Periodic Trajectories Research Articles

Periodic Trajectories for HIV Dynamics in a Seasonal Environment With a General Incidence Rate

In this paper, we propose a five-dimensional nonlinear system of differential equations for the human immunodeficiency virus (HIV) including the B-cell functions with a general nonlinear incidence rate. The compartment of infected cells was subdivided into three classes representing the latently infected cells, the short-lived productively infected cells, and the long-lived productively infected cells. The basic reproduction number was established, and the local and global stability of the equilibria of the model were studied. A sensitivity analysis with respect to the model parameters was undertaken. Finally, some numerical simulations are presented to illustrate the theoretical findings.

A Cyclic Differentiable Manifold Representation of Redundant Manipulator Kinematics

Abstract An inverse kinematic mapping for redundant serial manipulators is presented at the configuration level, for which periodic manipulator operational trajectories map into periodic input trajectories, i.e., for which serial manipulators are cyclic. The inverse kinematic mapping defines a differentiable manifold on which output and self-motion coordinates comprise operational coordinates that represent manipulator redundant degrees-of-freedom. The inverse kinematic mapping and differentiable manifold are defined in analytical form and a computational method for their evaluation is presented. Numerical examples are presented to illustrate the validity of the formulation on subsets of manipulator regular configuration space, or on the entire space.

Bifurcation analysis of ion-acoustic superperiodic waves in ultra-relativistic dense plasma

Abstract Bifurcation analysis of ion-acoustic solitary, periodic and superperiodic waves in ultra-relativistic quantum degenerate electron-positron-ion plasma is investigated based on Sagdeev’s pseudopotential approach. The phase plane plots for the superperiodic, periodic, and homoclinic trajectories for the dynamical system are developed. Rarefactive and compressive solitary waves are demonstrated to exist. The influence of plasma parameters on the characteristic properties of the periodic and superperiodic waves is studied. Also, the effect of different plasma parameters on the Sagdeev pseudopotential profile is investigated. The results of this study may be used to shed light on the fundamental properties of nonlinear waves in dense objects (e.g., white dwarfs and neutron stars). These results also may be beneficial in describing laboratory plasma (e.g., interaction experiments of intense laser-solid matter), and plasma applications such as microelectronic devices.

Solid-body trajectoids shaped to roll along desired pathways.

In everyday life, rolling motion is typically associated with cylindrical (for example, car wheels) or spherical (for example, billiard balls) bodies tracing linear paths. However, mathematicians have, for decades, been interested in more exotically shaped solids such as the famous oloids1, sphericons2, polycons3, platonicons4 and two-circle rollers5 that roll downhill in curvilinear paths (in contrast to cylinders or spheres) yet indefinitely (in contrast to cones, Supplementary Video1). The trajectories traced by such bodies have been studied in detail6-9, and can be useful in the context of efficient mixing10,11 and robotics, for example, in magnetically actuated, millimetre-sized sphericon-shaped robots12,13, or larger sphericon- and oloid-shaped robots translocating by shifting their centre of mass14,15. However, the rolling paths of these shapes are all sinusoid-like and their diversity ends there. Accordingly, we were intrigued whether a more general problem is solvable: given an infinite periodic trajectory, find the shape that would trace this trajectory when rolling down a slope. Here, we develop an algorithm to design such bodies-which we call 'trajectoids'-and then validate these designs experimentally by three-dimensionally printing the computed shapes and tracking their rolling paths, including those that close onto themselves such that the body's centre of mass moves intermittently uphill (Supplementary Video2). Our study is motivated largely by fundamental curiosity, but the existence of trajectoids for most paths has unexpected implications for quantum and classical optics, as the dynamics of qubits, spins and light polarization can be exactly mapped to trajectoids and their paths16.

The long-term behavior of the p-adic Sigmoid Beverton-Holt dynamical systems in the projective line [formula omitted

In this paper we study the long-term behavior of the p-adic dynamical systems associated with the Sigmoid Beverton-Holt model that arises in population dynamics. The corresponding fixed points, maximal Siegel disks, attractors, and periodic trajectories are analyzed in the projective line P1(Qp). Moreover, the Julia and Fatou sets associated with these dynamical systems are also examined.

On Almost Periodic Trajectories of Control Systems with Feedback in the Form of Sweeping Processes

On Almost Periodic Trajectories of Control Systems with Feedback in the Form of Sweeping Processes

Minimizing Energy Consumption of Elastic Robots in Repetitive Tasks

Energy consumption is an important issue in robotics. This article deals with the problem of reducing the energy consumption of compliant electro-mechanical systems while performing periodic tasks. After deriving performance indices to quantify the energy consumption of a mechanical system, we propose a method to determine both the optimal compliant actuation parameters, and link trajectories to minimize energy consumption. We show how this problem can be cast in a simpler one where the optimization regards only parameters that define the shape of periodic trajectories to be subsequently determined by using numerical optimization tools. Indeed, in our framework, the optimal stiffness and spring preload can be analytically obtained as a function of the desired link trajectories. We then provide simulations and experimental validations of the obtained results on a two-link compliant manipulator platform which performs a repetitive pick-and-place task. Our experiments show that the use of compliant actuators instead of rigid ones and the optimization of their compliant parameters give rise to an energy saving up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$62\%$</tex-math> </inline-formula> with respect to rigid actuation. Moreover, the simultaneous optimization of the compliant parameters and link trajectories provide an additional energy saving up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$20\%$</tex-math> </inline-formula> .

Covariance analysis of periodic and quasi-periodic orbits around Phobos with applications to the Martian Moons eXploration mission

Understanding the internal composition of a celestial body is fundamental for formulating theories regarding its origin. Deep knowledge of the distribution of mass under the body’s crust can be achieved by analyzing its moments of inertia and gravity field. In this regard, the two moons of the Martian system have not yet been closely studied and continue to pose questions regarding their origin to the space community; thus, they deserve further characterization. The Martian Moons eXploration mission will be the first of its kind to sample and study Phobos over a prolonged period. This study aims to demonstrate that the adoption of periodic and quasi-periodic retrograde trajectories would be beneficial for the scientific value of the mission. Here, a covariance analysis was implemented to compare the estimation of high-order gravitational field coefficients from different orbital geometries and for different sets of processed observables. It was shown that the adoption of low-altitude non-planar quasi-satellite orbits would help to refine the knowledge of the moon’s libration angle and gravitational field.

Memory Maps with Elliptical Trajectories

A family of maps with memory, parameterized by [Formula: see text], is shown to have either periodic trajectories or dense trajectories on ellipses which support absolutely continuous invariant measures. Furthermore, for [Formula: see text], i.e. [Formula: see text] with [Formula: see text] and [Formula: see text], all points except [Formula: see text] either go into a polygonal region centered at [Formula: see text] if [Formula: see text] is rational, or are attracted to an elliptical region having the same center, if [Formula: see text] is irrational. In the polygonal case, we examine a mechanism for the appearance of islands supporting absolutely continuous invariant measures.

Circuit analysis using monotone+skew splitting

It is shown that the behavior of an m-port circuit of maximal monotone elements can be expressed as a zero of the sum of a maximal monotone operator containing the circuit elements, and a structured skew-symmetric linear operator representing the interconnection structure, together with a linear output transformation. The Condat–Vũ algorithm solves inclusion problems of this form, and may be used to solve for the periodic steady-state behavior, given a periodic excitation at each port, using an iteration in the space of periodic trajectories.

Early development of electrophysiological activity: Contribution of periodic and aperiodic components of the EEG signal.

Brain function rapidly changes in the first 2 years of life. In the last decades, resting-state EEG has been widely used to explore those changes. Previous studies have focused on the relative power of the signal in established frequency bands (i.e., theta, alpha, and beta). However, EEG power is a mixture of a 1/f-like background power (aperiodic) in combination with narrow peaks that appear over that curve (periodic activity, e.g., alpha peak). Therefore, it is possible that relative power captures both, aperiodic and periodic brain activity, contributing to changes in electrophysiological activity observed in infancy. For this reason, we explored the early developmental trajectory of the relative power in theta, alpha, and beta frequency bands from infancy to toddlerhood and compared it with changes in periodic activity in a longitudinal study with three waves at age 6, 9, and 16 to 18 months. Finally, we tested the contribution of periodic activity and aperiodic components of the EEG to age changes in relative power. We found that relative power and periodic activity trajectories differed in this period in all the frequency bands but alpha. Furthermore, aperiodic EEG activity flattened between 6 and 18 months. More importantly, only alpha relative power was exclusively related to periodic activity, whereas aperiodic components of the signal significantly contributed to the relative power of activity in theta and beta bands. Thus, relative power in these frequencies is influenced by developmental changes of the aperiodic activity, which should be considered for future studies.

Dynamic soaring in wind turbine wakes

Dynamic soaring for UAVs is a flight technique that enables continuous, powerless periodic flight patterns in the presence of a wind gradient. However, sufficiently large wind gradients are uncommon over land, while at offshore locations the largest wind gradients are located close to the ocean surface, thereby limiting the scope of practical application. An intrinsic feature of wind turbines is that they inherently produce very sharp wind gradients in the near wake. Therefore, in this paper, we propose and investigate periodic stationary dynamic soaring trajectories in the near wake of wind turbines. We additionally consider the potential of dynamic soaring for revitalizing the wind turbine wake. To this end, we apply periodic optimal control based on a simplified model for the glider dynamics and the wind profile in the wake. The cost function maximizes the revitalization of the wake. We compute optimal orbits for a range of different wing spans and different mass-scaling assumptions. The largest glider configuration, with a wingspan of 10 m and a mass of 222.6 kg, achieves a wake revitalization of about 0.94% of the total turbine thrust.

Research on RBF neural network adaptive control of three-point contactless measuring device for CNC roller grinder

Abstract In order to improve the CNC roll grinder production efficiency, a novel three-point non-contact measurement device is proposed. An adaptive back-stepping control system that combines with a radial basis function neural network (RBFNN) was designed to control the measurement device working position and also to track the periodic reference movement trajectory. In the proposed control system, the RBFNN approximation and tracking ability are used to identify the unknown measuring device dynamic information. Then, the Lyapunov stability theorem is used to derive the adaptive online learning algorithm. All control algorithms are placed in the control chip based on the TMS320F28335 archive. The simulation results show that the proposed control system has a good control effect when applied to the three-point non-contact measuring device in CNC roll grinders.

Influence of sleep fragmentation in infancy and toddler period on emotional and behavioral problem at the age of 6 years: a birth cohort study

Objective: To investigate the influence of sleep fragmentation in infancy and toddler period on emotional and behavioral problems at the age of 6 years. Methods: Using a prospective cohort design, 262 children were extracted from mother-child birth cohort recruited from May 2012 to July 2013 in Renji Hospital, School of Medicine, Shanghai Jiao Tong University. Children's sleep and physical activities were assessed using actigraphy at 6, 12, 18, 24, and 36 months of age, from which the sleep fragmentation index (FI) at each follow-up point was calculated. Children's emotional and behavioral problems at 6 years of age were assessed using the strengths and difficulties questionnaire. Group-based trajectory model was applied to determine sleep FI in infancy and toddler period trajectory groups with Bayesian information criteria being used to determine the best fitting model. Children's emotional and behavioral problems between groups were examined with independent t test and linear regression models, etc. Results: A total of 177 children, with 91 boys and 86 girls, were included in the final analysis and were divided into 2 groups: high FI group (n=30) and low FI group (n=147). Compared with children in the low FI group, those in the high FI group presents with higher total difficulties score and higher hyperactivity or inattention score ((11.0±4.9) vs. (8.9±4.1), (4.9±2.7) vs. (3.7±2.3) scores, t=2.17, 2.23, both P<0.05, respectively), with the differences remaining significant after adjusting for covariates (t=2.08, 2.09, both P<0.05 respectively). Conclusion: High sleep fragmentation in infancy and toddler period is associated with more emotional and behavioral problems, especially hyperactivity or inattention problems, at 6 years of age.

Age of depression, age of melancholy? age, period, and cohort trajectories of mental health in ten countries between 1991-2020

Age of depression, age of melancholy? age, period, and cohort trajectories of mental health in ten countries between 1991-2020

Forecasting and stabilizing chaotic regimes in two macroeconomic models via artificial intelligence technologies and control methods

One of the key tasks in the economy is forecasting the economic agents’ expectations of the future values of economic variables using mathematical models. The behavior of mathematical models can be irregular, including chaotic, which reduces their predictive power. In this paper, we study the regimes of behavior of two economic models and identify irregular dynamics in them. Using these models as an example, we demonstrate the effectiveness of evolutionary algorithms and the continuous deep Q-learning method in combination with Pyragas control method for deriving a control action that stabilizes unstable periodic trajectories and suppresses chaotic dynamics. We compare qualitative and quantitative characteristics of the model’s dynamics before and after applying control and verify the obtained results by numerical simulation. Proposed approach can improve the reliability of forecasting and tuning of the economic mechanism to achieve maximum decision-making efficiency.

Nonautonomous Breather and Rogue Wave in Spinor Bose–Einstein Condensates with Space-Time Modulated Potentials

To study controlled evolution of nonautonomous matter-wave breathers and rogue waves in spinor Bose–Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross–Pitaevskii equations with spacetime-dependent external potentials and temporally modulated gain-loss distributions. With different external potentials and gain-loss distributions, various solutions for controlled nonautonomous matter-wave breathers and rogue waves are derived by the Darboux transformation method, such as breathers and rogue waves on arched and constant backgrounds which have the periodic and parabolic trajectories. Effects of the gain-loss distribution and linear potential on the breathers and rogue waves are studied. Nonautonomous two-breathers on the arched and constant backgrounds are also derived.

A Dynamical Systems Approach to the Theory of Circumbinary Orbits in the Circular Restricted Problem

To better understand the orbital dynamics of exoplanets around close binary stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical systems theory to a physically motivated set of solutions in the Circular Restricted Three-Body Problem (CR3BP). We applied Floquet theory to characterize the linear dynamical behavior—static, oscillatory, or exponential—surrounding planar circumbinary periodic trajectories (limit cycles). We computed prograde and retrograde limit cycles and analyzed their geometries, stability bifurcations, and dynamical structures. Orbit and stability calculations are exact computations in the CR3BP and reproducible through the open-source Python package pyraa. The periodic trajectories (doi.org/10.5281/zenodo.7532982) produce a set of noncrossing, dynamically cool circumbinary orbits conducive to planetesimal growth. For mass ratios μ ∈ [0.01, 0.50], we found recurring features in the prograde families. These features include (1) an innermost near-circular trajectory, inside which solutions have resonant geometries, (2) an innermost stable trajectory (a c ≈ 1.61 − 1.85 a bin) characterized by a tangent bifurcating limit cycle, and (3) a region of dynamical instability (a ≈ 2.1 a bin; Δa ≈ 0.1 a bin), the exclusion zone, bounded by a pair of critically stable trajectories bifurcating limit cycles. The exterior boundary of the exclusion zone is consistent with prior determinations of a c around a circular binary. We validate our analytic results with N-body simulations and apply them to the Pluto–Charon system. The absence of detected CBPs in the inner stable region, between the prograde exclusion zone and a c , suggests that the exclusion zone may inhibit the inward migration of CBPs.

Quantum scar affecting the motion of three interacting particles in a circular trap

We propose theoretically a quantum scar affecting the motion of three interacting particles in a circular trap. We numerically calculate the quantum eigenstates of the system and show that some of them are scarred by a classically unstable periodic trajectory, in the vicinity of which the classical analog exhibits chaos. The few-body scar we consider is stabilized by quantum mechanics, and we analyze it along the lines of the original quantum scarring mechanism [Phys. Rev. Lett. 53, 1515 (1984)]. In particular, we identify towers of scarred quantum states which we fully explain in terms of the unstable classical trajectory underlying the scar. Our proposal is within experimental reach owing to very recent advances in Rydberg atom trapping.

Locally-symplectic neural networks for learning volume-preserving dynamics

We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector field and the splitting methods based on symplectic integrators. Symplectic gradient modules of the recently proposed symplecticity-preserving neural networks SympNets are used to construct invertible locally-symplectic modules, which compositions result in volume-preserving neural networks LocSympNets. To further preserve properties of the flow of a dynamical system LocSympNets are extended to symmetric locally-symplectic neural networks SymLocSympNets, such that the inverse of SymLocSympNets is equal to the feed-forward propagation of SymLocSympNets with the negative time step, which is a general property of the flow of a dynamical system. LocSympNets and SymLocSympNets are studied numerically considering learning linear and nonlinear volume-preserving dynamics. In particular, we demonstrate learning of linear traveling wave solutions to the semi-discretized advection equation, periodic trajectories of the Euler equations of the motion of a free rigid body, and quasi-periodic solutions of the charged particle motion in an electromagnetic field. LocSympNets and SymLocSympNets can learn linear and nonlinear dynamics to a high degree of accuracy even when random noise is added to the training data. In all numerical experiments, SymLocSympNets have produced smaller errors in long-time predictions compared to the LocSympNets. When learning a single trajectory of the rigid body dynamics locally-symplectic neural networks can learn both quadratic invariants of the system with absolute relative errors below 1%. In addition, SymLocSympNets produce qualitatively good long-time predictions, when the learning of the whole system from randomly sampled data is considered. LocSympNets and SymLocSympNets can produce accurate short-time predictions of quasi-periodic solutions, which is illustrated in the example of the charged particle motion in an electromagnetic field.