Orbits In Phase Space Research Articles

Dynamics of the epidemiological Predator-Prey system in advective environments.

This paper aims to establish the existence of traveling wave solutions connecting different equilibria for a spatial eco-epidemiological predator-prey system in advective environments. After applying the traveling wave coordinates, these solutions correspond to heteroclinic orbits in phase space. We investigate the existence of the traveling wave solution connecting from a boundary equilibrium to a co-existence equilibrium by using a shooting method. Different from the techniques introduced by Huang, we directly prove the convergence of the solution to a co-existence equilibrium by constructing a special bounded set. Furthermore, the Lyapunov-type function we constructed does not need the condition of bounded below. Our approach provides a different way to study the existence of traveling wave solutions about the co-existence equilibrium. The existence of traveling wave solutions between co-existence equilibria are proved by utilizing the qualitative theory and the geometric singular perturbation theory. Some other open questions of interest are also discussed in the paper.

Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games.

Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of asymmetric games. In this paper, we explore a special class of asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of asymmetric games' evolutionary dynamics.

Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the x−y plane, with parameters vx=vy, it is possible to find exact Landau levels, indexed by the quantum number n, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of kz=0) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as n→∞, we find energies En∝n3/2, which means the nth step as a function of the chemical potential roughly occurs at a value μ∝n3/2. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case vy=λvx, with λ≠1. First, we consider a perturbative treatment around λ≈1 and find that energies En∝n3/2 still hold as n→∞. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints. Published by the American Physical Society 2024

Lagrangian approach for analysis of acoustic energy transport in open cavity flows

The energy transport in aero-acoustics is investigated in the Lagrangian frame. First, based on finite-time Lyapunov exponent (FTLE) and momentum potential theory, a Lagrangian approach is proposed to identify transport barriers of acoustic energy. Specifically, the method, named relative flux gradient (RFG), is presented in detail. Then, to verify the method, it is applied to analytical fields, showing that it could reveal the wavefronts and energy transport barriers depending on the time interval of computation. Moreover, RFG is applied to analyze a simulated flow field of an open cavity flow, and the results are compared with the Lagrangian coherent structures identified by FTLE, demonstrating great similarity. Importantly, the differences between the structures are further analyzed, illustrating several transport channels that correspond to the Rossiter mode and showing a complex interaction between acoustic and vorticity modes. Finally, the relationship between the identified transport barriers and the acoustic behaviors in Eulerian frame is studied in detail. The results show that the transport barriers identified by RFG significantly impact the orbits in phase space, and in particular, RFG has the potential to illustrate and analyze the transport of acoustic energy in complex flow fields in a quantitative way: one method for direct analysis of acoustic phenomena in complex flow regions.

Constructing n-dimensional discrete non-degenerate hyperchaotic maps using QR decomposition

Lyapunov exponents (LEs) characterize the average exponential rate of convergence or divergence between adjacent orbits in phase space. Thus, the number of positive LEs can reflect the complexity of chaotic systems from a certain point of view. To resist the dynamic degradation of digital chaos, we propose a novel universal method that is based on QR decomposition for constructing non-degenerate hyperchaotic maps. A large number of positive LEs can be generated to increase the complexity of chaotic systems. Furthermore, we construct a 4-D discrete non-degenerate hyperchaotic map as an example to demonstrate the adaptability and efficacy of the proposed scheme. For the proposed method, the related control parameters can not only effectively adjust Lyapunov exponents, but also carry out chaotic regulation on the discrete map, such as amplitude control and offset boosting. In addition, a pseudorandom number generator (PRNG) is designed with desirable statistical properties. Then, a microcontroller-based platform was developed to implement the proposed chaotic map. This study is interesting and has a potential for real-world applications, such as secure communication and cryptography.

A hydrodynamic approach to Stark localization

When a free Fermi gas on a lattice is subject to the action of a linear potential it does not drift away, as one would naively expect, but it remains spatially localized. Here we revisit this phenomenon, known as Stark localization, within the recently proposed framework of generalized hydrodynamics. In particular, we consider the dynamics of an initial state in the form of a domain wall and we recover known results for the particle density and the particle current, while we derive analytical predictions for relevant observables such as the entanglement entropy and the full counting statistics. Then, we extend the analysis to generic potentials, highlighting the relationship between the occurrence of localization and the presence of peculiar closed orbits in phase space, arising from the lattice dispersion relation. We also compare our analytical predictions with numerical calculations and with the available results, finding perfect agreement. This approach paves the way for an exact treatment of the interacting case known as Stark many-body localization.

Nonlinear dynamics and onset of chaos in a physical model of a damper pressure relief valve

Hydraulic valves, for the purpose of targeted pressure relief and damping, are a ubiquitous part of modern suspension systems. This paper deals with the analytical computation of fixed points of the dynamical system of a single-stage shock absorber valve, which can be applied for the direct computation of its system variables at equilibrium without brute-force numerical integration. The obtained analytical expressions are given for the original dimensional version of the model derived from continuity and motion equations for a realistic valve setup. Furthermore, a large part of the study is devoted to a systematic sensitivity analysis of the model towards crucial parameter changes. Numerical investigation of a potential loss of stability and following nonlinear oscillations is performed in multi-dimensional parameter spaces, which reveals sustained valve vibrations at increased valve mass and applied pretension force. The dynamical behavior is analyzed by phase space orbits, as well as Fourier-transformed valve displacement data to identify dominant frequencies. Chaotic indicators, such as Lyapunov exponents and the Smaller Alignment Index (SALI), are utilized to understand the nature of the oscillatory motion and to distinguish between order and chaos.

Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link, and propose a new six-dimensional system, which exhibits some hidden attractors. The parameter switching algorithm is used to numerically study the dynamic behaviors of the system. Moreover, it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors. A self-excited attractor with the change of its parameters is also recognized. In addition, numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra, Lyapunov dimensions, bifurcation diagrams, phase space orbits, and basins of attraction. Consequently, the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable. These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems, and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.

Generation of multicavity maps with different behaviours and its DSP implementation

In this paper, based on the mathematical expression of the rotating body in the cylindrical coordinate, the empty rotating-body chaotic model (ERCM) and the full rotating-body chaotic model (FRCM) are constructed. These two models have a pair of coexisting attractors with strictly symmetric phase space orbits. The two models can be used to construct multi-cavity chaotic systems with different attractors. The cylindrical ERCM and FRCM were analyzed. The results show that the two systems have wide chaotic range, large Lyapunov exponent, and high complexity. The DSP implementation of the proposed chaotic systems indicate that it has a good application prospect in engineering.

Orbits of charged particles trapped in a dipole magnetic field.

Motion of a test charged particle in a dipole magnetic field can be reduced to a two degree-of-freedom Hamiltonian system due to the axisymmetry of the dipole field. We carried out a systematic study of orbits of low-energy trapped charged particles in the dipole field via calculation of their Lyapunov characteristic exponents (LCEs) with random initial conditions in the four-dimensional phase space. Since there is at most one positive LCE, these orbits are classified as chaotic ones with one positive LCE and quasi-periodic ones with vanishing LCEs. The dependence of the fraction of quasi-periodic orbits in the phase space on the particle energy is given, which reveals a discrete spectrum, confirming the results of earlier studies. It is also found that most quasi-periodic orbits are confined near the equatorial plane and away from the dipole except for some at very low energies, while chaotic ones are ergodic. The distribution of the maximum LCE (mLCE) appears to vary gradually in the phase space and chaotic orbits with very low values of the mLCE wander near quasi-periodic orbits for a significant amount of time before merging into the sea of chaos.

Triggering recollisions with XUV pulses: Imprint of recolliding periodic orbits

We consider an electron in an atom driven by an infrared (IR) elliptically polarized laser field after its ionization by an ultrashort extreme ultraviolet (XUV) pulse. We find that, regardless of the atom species and the laser ellipticity, there exists XUV parameters for which the electron returns to its parent ion after ionizing, i.e., undergoes a recollision. This shows that XUV pulses trigger efficiently recollisions in atoms regardless of the ellipticity of the IR field. The XUV parameters for which the electron undergoes a recollision are obtained by studying the location of recolliding periodic orbits (RPOs) in phase space. The RPOs and their linear stability are followed and analyzed as a function of the intensity and ellipticity of the IR field. We determine the relation between the RPOs identified here and the ones found in the literature and used to interpret other types of highly nonlinear phenomena for low elliptically and circularly polarized IR fields.

Molecular dynamics simulation of synchronization of a driven particle

Synchronization plays an important role in many physical processes. We discuss synchronization in a molecular dynamics simulation of a single particle moving through a viscous liquid while being driven across a washboard potential energy landscape. Our results show many dynamical patterns as the landscape and driving force are altered. For certain conditions, the particle's velocity and location are synchronized or phase-locked and form closed orbits in phase space. Quasi-periodic motion is common, for which the dynamical center of motion shifts the phase space orbit. By isolating synchronized motion in simulations and table-top experiments, we can study complex natural behaviors important to many physical processes.

Predicting Limit Cycle Boundaries Deep Inside Parameter Space of a 2D Biochemical Nonlinear Oscillator Using Renormalization Group

Formation and study of periodic orbits in phase space in the case of nonlinear oscillators have been a topic of much interest in the recent past. In the current work, a method to go deep inside the limit cycle zone on one side of the bifurcation curve of a 2D non-Lienard biochemical oscillator has been introduced. It is discussed how such an introduction facilitates predicting the boundaries of limit cycles at various points of parameter space, nearly accurately, by the use of perturbative Renormalization Group. Sel’kov model of Glycolytic oscillator has been chosen as the base model to introduce the method.

Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length [Formula: see text] is investigated. Fundamental physical constants such as [Formula: see text], [Formula: see text], and [Formula: see text] are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles [Formula: see text] of radii [Formula: see text] within the discrete [Formula: see text]-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities [Formula: see text] and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion [Formula: see text] matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein–Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group [Formula: see text].

Chaotic Dynamics of Piezoelectric MEMS Based on Maximum Lyapunov Exponent and Smaller Alignment Index Computations

We characterize the dynamical states of a piezoelectric micrcoelectromechanical system (MEMS) using several numerical quantifiers including the maximum Lyapunov exponent, the Poincaré Surface of Section and a chaos detection method called the Smaller Alignment Index (SALI). The analysis makes use of the MEMS Hamiltonian. We start our study by considering the case of a conservative piezoelectric MEMS model and describe the behavior of some representative phase space orbits of the system. We show that the dynamics of the piezoelectric MEMS becomes considerably more complex as the natural frequency of the system’s mechanical part decreases. This refers to the reduction of the stiffness of the piezoelectric transducer. Then, taking into account the effects of damping and time-dependent forces on the piezoelectric MEMS, we derive the corresponding nonautonomous Hamiltonian and investigate its dynamical behavior. We find that the nonconservative system exhibits a rich dynamics, which is strongly influenced by the values of the parameters that govern the piezoelectric MEMS energy gain and loss. Our results provide further evidences of the ability of the SALI to efficiently characterize the chaoticity of dynamical systems.

Traveling wave oscillatory patterns in a signed Kuramoto–Sivashinsky equation with absorption

In this paper, a proof of a part of a conjecture raised in Galaktionov and Svirshchevskii (2007) concerning existence and global uniqueness of an asymptotically stable periodic orbit in a fourth-order piecewise linear ordinary differential equation is presented. The fourth-order equation comes from the study of traveling wave patterns in a signed Kuramoto–Sivashinsky equation with absorption. The proof is twofold. First, the problem of solving for the periodic orbit is transformed into a zero finding problem on R4, which is solved with a computer-assisted proof based on Newton’s method and the contraction mapping theorem. Second, the rigorous bounds about the periodic orbit in phase space are combined with the theory of discontinuous dynamical systems to prove that the orbit is asymptotically stable.

On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

Lyapunov exponents are related to the exponentially fast divergence or convergence of nearby orbits in phase space, and they can be used to evaluate the Kaplan–Yorke dimension. In this manner, since the existence of a positive Lyapunov exponent (LE+) is taken as an indication that chaotic behavior exists, and due to the huge search spaces of the design variables of chaotic oscillators, we show the application of differential evolution (DE) and particle swarm optimization (PSO) algorithms to maximize LE+. Four chaotic oscillators are optimized herein, for which we detail the evaluation of their equilibrium points and their eigenvalues that are used to estimate the step-size h to perform appropriate numerical simulation. Both DE and PSO are calibrated to perform different number of generations with three different sizes of individuals in the populations, and with search spaces around the values already published for the four chaotic oscillators. As a result, we show that both DE and PSO algorithms provide higher values of LE+ and Kaplan–Yorke dimension compared to the ones already published in the literature.

Early Fault Diagnosis of Rolling Bearing Based on Lyapunov Exponent

Lyapunov exponent is an important quantitative index to measure the dynamic characteristics of the system. It represents the average exponential rate of convergence or divergence between adjacent orbits in phase space. Whether there is dynamic chaos in the system can be judged intuitively from whether the maximum Lyapunov exponent is greater than zero. The Lyapunov exponent obtained by the small data method has the characteristics of simple calculation process and accurate calculation results, and it is applied to the fault diagnosis of rolling bearings. By calculating the Lyapunov exponents of bearing rollers, bearing outer rings, bearing inner rings and normal state bearings, and comparing the calculation results under different working conditions, it is concluded that Lyapunov exponents are of great significance in judging the early failure of rolling bearings.

Bifurcation Analysis and Vibration Signal Identification for a Motorized Spindle with Random Uncertainty

A nonparametric model is established to investigate the vibration characteristics of a motorized spindle system, where the bearing restoring force, the unbalanced magnetic pull, and the external bounded noise excitation are taken into account. Based on the Monte Carlo simulation, several numerical examples are used to study the effect of dispersion parameters from model uncertainty and strength of external bounded noise excitation on the whirl frequency and bifurcation behavior of the spindle. It is shown that the whirl frequency fluctuates due to the random uncertainty, and the fluctuation range grows wider as the dispersion parameters increase as anticipated. In the randomly uncertain case, the vibration of the spindle also exhibits periodic-dominant and bifurcation phenomena, of particular interest is the delay of bifurcation point following the increase of the rotation speed. Towards the vibration condition monitoring on the spindle system, the phase space reconstruction technique is also applied to identify the vibration signals by comparing the reconstructed deterministic phase-space orbits with those from the random uncertainty and the bounded noise excitation, and the inherent dynamical behavior can be revealed from the irregular one-dimension time series effectively.

Examples of Hamiltonians isochronous in configuration space only and their quantization

Examples of many-body Hamiltonians are identified which yield time-evolutions, having the property that the projection of an arbitrary orbit in phase space onto configuration space is periodic with a period independent of initial data (“isochrony”), while the evolution in phase space is not periodic. These include a variation on the harmonic oscillator as well as a Hamiltonian yielding the “goldfish” equations of motion. We then investigate a one-body version in the complex plane of this goldfish Hamiltonian. Its quantization is studied, and it is shown that the spectrum is continuous and infinitely degenerate; the general solution of the time-dependent Schrödinger equation in the position representation is not time-periodic, but its square modulus does evolve isochronously.