Periodic Trajectories Research Articles

Convergence aspects on internal model principle and tracking control in linear dynamical systems: classification and properties

This paper talks about asymptotic convergence aspects about the internal model principle (IMP) and trajectory tracking control (TTC) in linear dynamical systems. First, IMP and asymptotic TTC are re-formulated and explicated. Second, TTC under internal models of trajectory and disturbance is classified in terms of tracking error bias into bounded; nonzero steady-state; zero steady-state; unbounded (tracking failure). Third, TTC attainability and tracking error biases are examined via asymptotic convergence analysis. In particular, bounded TTC is attainable if trajectory and disturbance are essentially bounded; zero/nonzero-steady-state TTC is attainable if trajectory and disturbance limits are well-defined so that the Laplace ultimate value theorem holds; zero-steady-state TTC is realisable if internal models exactly reflect all unstable modes of trajectory and disturbance. Fourth, when internal models are subject to mismatches and sampled-data approximates, several interesting convergence facts are revealed. More specifically, for addressing zero-steady-state TTC to periodic trajectory and disturbance, stabilisation may be insufficient; for addressing TTC under sampled-data internal models, unknown convergence features are claimed. Finally, TTC controller parametrisation via pole assignment stabilisation is scrutinised under static output and state feedback. Numerical examples are included to illustrate the main results.

Dynamical phenomena connected with stability loss of equilibria and periodic trajectories

This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.

Trajectory Optimization of Hypersonic Periodic Cruise Using an Improved PSO Algorithm

Periodic cruise has the potential to improve the fuel-saving efficiency of hypersonic cruise vehicles but is difficult to optimize. In this paper, hypersonic periodic cruise trajectory is analyzed theoretically and optimized by an improved Particle Swarm Optimization algorithm. Firstly, through theoretical analysis, it is determined that the optimal throttle curve can be parameterized as a switching function. Considering the optimization direction of algorithm, a new penalty function for the constraints of periodic cruise is proposed. Then, PSO algorithm is improved and applied in periodic cruise trajectory optimization. Numerical results demonstrate that optimized periodic cruise trajectory costs less fuel compared with steady-state cruise trajectory, and without computing gradient information, the proposed method is also robust. Finally, the fuel-saving mechanism of periodic cruise is explored by comparing with steady-state cruise, which reveals that periodic cruise trajectory has higher impulse and lift-drag ratio, but lower mechanical energy loss rate.

Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma

Abstract A wide class of nonlinear excitations and the dynamics of wave groups of finite amplitude ion-acoustic waves are investigated in multicomponent magnetized plasma system comprising warm ions, and superthermal electrons as well as positrons in presence of negatively charged impurities or dust particles. Employing the reductive perturbation technique (RPT), the Korteweg–de-Vries (KdV) equation, and extended KdV equation are derived. The presence of excess superthermal electrons as well as positrons and other plasma parameters are shown to influence the characteristics of both compressive and rarefactive solitons as well as double layers (DLs). Also, we extend our investigation by deriving the nonlinear Schrödinger equation from the extended KdV equation employing a suitable transformation to study the wave group dynamics for long waves. The analytical and numerical simulation results demonstrate that nonlinear wave predicts solitons, “table-top” solitons, DLs, bipolar structure, rogue waves, and breather structures. Moreover, implementing the concept of dynamical systems, phase portraits of nonlinear periodic, homoclinic trajectories, and supernonlinear periodic trajectories are presented through numerical simulation.

Controlling reaction dynamics in chemical model systems through external driving

The rate of a chemical reaction can often be determined by the properties of a rank-1 saddle and the associated transition state separating reactants and products. We have found evidence that such rates can be controlled and even enhanced by external driving in at least one such system. Specifically, we analyze a reactive model in two degrees of freedom that has been used earlier to describe driven chemical reactions. Therein, changes in the external driving can lead to a local maximum of the decay rate constant or even to bifurcations of periodic trajectories on the normally hyperbolic invariant manifold (NHIM) corresponding to the transition state. Inspired by these bifurcations, we show that in this case, the dynamics on the NHIM can be connected to the geometry of reactive trajectories and to reaction probabilities of Maxwell–Boltzmann distributed reactant ensembles.

Ergodic edge modes in the 4D quantum Hall effect

The gapless modes on the edge of four-dimensional (4D) quantum Hall droplets are known to be anisotropic: they only propagate in one direction, foliating the 3D boundary into independent 1D conduction channels. This foliation is extremely sensitive to the confining potential and generically yields chaotic flows. Here we study the quantum correlations and entanglement of such edge modes in 4D droplets confined by harmonic traps, whose boundary is a squashed three-sphere. Commensurable trapping frequencies lead to periodic trajectories of electronic guiding centers; the corresponding edge modes propagate independently along S^1S1 fibers, forming a bundle of 1D conformal field theories over a 2D base space. By contrast, incommensurable frequencies produce quasi-periodic, ergodic trajectories, each of which covers its invariant torus densely; the corresponding correlation function of edge modes has fractal features. This wealth of behaviors highlights the sharp differences between 4D Hall droplets and their 2D peers; it also exhibits the dependence of 4D edge modes on the choice of trap, suggesting the existence of observable bifurcations due to droplet deformations.

Computational assessment of select antiviral phytochemicals as potential SARS-Cov-2 main protease inhibitors: molecular dynamics guided ensemble docking and extended molecular dynamics.

Graphic abstractCovid-19 caused by novel coronavirus, 2019-nCoV or SARS-CoV-2 has become most severe pandemic of this century. No specific therapies are available to treat Covid-19 so far. Recently, main protease (Mpro), a potential drug target from SARS-CoV-2 has been successfully crystallised. The present study is aimed at assessment of bioactive antiviral phytochemicals as potential SARS-COV-2 Mpro inhibitors, using ensemble docking, molecular dynamics and MM-PBSA calculations. Ensemble docking studies were performed with Autodock vina program. The top 5 compounds having highest binding free energy were subjected to 100 ns molecular dynamics simulations with Gromacs. The resulting trajectories of converged period of MD were further exploited in MM-PBSA calculations to derive accurate estimates of binding free energies. The MD results were analysed with respect to RMSD, RMSF and hydrogen bond formation and occupancy parameters. The drugs remdesivir and nelfinavir were used as standard drugs for comparative studies. In the docking studies five phytochemicals, dalpanitin, amentoflavone, naringin, hinokiflavone, and rutin were found having lowest binding free energies (< − 10 kcal mol−1) which is lower than standard drugs. MD studies suggested that the complexes of these five phytochemicals with Mpro stabilize with well accepted RMSD. Amongst these phytochemicals, hinokiflavove, amentoflavone and naringin were found having better binding affinity with ΔGbinging than the standard drug remdesivir. Investigations and validation of these inhibitors against SARS-CoV-2 would be helpful in bring these molecules at the clinical settings. Supplementary InformationThe online version contains supplementary material available at 10.1007/s40203-021-00107-9.

A Two-Level Optimization Method for Hypersonic Periodic Cruise Trajectory

Periodic cruise has the potential to improve the fuel-saving efficiency of hypersonic cruise vehicle but is difficult to optimize. In this paper, a two-level optimization method for the trajectory of periodic cruise is proposed. Due to that the periodic cruise trajectory can be divided into an acceleration phase where engine works and a glide phase where engine is off, the two-level optimization method is proposed to optimize the trajectory in each phase by the corresponding level. In the first level, Downhill Simplex Method (DSM) is employed to find an optimal angle of attack in the acceleration phase. Subsequently, the optimal trajectory in glide phase is obtained by the Pseudo-Spectral Method (PSM) in the second optimization level. Numerical results demonstrate the effectiveness of the proposed method. Finally, through comparing with steady-state cruise, it is concluded that periodic cruise makes full use of the change of atmospheric density and lift-drag ratio; thus, fuel saving is achieved.

Periodic trajectories for an age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion

This paper studies the periodic trajectories of a novel age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion. To begin with, the system is turned into an abstract non-densely defined Cauchy problem, and a time-lag effect in their interaction is investigated. Next, we acquire that this system appears a periodic orbit near the positive steady state by employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, which is also the main result of this article. Finally, in order to illustrate our theoretical analysis more vividly, we make some numerical simulations and give some discussions.

Modular symbols for Teichmüller curves

Abstract This paper introduces a space of nonabelian modular symbols 𝒮 ⁢ ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ω ω + 1 {\omega^{\omega}+1} .

New bracket and motion control system for distributed photovoltaic power stations

In view of the existing solar panel blackout, affecting the ecological environment, unreasonable spatial distribution, low power generation efficiency, high failure rate, difficult to operate and other issues, design a mechanical uniform solar power bracket: weather conditions, temperature, light strength and other multi-factor evaluation of the way to monitor the state of photovoltaic panels to adjust. According to the latitude and longitude and terrain of photovoltaic plate installation, the periodic movement trajectory is automatically planned, the operation is monitored centrally, and the failure point is located quickly and the operational efficiency is improved by comparing with big data speculation.

Multistability and dynamical properties of quantum ion-acoustic flow

Multistability and dynamical properties of ion-acoustic flow are studied in a quantum plasma containing positive beam ions, positive ions and electrons. A four dimensional conservative dynamical system has been proposed for the considered plasma system and is analyzed by considering effects of Mach number and quantum diffraction parameter. Coexistences of multiple chaotic trajectories, chaotic with quasiperiodic and multiperiodic trajectories and chaotic with quasi-periodic and periodic trajectories for ion-acoustic waves are established. The results are suitable for application in comprehending the beam-plasma interaction and studying dynamics of coexisting features in extreme astrophysical plasmas, such as, neutron stars.

Dicke Model Semiclassical Dynamics in Superradiant Dipolar Phase in the “Bound Luminosity” State

Analytic solution is presented of the nonlinear semiclassical dynamics of superradiant photonic condensate that arises in the Dicke model of two-level atoms dipolar coupled to the electromagnetic field in the microwave cavity. In adiabatic limit with respect to photon degree of freedom the system is approximately integrable and its evolution is expressed via Jacobi elliptic functions of real time. Periodic trajectories of the semiclassical coordinate of photonic condensate either localise around two degenerate minima of the condensate ground state energy or traverse between them over the saddle point. An exact mapping of the semiclassical dynamics of photonic condensate on the motion of unstable Lagrange 'sleeping top' is found. Analytic expression is presented for the frequency dependence of transmission coefficient along a transmission line inductively coupled to the resonant cavity with superradiant condensate. Sharp transmission drops reflect Fourier spectrum of the semiclassical motion of photonic condensate and of 'sleeping top' nodding.

Pointwise rotation sets of dispersing billiards

In this paper, we obtain sufficient conditions for a dispersing billiards so that the closure of the pointwise rotation set is convex, and the rotation vectors of periodic trajectories are dense in it. We also provide an example showing what the exact pointwise rotation set is for a class of dispersing billiards.

Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Nervous Activity of the Brain in Five Dimensions

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

Combinatorics of periodic ellipsoidal billiards

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.

Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive.

Analysis of P-time event graphs in (max,+) and (min,+) semirings

In this paper, we investigate the behaviour of P-time event graphs, a class of time Petri nets with non-deterministic timing of places. Our approach is based on combined linear descriptions in both (max,+) and (min,+) semirings, where lower bounds on the state vector are (max,+)-linear and upper bounds are (min,+)-linear. We present necessary and sufficient conditions for the existence of extremal (fastest and slowest) periodic trajectories that are derived from the new description. The results are illustrated by a realistic example of an electroplating process.

Adaptive Feedforward Neural Network Control With an Optimized Hidden Node Distribution

Composite adaptive radial basis function neural network (RBFNN) control with a lattice distribution of hidden nodes has three inherent demerits: 1) the approximation domain of adaptive RBFNNs is difficult to be determined a priori; 2) only a partial persistence of excitation (PE) condition can be guaranteed; and 3) in general, the required number of hidden nodes of RBFNNs is enormous. This paper proposes an adaptive feedforward RBFNN controller with an optimized distribution of hidden nodes to suitably address the above demerits. The distribution of the hidden nodes calculated by a K-means algorithm is optimally distributed along the desired state trajectory. The adaptive RBFNN satisfies the PE condition for the periodic reference trajectory. The weights of all hidden nodes will converge to the optimal values. This proposed method considerably reduces the number of hidden nodes, while achieving a better approximation ability. The proposed control scheme shares a similar rationality to that of the classical PID control in two special cases, which can thus be seen as an enhanced PID scheme with a better approximation ability. For the controller implemented by digital devices,the proposed method, for a manipulator with unknown dynamics, potentially achieves better control performance than model-based schemes with accurate dynamics.Simulation results demonstrate the effectiveness of the proposed scheme. This result provides a deeper insight into the coordination of the adaptive neural network control and the deterministic learning theory.