Phase Space Structure Research Articles

Nonlinear dynamics of coupled axion-Josephson junction systems

We study the classical dynamics of an axion field (the signal) that is coupling into a Josephson junction (the detector) by means of a capacitive coupling of arbitrary size. Depending on the size of the coupling constant and the initial conditions, we find a rich phase space structure of this nonlinear problem. We present general analytic solutions of the equations of motion in the limit of small amplitudes of the angle variables, and discuss both the case of no dissipation and the case of dissipation in the system. The effect of a magnetic field is investigated as well, leading to topological phase transitions in the phase space structure.

Phase Space Analysis of Relaxation Dynamics of Isoscalar Giant Monopole Resonances

A classical model based on the independent particle approach to the nuclear dynamics is used to study the influence of the phase space structures on the onebody dissipation of isoscalar Giant Monopole Resonances. The model consists of a harmonic oscillator describing the collective excitation coupled with a nonlinear (Woods-Saxon) oscillator representing the motion of each nucléon. We are particulary interested in the dependence of relaxation on the energy of the system. We have found that in a rather broad region of parameter space, contrary to the common expectation, both Lyapunov exponent and relaxation time increase as a function of the total energy. We examine the conditions required for this effect to occur and demonstrate the key role of the dispersion relation of the nonlinear oscillator.

Phase space structures causing the reaction rate decrease in the collinear hydrogen exchange reaction

The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with 2 degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood—the reaction rate decreases as a function of energy beyond a critical value. Using lobe dynamics we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive upper and lower bounds of the reaction rate, which are desirable for practical reasons.

Roaming at Constant Kinetic Energy: Chesnavich’s Model and the Hamiltonian Isokinetic Thermostat

We consider the roaming mechanism for chemical reactions under the nonholonomic constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian isokinetic thermostat applied to Chesnavich's model for an ion-molecule reaction. Through an analysis of phase space structures we show that imposing the nonholonomic constraint does not prevent the system from exhibiting roaming dynamics, and that the origin of the roaming mechanism turns out to be analogous to that found in the previously studied Hamiltonian case.

A Dynamical Model for Clustered Star Formation in the Galactic Disk

The clustered nature of star formation should produce a high degree of structure in the combined phase and chemical space in the Galactic disk. To date, observed structure of this kind has been mostly limited to bound clusters and moving groups. In this paper we present a new dynamical model of the Galactic disk that takes into account the clustered nature of star formation. This model predicts that the combined phase and chemical space is rich in substructure, and that this structure is sensitive to both the precise nature of clustered star formation and the large-scale properties of the Galaxy. The model self-consistently evolves 4 billion stars over the last 5 Gyr in a realistic potential that includes an axisymmetric component, a bar, spiral arms, and giant molecular clouds (GMCs). All stars are born in clusters with an observationally-motivated range of initial conditions. As direct \textit{N}-body calculations for billions of stars is computationally infeasible, we have developed a method of initializing star cluster particles to mimic the effects of direct \textit{N}-body effects, while the actual orbit integrations are treated as test particles within the analytic potential. We demonstrate that the combination of chemical and phase space information is much more effective at identifying truly co-natal populations than either chemical or phase space alone. Furthermore, we show that co-moving pairs of stars are very likely to be co-natal if their velocity separation is $< 2$ km s$^{-1}$ and their metallicity separation is $< 0.05$ dex. The results presented here bode well for harnessing the synergies between \textit{Gaia} and spectroscopic surveys to reveal the assembly history of the Galactic disk.

Dynamical system analysis of generalized energy-momentum-squared gravity

In this work we have investigated the dynamics of a recent modification to the general theory of relativity, the energy-momentum squared gravity model $f(R,\mathbf{T^2})$, where $R$ represents the scalar curvature and $\mathbf{T^2}$ the square of the energy-momentum tensor. By using dynamical system analysis for various types of gravity functions $f(R,\mathbf{T^2})$, we have studied the structure of the phase space and the physical implications of the energy-momentum squared coupling. In the first case of functional where $f(R,\mathbf{T^2})=f_0 R^n(\mathbf{T^2})^m$, with $f_0$ constant, we have shown that the phase space structure has a reduced complexity, with a high sensitivity to the values of the $m$ and $n$ parameters. Depending on the values of the $m$ and $n$ parameters, the model exhibits various cosmological epochs, corresponding to matter eras, solutions associated with an accelerated expansion, or decelerated periods. The second model studied corresponds to the $f(R,\mathbf{T^2})=\alpha R^n+\beta\mathbf{(T^2)}^m$ form with $\alpha, \beta$ constant parameters. In this case, a richer phase space structure is obtained which can recover different cosmological scenarios, associated to matter eras, de--Sitter solutions, and dark energy epochs. Hence, this model represents an interesting cosmological model which can explain the current evolution of the Universe and the emergence of the accelerated expansion as a geometrical consequence.

Dynamics and computation in mixed networks containing neurons that accelerate towards spiking.

Networks in the brain consist of different types of neurons. Here we investigate the influence of neuron diversity on the dynamics, phase space structure, and computational capabilities of spiking neural networks. We find that already a single neuron of a different type can qualitatively change the network dynamics and that mixed networks may combine the computational capabilities of ones with a single-neuron type. We study inhibitory networks of concave leaky (LIF) and convex "antileaky" (XIF) integrate-and-fire neurons that generalize irregularly spiking nonchaotic LIF neuron networks. Endowed with simple conductance-based synapses for XIF neurons, our networks can generate a balanced state of irregular asynchronous spiking as well. We determine the voltage probability distributions and self-consistent firing rates assuming Poisson input with finite-size spike impacts. Further, we compute the full spectrum of Lyapunov exponents (LEs) and the covariant Lyapunov vectors (CLVs) specifying the corresponding perturbation directions. We find that there is approximately one positive LE for each XIF neuron. This indicates in particular that a single XIF neuron renders the network dynamics chaotic. A simple mean-field approach, which can be justified by properties of the CLVs, explains the finding. As an application, we propose a spike-based computing scheme where our networks serve as computational reservoirs and their different stability properties yield different computational capabilities.

A cartographic study of spin-orbit coupling in binary asteroids

AbstractIn the spin-orbit resonances, we assume that the orbit of the secondary asteroid around the primary is invariant, which is a reasonable assumption at first glance. Owing to the irregularity of asteroids’ geometry and their effect on the mutual orbit, this assumption should be revised. Therefore, we focus on a binary asteroid with a spherical primary and a secondary with an irregular shape. When the shape of a secondary asteroid is not a sphere, the gravitational interaction is important, and we should consider the interaction of orbit and spin. We generate fast Lyapunov indicator (FLI) maps for both spin-orbit resonance and spin-orbit coupling problems and investigate the effect of orbit alternation on the structure of phase space.

New results on orbital resonances.

Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently reexamined the nature of these divergences using non-perturbative numerical analyses, making use of Poincaré sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant angle. At low eccentricity, first order resonances do not have diverging widths but have two asymmetric branches leading away from the nominal resonance location. A sequence of structures called "low-eccentricity resonant bridges" connecting neighboring resonances is revealed. At planet-grazing eccentricity, the true resonance width is non-divergent. At higher eccentricities, the new results reveal hitherto unknown resonant structures and show that these parameter regions have a loss of some - though not necessarily entire - resonance libration zones to chaos. The chaos at high eccentricities was previously attributed to the overlap of neighboring resonances. The new results reveal the additional role of bifurcations and co-existence of phase-shifted resonance zones at higher eccentricities. By employing a geometric point of view, we relate the high eccentricity phase space structures and their transitions to the shapes of resonant orbits in the rotating frame. We outline some directions for future research to advance understanding of the dynamics of mean motion resonances.

The bifurcation of periodic orbits and equilibrium points in the linked restricted three-body problem with parameter ω.

This paper is devoted to the bifurcation of periodic orbits and libration points in the linked restricted three-body problem (LR3BP). Inherited from the classic circular restricted three-body problem (CR3BP), it retains most of the dynamical structure of CR3BP, while its dynamical flow is dominated by angular velocity ω and Jacobi energy C. Thus, for the first time, the influence of the angular velocity in the three-body problem is discussed in this paper based on ω-motivated and C-motivated bifurcation. The existence and collision of equilibrium points in the LR3BP are investigated analytically. The dynamic bifurcation of the LR3BP under angular velocity variation is obtained based on three typical kinds of periodic orbits, i.e., planar and vertical Lyapunov orbits and Halo orbits. More bifurcation points are supplemented to Doedel's results in the CR3BP for a global sketch of bifurcation families. For the first time, a new bifurcation phenomenon is discovered that as ω approaches to 1.4, two period-doubling bifurcation points along the Halo family merge together. It suggests that the number and the topological type of bifurcation points themselves can be altered when the system parameter varies in LR3BP. Thus, it is named as "bifurcation of bifurcation" or "secondary bifurcation" in this paper. At selected values of ω, the phase space structures of equilibrium points L2 and L3 are revealed by Lie series method numerically, presenting the center manifolds on the Poincaré section and detecting three patterns of evolution for center manifolds in LR3BP.

Degenerate Hamiltonian operator in higher-order canonical gravity — The problem and a remedy

Different routes towards canonical formulation of a classical theory result in different canonically equivalent Hamiltonians, while their quantum counterparts are related through appropriate unitary transformation. However, for higher-order theory of gravity although two Hamiltonians emerging from the same action differing by total derivative terms are related through canonical transformation, the difference transpires while attempting canonical quantization, which is predominant in non-minimally coupled higher-order theory of gravity. We follow Dirac’s constraint analysis to formulate phase-space structures, in the presence (case-I) and absence (case-II) of total derivative terms. While the coupling parameter plays no significant role as such for case-I, quantization depends on its form explicitly in case-II, and as a result unitary transformation relating the two is not unique. We also find certain mathematical inconsistencies in case-I, for modified Gauss–Bonnet-Dilatonic coupled action, in particular. Thus, we conclude that total derivative terms indeed play a major role in the quantum domain and should be taken care of a-priori, for consistency.

Simulating cosmological substructure in the solar neighbourhood

ABSTRACT We explore the predictive power of cosmological, hydrodynamical simulations for stellar phase-space substructure and velocity correlations with the auriga simulations and aurigaia mock Gaia catalogues. We show that at the solar circle the auriga simulations commonly host phase-space structures in the stellar component that have constant orbital energies and arise from accreted subhaloes. These structures can persist for a few Gyr, even after coherent streams in position space have been erased. We also explore velocity two-point correlation functions and find this diagnostic is not deterministic for particular clustering patterns in phase space. Finally, we explore these structure diagnostics with the aurigaia catalogues and show that current catalogues have the ability to recover some structures in phase space but careful consideration is required to separate physical structures from numerical structures arising from catalogue generation methods.

Structure of the centre manifold of the $$L_1,L_2$$ collinear libration points in the restricted three-body problem

We present a global analysis of the center manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff-Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

Mapping the Stellar Halo with the H3 Spectroscopic Survey

Abstract Modern theories of galaxy formation predict that the Galactic stellar halo was hierarchically assembled from the accretion and disruption of smaller systems. This hierarchical assembly is expected to produce a high degree of structure in the combined phase and chemistry space; this structure should provide a relatively direct probe of the accretion history of our Galaxy. Revealing this structure requires precise 3D positions (including distances), 3D velocities, and chemistry for large samples of stars. The Gaia satellite is delivering proper motions and parallaxes for &gt;1 billion stars to G ≈ 20. However, radial velocities and metallicities will only be available to G ≈ 15, which is insufficient to probe the outer stellar halo (≳10 kpc). Moreover, parallaxes will not be precise enough to deliver high-quality distances for stars beyond ∼10 kpc. Identifying accreted systems throughout the stellar halo therefore requires a large ground-based spectroscopic survey to complement Gaia. Here we provide an overview of the H3 Stellar Spectroscopic Survey, which will deliver precise stellar parameters and spectrophotometric distances for ≈200,000 stars to r = 18. Spectra are obtained with the Hectochelle instrument at the MMT, which is configured for the H3 Survey to deliver resolution R ≈ 23,000 spectra covering the wavelength range 5150–5300 Å. The survey is optimized for stellar halo science and therefore focuses on high Galactic latitude fields ( ), sparsely sampling 15,000 sq. degrees. Targets are selected on the basis of Gaia parallaxes, enabling very efficient selection of bona fide halo stars. The survey began in the fall of 2017 and has collected 88,000 spectra to-date. All of the data, including the derived stellar parameters, will eventually be made publicly available via the survey website: h3survey.rc.fas.harvard.edu.

Noncommutative phase-space effects in thermal diffusion of Gaussian states

Noncommutative phase-space and its effects have been studied in different settings in physics, in order to unveil a better understanding of phase-space structures. Here, we use the thermal diffusion approach to study how noncommutative effects can influence the time evolution of a one-mode Gaussian state when in contact with a thermal environment obeying the Markov approximation. Employing the cooling process and considering the system of interest as a one-mode Gaussian state, we show that the fidelity comparing the Gaussian state of the system in different times and the asymptotic thermal state is useful to sign noncommutative effects. In addition, by using the monotonicity behavior of the fidelity, we discuss some aspects of non-Markovianity during the dynamics.

Dynamics of a discrete-time stage-structured predator–prey system with Holling type II response function

A discrete-time model of predator–prey dynamics with Holling type II response function is proposed. Each species considered has its age structure which is typical for natural communities. The bifurcations, dynamic modes and a possibility of its shifting are studied for the proposed model. The stability loss of a non-trivial fixed point is realised according to Neimark–Sacker and Feigenbaum scenarios. The phase space structure of multistability areas in which a variation of current numbers of prey or predator can shift the dynamic modes is analysed using basins of attraction. Particular attention is paid to the investigation of the feasible values of the parameters in which the model has a biological meaning. To understand the mechanisms of interspecific interaction influence on each species dynamics in a community, we compare cases with and without interspecific interaction. The presence of interspecific interaction in the community increases the variety of emerging dynamic modes of the predator population size.

Butterfly in a Cocoon, Understanding the Origin and Morphology of Globular Cluster Streams: The Case of GD-1

Abstract Tidally disrupted globular cluster (GC) streams are usually observed, and therefore perceived, as narrow, linear, and one-dimensional structures in the 6D phase space. Here, we show that the GD-1 stellar stream, which is the tidal debris of a disrupted GC, possesses a secondary diffuse and extended stellar component (∼100 pc wide) around it, detected at the &gt;5σ confidence level. Similar morphological properties are seen in synthetic streams that are produced from star clusters that are formed within dark matter sub-halos and then accrete onto a massive host galaxy. This lends credence to the idea that the progenitor of the highly retrograde GD-1 stream was originally formed outside of the Milky Way in a now defunct dark satellite galaxy. We deem that in future studies, this newly found cocoon component may serve as a structural hallmark to distinguish between the in situ and ex situ (accreted) formed GC streams.

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential.

We present a method based on a Lagrangian descriptor for revealing the high-dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include a normally hyperbolic invariant manifold and its stable and unstable manifolds, which act as codimension-1 barriers to phase space transport. In this article, finding the invariant manifolds in high-dimensional phase space will constitute identifying coordinates on these invariant manifolds. The method of Lagrangian descriptor is demonstrated by applying to classical two and three degrees of freedom Hamiltonian systems which have implications for myriad applications in chemistry, engineering, and physics.

Secular dynamics of binaries in stellar clusters – II. Dynamical evolution

AbstractDense stellar clusters are natural sites for the origin and evolution of exotic objects such as relativistic binaries (potential gravitational wave sources) and blue stragglers. We investigate the secular dynamics of a binary system driven by the global tidal field of an axisymmetric stellar cluster in which the binary orbits. In a companion paper we developed a general Hamiltonian framework describing such systems. The effective (doubly-averaged) Hamiltonian derived there encapsulates all information about the tidal potential experienced by the binary in its orbit around the cluster in a single parameter Γ. Here we provide a thorough exploration of the phase-space of the corresponding secular problem as Γ is varied. We find that for Γ &gt; 1/5 the phase-space structure and the evolution of binary orbital elements are qualitatively similar to the Lidov–Kozai problem. However, this is only one of four possible regimes, because the dynamics are qualitatively changed by bifurcations at Γ = 1/5, 0, −1/5. We show how the dynamics are altered in each regime and calculate characteristics such as the secular evolution time-scale and maximum possible eccentricity. We verify the predictions of our doubly-averaged formalism numerically and find it to be very accurate when its underlying assumptions are fulfilled, typically meaning that the secular time-scale should exceed the period of the binary around the cluster by ≳10–102 (depending on the cluster potential and binary orbit). Our results may be relevant for understanding the nature of a variety of exotic systems harboured by stellar clusters.

Transport in Hamiltonian systems with slowly changing phase space structure

Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the one-dimensional case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called ”transport by migration with the mixing regions” - is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates.