phase spaces; or, catastrophe theory, applied in solving multiparametric stability problems; or, the theory of phase transitions and critical phenomena, developed in special areas of physics; or, a theory considering nonequilibrium phase transitions and self-organization complex spatial and temporal ordered structures (synergetics).

Sustainability of Macroeconomic Policies, Inflation Targeting, and Crowding out

Sustainability of Macroeconomic Policies, Inflation Targeting, and Crowding out

Theoretical studies of ion trajectories in quadrupole systems

AbstractA brief introduction to the mathematical theory of quadrupole systems is given and the configuration of hyperbolic and circular cross‐section rods is considered. Application of the theory of electrostatics to the latter case indicates that multipole contributions to the field occur which influence the ion trajectories. We have considered motion in the x‐z plane of a static quadrupole (DC‐only) for the simplest case of a 1:1 combination of quadrupole and dodecapole contributions. The quadrupole contribution to ion motion is simple harmonic and may be described as a sine function, whereas the dodecapole contribution may only be obtained by solving the appropriate elliptic integral. This solution gives rise to a sine (amplitude) or sn function. In the case of collision cells (RF‐only) we have attempted to solve the Mathieu equation both analytically and numerically. Both stable and unstable trajectories may be calculated from these solutions and are presented in various formats. Some consideration is also given to the problems of collisional interaction and the associated difficulties likely to be encountered when the study is expanded to include this phenomenon.

Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.

Dynamics of generalised spatial interaction models

This paper analyses dynamic properties of generalised spatial interaction models, with particular emphasis on Alonso's general theory of movements. Although the application of this theory addressed in the paper is a multiregional demographic stock–flow model, the approach can also be adopted to other types of spatial interaction phenomena. Equilibrium and stability conditions are examined by rewriting the generalised spatial interaction model as a general non-linear dynamic Volterra–Lotka model. More insight into local and global stability is obtained by means of simulation experiments which identify a variety of stable and unstable time trajectories.

Poincare map for scattering states

For scattering states in classical mechanics a map is constructed, which is an analogue to the Poincare map for bound state trajectories. Iteration of this map indicates which parts of the phase space are filled by unstable trajectories that are sensitive to small changes in the initial conditions, and which regions are filled by stable trajectories. Examples for this map are given by numerical calculations for scattering off a simple two-dimensional model potential.

Electron Trajectories in a Combined Wiggler and Alternating Gradient Quadrupole Field

This paper reports on studies of electron trajectories in a combined wiggler and alternating gradient quadrupole field. The quadrupole field is assumed to vary continuously along the symmetry axis. The linearized equations of electron motion are solved analytically for a plane-polarized wiggler by using the two-scale perturbation method. A comparison with the numerical solution is presented, and the conditions for unstable trajectories are discussed.

Instability of relativistic-electron helical trajectories in combined uniform axial and helical wiggler magnetic fields

The stability is investigated of helical trajectories of relativistic electrons in combined axial guide and helical wiggler magnetic fields appropriate for studies of free-electron lasers without using the frequently invoked assumption of evaluating the wiggler field on the laser axis. This is the first analysis of such trajectories which is free of this approximation. It is found that the helical trajectories are unstable with the possible exception of one value of the orbital radius. The instabilities are of two types corresponding to different properties of the eigenvalues of the matrix associated with the linearization of the equations of motion. These properties, in turn, depend upon the relative signs and magnitudes of the physical parameters that occur in the problem. At the exceptional radial value the trajectories can be either unstable or linearly stable depending upon the values of these parameters. For the linearly stable situations it is not known whether the trajectories are nonlinearly stable or unstable. We also prove that the trajectories of one of the instability types are of a particular kind called conditionally stable. This type of unstable trajectory has the property that it is approached by some (but not all) solutions of the equations of motion as time approaches positive or negative infinity. This type of instability is found to be more severe for larger values of the orbital radius. Even so, it still appears to be significant for small radii, of the order of $\frac{1}{100}$ of the wiggler wavelength.

Theoretical determination of ionization efficiency for an orbitron device and variation with the main working parameters

A knowledge of the distribution function for electrons describing stable and unstable trajectories, respectively, and the evaluation of the stable electrons’ mean lifetime deduced from the properties of the ‘‘electrostatic mirror’’ corresponding to the endplates, allow us to perform a theoretical calculation of ionization efficiency for an orbitron device. The ionization efficiency of the device is calculated for various geometrical and electrical parameters using the preceding theoretical considerations. Comparison between experiments and calculations allow us to validate the theoretical model.

Dynamical instability and external perturbations: Bimolecular collisions in laser fields

The detailed dynamics of F+H2, and to a lesser extent H+H2, in the presence of laser fields are examined in order to establish the relationship between field-free unstable trajectories and the effect of external perturbations. Weak fields are found to alter the dynamics of individual unstable trajectories and, in particular, to induce transitions amongst the unstable trajectory manifold. As a result, the field has little effect, on the average, on gross collision properties. Intense fields substantially alter the qualitative stable/ unstable structure of phase space and hence the reaction probability. An interesting high field limit is also discussed.