Global Structure Of Phase Space Research Articles

Charged Particle Motions near Non-Schwarzschild Black Holes with External Magnetic Fields in Modified Theories of Gravity

A small deformation to the Schwarzschild metric controlled by four free parameters could be referred to as a nonspinning black hole solution in alternative theories of gravity. Since such a non-Schwarzschild metric can be changed into a Kerr-like black hole metric via a complex coordinate transformation, the recently proposed time-transformed, explicit symplectic integrators for the Kerr-type spacetimes are suitable for a Hamiltonian system describing the motion of charged particles around the non-Schwarzschild black hole surrounded with an external magnetic field. The obtained explicit symplectic methods are based on a time-transformed Hamiltonian split into seven parts, whose analytical solutions are explicit functions of new coordinate time. Numerical tests show that such explicit symplectic integrators for intermediate time steps perform well long-term when stabilizing Hamiltonian errors, regardless of regular or chaotic orbits. One of the explicit symplectic integrators with the techniques of Poincaré sections and fast Lyapunov indicators is applied to investigate the effects of the parameters, including the four free deformation parameters, on the orbital dynamical behavior. From the global phase-space structure, chaotic properties are typically strengthened under some circumstances, as the magnitude of the magnetic parameter or any one of the negative deformation parameters increases. However, they are weakened when the angular momentum or any one of the positive deformation parameters increases.

Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole

In a recent work of Wu, Wang, Sun and Liu, a second-order explicit symplectic integrator was proposed for the integrable Kerr spacetime geometry. It is still suited for simulating the nonintegrable dynamics of charged particles moving around the Kerr black hole embedded in an external magnetic field. Its successful construction is due to the contribution of a time transformation. The algorithm exhibits a good long-term numerical performance in stable Hamiltonian errors and computational efficiency. As its application, the dynamics of order and chaos of charged particles is surveyed. In some circumstances, an increase of the dragging effects of the spacetime seems to weaken the extent of chaos from the global phase-space structure on Poincaré sections. However, an increase of the magnetic parameter strengthens the chaotic properties. On the other hand, fast Lyapunov indicators show that there is no universal rule for the dependence of the transition between different dynamical regimes on the black hole spin. The dragging effects of the spacetime do not always weaken the extent of chaos from a local point of view.

Effect of chaos on the simulation of quantum critical phenomena in analog quantum simulators

We study how chaos, introduced by a weak perturbation, affects the reliability of the output of analog quantum simulation. As a toy model, we consider the Lipkin-Meshkov-Glick model. Inspired by the semiclassical behavior of the order parameter in the thermodynamic limit, we propose a protocol to measure the quantum phase transition in the ground state and the dynamical quantum phase transition associated with quench dynamics. We show that the presence of a small time-dependent perturbation can render the dynamics of the system chaotic. We then show that the estimates of the critical points of these quantum phase transitions, obtained from the quantum simulation of its dynamics, are robust to the presence of this chaotic perturbation, while other aspects of the system, such as the mean magnetization, are fragile and therefore cannot be reliably extracted from this simulator. This can be understood in terms of the simulated quantities that depend on the global structure of phase space vs those that depend on local trajectories.

Construction of Explicit Symplectic Integrators in General Relativity. II. Reissner–Nordström Black Holes

Abstract In a previous paper, second- and fourth-order explicit symplectic integrators were designed for a Hamiltonian of the Schwarzschild black hole. Following this work, we continue to trace the possibility of construction of explicit symplectic integrators for a Hamiltonian of charged particles moving around a Reissner–Nordström black hole with an external magnetic field. Such explicit symplectic methods are still available when the Hamiltonian is separated into five independently integrable parts with analytical solutions as explicit functions of proper time. Numerical tests show that the proposed algorithms share desirable properties in their long-term stability, precision, and efficiency for appropriate choices of step size. For the applicability of one of the new algorithms, the effects of black hole’s charge, the Coulomb part of the electromagnetic potential and the magnetic parameter on the dynamical behavior are surveyed. Under some circumstances, the extent of chaos becomes strong with an increase of the magnetic parameter from a global phase-space structure. No variation of the black hole’s charge other than the Coulomb part affects the regular and chaotic dynamics of the particles’ orbits. A positive Coulomb part more easily induces chaos than a negative one.

A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space

The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum or, equivalently, the flux of the vorticity or magnetic fields. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoint orbits have a natural symplectic structure, the global geometry of the degenerate (constrained) Poisson manifold can be very complex. Some invariants are represented by the center of the Poisson algebra (i.e., the Casimir elements such as the helicities), and then, the global structure of phase space is delineated by Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir element; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce mock fields and extend the Poisson algebra so that the mock fields are Lie-dragged by the flow vector. A mock field defines a new Casimir element, a cross helicity, which represents topological constraints including the circulation. Unearthing a Casimir element brings about immense advantage in the study of dynamics and equilibria—the so-called energy-Casimir method becomes readily available. Yet, a mock field does not a priori have a physical meaning. Here we proffer an interpretation of a Casimir element obtained, e.g., by such a construction as an adiabatic invariant associated with a hidden ‘microscopic’ angle variable, and in this way give the mock field a physical meaning. We proceed further and consider a perturbation of the Hamiltonian by a canonical pair, composed of the Casimir element and the angle, that causes the topological constraint to be unfrozen. The theory is applied to the tearing modes of magneto-hydrodynamics.

Local integrals and their globally connected invariant structure in phase space giving rise to a promoting mode of chemical reaction

We develop a method to extract local integrals, that is, integrals defined locally in the linear regime of an arbitrary point in phase space. The individual integral represents a vibrational mode. We also propose an index that quantifies the extent of connection between neighboring local integrals. Those pieces that are smoothly connected over a wide range represent a global structure of phase space. With a six-atomic Lennard-Jones cluster, we show that it is possible to identify which vibrational mode in the potential basin correlates smoothly to that in the area of transition state, which is nothing but a reactive mode. As an application of the method, we attempt to enhance the structural transition by exciting the reactive mode thus found. This method works successfully as shown in numerical calculations.

FRONTIER DIAGRAMS: PARTITION OF PHASE SPACE ACCORDING TO THE SIGNS OF EIGENVALUES OR SIGN PATTERNS OF THE CIRCUITS

Nonlinear dynamic systems can be considered in terms of feedback circuits (for short, circuits), which are circular interactions between variables. Each circuit can be identified without ambiguity from the Jacobian matrix of the system. Of special relevance are those circuits (or unions of disjoint circuits) that involve all the variables of the system. We call them "nuclei" because, in the same way as, in Biology, the cell nucleus contains essential genetic information, in nonlinear dynamics, nuclei are crucial elements in the genesis of steady states. Indeed, each nucleus taken alone can generate one or more steady states, whose nature is determined by the sign patterns of the nucleus. There can be up to two nuclei in 2D systems, six in 3D systems, … n! in nD systems. However, many interesting systems of high dimensionality have only two, sometimes even one, nucleus. This paper is based on an extensive exploitation of the Jacobian matrix of systems, in order to figure the global structure of phase space. In nonlinear systems, the value, and often the sign, of terms of the Jacobian matrix depend on the location in phase space. In contrast with a current usage, we consider this matrix, as well as its eigenvalues and eigenvectors, not only in close vicinity of steady states of the system, but also everywhere in phase space. Two distinct, but complementary approaches are used here. In Sec. 2, we define frontiers that partition phase space according to the signs of the eigenvalues (or of their real part if they are complex), and where required, to the slopes of the eigenvectors. From then on, the exact nature of any steady state that might be present in a domain can be identified on the sole basis of its location in that domain and, in addition, one has at least an idea of the possible number of steady states. In Sec. 3, we use a more qualitative approach based on the theory of circuits. Here, phase space is partitioned according to the sign patterns of elements of the Jacobian matrix, more specifically, of the nuclei. We feel that this approach is more generic than that described in Sec. 2. Indeed, it provides a global view of the structure of phase space, and thus permits to infer much of the dynamics of the system by a simple analysis of sign patterns within the Jacobian matrix. This approach also turned out to be extremely useful for synthesizing systems with preconceived properties. For a wide variety of systems, once the partition process has been achieved, each domain comprises at most one steady state. We found, however, a family of systems in which two steady states (typically, two stable or two unstable nodes) differ by neither of the criteria we use, and are thus not separated by our partition processes. This counter-example implies non-polynomial functions. It is essential to realize that the frontier diagrams permit a reduction of the dimensionality of the analysis, because only those variables that are involved in nonlinearities are relevant for the partition process. Whatever the number of variables of a system, its frontier diagram can be drawn in two dimensions whenever no more than two variables are involved in nonlinearities. Pre-existing conjectures concerning the necessary conditions for multistationarity are discussed in terms of the partition processes.

Cold phase fluid model of the longitudinal dynamics of space-charge-dominated beams

The dynamics of a longitudinally cold, charged-particle beam can be simulated by dividing the beam into slices and calculating the motion of the slice boundaries due to the longitudinal electric field generated by the beam. On each time step, the beam charge is deposited onto an (r, z) grid, and an existing (r, z) electrostatic field solver is used to find the longitudinal electric field. Transversely, the beam envelope equation is used for each slice boundary separately. In contrast to the g-factor model, it can be shown analytically that the repulsive electric field of a slice compressed to zero length is bounded. Consequently, this model allows slices to overtake their neighbors, effectively incorporating mixing. The model then effectively describes a cold fluid in longitudinal z, vz phase space. Longitudinal beam compression calculations based on this cold phase fluid model showed that slice overtaking reflects local mixing, while the global phase space structure is preserved.

Stochastic dynamics in a two-dimensional oscillator near a saddle-node bifurcation.

We study the oscillator equations describing a particular class of nonlinear amplifier, exemplified in this work by a two-junction superconducting quantum interference device. This class of dynamic system is described by a potential energy function that can admit minima (corresponding to stable solutions of the dynamic equations), or "running states" wherein the system is biased so that the potential minima disappear and the solutions display spontaneous oscillations. Just beyond the onset of the spontaneous oscillations, the system is known to show significantly enhanced sensitivity to very weak magnetic signals. The global phase space structure allows us to apply a center manifold technique to approximate analytically the oscillatory behavior just past the (saddle-node) bifurcation and compute the oscillation period, which obeys standard scaling laws. In this regime, the dynamics can be represented by an "integrate-fire" model drawn from the computational neuroscience repertoire; in fact, we obtain an "interspike interval" probability density function and an associated power spectral density (computed via Renewal theory) that agree very well with the results obtained via numerical simulations. Notably, driving the system with one or more time sinusoids produces a noise-lowering injection locking effect and/or heterodyning.

Oscillatory dynamics of a nonlinear amplifier in the high-gain regime: Exploiting a global connection

We study the oscillator equations describing a type of nonlinear amplifier, exemplified by a two-junction superconducting quantum interference device. Just beyond the onset of spontaneous oscillations, the system is known to show significantly enhanced sensitivity to very weak magnetic signals. The global phase-space structure allows us to apply a center manifold technique to calculate the frequency of spontaneous oscillations as a function of the natural control parameters. The derived scaling form compares very well with numerical simulations. The ability to quantify the oscillation frequency permits its exploitation as a detection/analysis tool in remote sensing applications, and could also provide a pathway to a dynamic lowering of the low-frequency noise floor in oscillators exhibiting this class of dynamical behavior.

Hamiltonian moment reduction for describing vortices in shear

This paper discusses a general method for approximating two-dimensional and quasigeostrophic three-dimensional fluid flows that are dominated by coherent lumps of vorticity. The method is based upon the noncanonical Hamiltonian structure of the ideal fluid and uses special functionals of the vorticity as dynamical variables. It permits the extraction of exact or approximate finite degree-of-freedom Hamiltonian systems from the partial differential equations that describe vortex dynamics. We give examples in which the functionals are chosen to be spatial moments of the vorticity. The method gives rise to constants of motion known as Casimir invariants and provides a classification scheme for the global phase space structure of the reduced finite systems, based upon Lie algebra theory. The method is illustrated by application to the Kida vortex [S. Kida, J. Phys. Soc. Jpn. 50, 3517 (1981)] and to the problem of the quasigeostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and vertical shear [Meacham et al., Dyn. Atmos. Oceans 14, 333 (1994)]. The approach provides a simple way of visualizing the structure of the phase space of the Kida problem that allows one to easily classify the types of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are represented, without further approximation beyond the assumption of quasigeostrophy, by a finite degree-of-freedom system in canonical variables. The derivation presented here is simpler and more complete than the previous derivation which led to a finite degree-of-freedom system that governs the semi-axes and orientation of the ellipsoid. Using the reduced Hamiltonian description, it is shown that one of the possible modes of evolution of the ellipsoidal vortex is chaotic. These chaotic solutions are noteworthy in that they are exact chaotic solutions of a continuum fluid governing equation, the quasigeostrophic potential vorticity equation.

Frequency Analysis of 3D Electronic1/rDynamics: Tuning between Order and Chaos

Frequency analysis is used to chart the global phase space structure for the most general case of an electron bound by Coulombic and harmonic potentials and perturbed by tilted electric and magnetic fields. Distinct order-chaos transitions are found together with an example of a wave function scarred by a 3D ``exotic'' periodic orbit, thereby providing compelling evidence for the power of frequency analysis in studying multidimensional chaos.

Dynamics of homogeneous magnetizations in strong transverse driving fields

Spatially homogeneous solutions of the Landau--Lifshitz--Gilbert equation are analysed. The conservative as well as the dissipative case is considered explicitly. For the linearly polarized driven Hamiltonian system we apply canonical perturbation theory to uncover the main resonances as well as the global phase space structure. In the case of circularly polarized driven dissipative motion we present the complete bifurcation diagram including bifurcations up to codimension three.

Chaotic Dynamics of Nonideally Driven Simple Oscillators

We discuss the new chaotic phenomena arising from the feedback coupling of the response of a nonlinear oscillator to the driving force. The dynamics is described by two coupled autonomous second order differential equations. The fourth dimension in the state space is found to be important. For example, the simple tangent bifurcation of forced oscillations is replaced by a Feigenbaum sequence followed by a crisis, which can be understood in terms of two-dimensional Henon return map. We also argue that cross over from period doubling to intermittency scaling could be found in these systems. We concentrate in this paper to the global phase space structure and suggest the detailed study on universality and scaling of the routes to chaos as an object for further investigations.