Nonperiodic Trajectories Research Articles

Non-periodic not everywhere dense trajectories in triangular billiards

Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.

Newborn vaccination game model with potentially periodic behavior

We explore the complex dynamics of epidemics, considering the significant effects of disease transmission, newborn individuals, and mortality rate. Employing an evolutionary game theory approach, we model the vaccination aspect for newborns, incorporating potential periodic behavior. The goal is to create an engaging simulation that underscores the criticality of timely infant vaccinations. By introducing periodic elements, the game adds complexity, fostering player participation and awareness of vaccination schedules. In contrast to traditional intervention game models focused on adult vaccine behavior, our approach recognizes the unique challenges posed by newborn vaccination. We address the limitations of existing models by developing a generalized compartmental model, accommodating infection duration distribution, and accurately capturing both periodic and non-periodic disease trajectories. The results show that higher efficacy in a newborn vaccine program can potentially eradicate the disease, but higher effectiveness and cost may deter vaccination. Increased motivation cost boosts vaccination rates without impacting epidemic size, whereas lower costs stabilize vaccination rates. Lower vaccine costs call for increased government investment to counter hesitancy. However, reduced efficacy and lower motivation lead to higher Vaccine Fear Scores, discouraging participation. Our work also offers a unique perspective on the dilemma of newborn vaccination, providing solutions with closed-form expressions that may exhibit periodic or non-periodic behavior. This approach facilitates disease modelers in exploring the potential periodicity of newborn vaccination programs, aiding efforts to prevent recurrent outbreaks.

Periodic Solutions in Kolmogorov-Type Predator–Prey Systems

AbstractWe consider a class of prey-predator models, i.e., a Kolmogorov-type system. We are interested in their dynamics when a certain parameter (that can be viewed as the death rate of the predator) changes from zero value to positive. By utilizing alternative but simple techniques, including a sub- and super-solutions method, we establish the existence of periodic solutions when some conditions are satisfied. We also prove that the solutions are bounded by a non-periodic trajectory when the parameter vanishes. We give an example to illustrate our results.

A generalized ODE susceptible-infectious-susceptible compartmental model with potentially periodic behavior

Differential equation compartmental models are crucial tools for forecasting and analyzing disease trajectories. Among these models, those dealing with only susceptible and infectious individuals are particularly useful as they offer closed-form expressions for solutions, namely the logistic equation. However, the logistic equation has limited ability to describe disease trajectories since its solutions must converge monotonically to either the disease-free or endemic equilibrium, depending on the parameters. Unfortunately, many diseases exhibit periodic cycles, and thus, do not converge to equilibria. To address this limitation, we developed a generalized susceptible-infectious-susceptible compartmental model capable of accurately incorporating the duration of infection distribution and describing both periodic and non-periodic disease trajectories. We characterized how our model's parameters influence its behavior and applied the model to predict gonorrhea incidence in the US, using Akaike Information Criteria to inform on its merit relative to the traditional SIS model. The significance of our work lies in providing a novel susceptible-infected-susceptible model whose solutions can have closed-form expressions that may be periodic or non-periodic depending on the parameterization. Our work thus provides disease modelers with a straightforward way to investigate the potential periodic behavior of many diseases and thereby may aid ongoing efforts to prevent recurrent outbreaks.

Resonance of ellipsoidal billiard trajectories and extremal rational functions

We study resonant billiard trajectories within quadrics in the d-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of d intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to s-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.

High performance raster scanning of atomic force microscopy using Model-free Repetitive Control

The image quality of an atomic force microscope highly depends on the tracking performance of the lateral X–Y axis scanner. To reduce the requirement for accurate system models, this article describes a method based on Model-free Repetitive Control (MFRC) for high performance control of fast triangular trajectories in the X-axis while simultaneously achieving coupling compensation from the X-axis to Y-axis. The design and stability analysis of the MFRC scheme are presented in detail. The tracking results are experimentally evaluated with a range of different load conditions, showing the efficacy of the method with large variations in plant dynamics. To address the coupling from the X-axis to the Y-axis while tracking the non-periodic staircase trajectories, a pre-learning step is used to generate the compensation signals, which are combined with the baseline Proportional–Integral (PI) control in a feedforward manner in real-time implementations. This approach is also applied to address the problem of longer convergence if needed. Experimental tracking control and coupling compensation are demonstrated on a commercially available piezo-actuated scanner. The proposed method reduces the root-mean-square coupled tracking errors in Y-axis from 191.4 nm in open-loop control or 194.6 nm with PI control, to 2.8 nm with PI+MFRC control at 100 Hz tracking frequency, which demonstrates the significant improvement achieved by the proposed method.

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential-algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

Uniqueness of billiard coding in polygons

We consider polygonal billiards and we show that every nonperiodic billiard trajectory hits a unique sequence of sides if all the holes of the polygonal table have non-zero minimal diameters, generalizing a classical theorem of Galperin, Krüger and Troubetzkoy. Our approach uses symbolic dynamics and elementary geometry. We review some classical constructions in polygonal billiards and we introduce, as one of our main tools, a method to code pairs of parallel billiard trajectories in non-simply connected polygons. We also discuss some useful properties of ‘generalized trajectories’, which can be uniquely constructed from the limits of converging sequences of billiard codings.

The order of coexistence of homoclinic trajectories for the maps of an interval

A nonperiodic trajectory of a discrete dynamical system is called n-homoclinic if its α- and ω-limit sets coincide and form the same cycle of period n. We prove the statement that the ordering 1 ⊳ 3 ⊳ 5 ⊳ 7 ⊳ ... ⊳ 2 ⋅ 1 ⊳ 2 ⋅ 3 ⊳ 2 ⋅ 5 ⊳ ... ⊳ 22 ⋅ 1 ⊳ 22 ⋅ 3 ⊳ 22 ⋅ 5 ⊳ ... determines the coexistence of homoclinic trajectories of one-dimensional systems in a sense that if a onedimensional dynamical system possesses an n-homoclinic trajectory, then it also has an m-homoclinic trajectory for each m such that n ⊳ m. It is also proved that every one-dimensional dynamical system with a cycle of period n ≠ 2i also possesses an n-homoclinic trajectory.

Non-periodic phase-space trajectories of roughness-driven secondary flows in high- boundary layers and channels

Turbulent flows respond to bounding walls with a predominant spanwise heterogeneity – that is, a heterogeneity parallel to the prevailing transport direction – with formation of Reynolds-averaged turbulent secondary flows. Prior experimental and numerical work has determined that these secondary rolls occur in a variety of arrangements, contingent only upon the existence of a spanwise heterogeneity (i.e. from complex, multiscale roughness with a predominant spanwise heterogeneity, to canonical step changes, to different roughness elements). These secondary rolls are known to be a manifestation of Prandtl’s secondary flow of the second kind: driven and sustained by the existence of spatial heterogeneities in the Reynolds (turbulent) stresses, all of which vanish in the absence of spanwise heterogeneity. Herein, we show results from a suite of large-eddy simulations and complementary experimental measurements of flow over spanwise-heterogeneous surfaces. Although the resultant secondary cell location is clearly correlated with the surface characteristics, which ultimately dictates the Reynolds-averaged flow patterns, we show the potential for instantaneous sign reversals in the rotational sense of the secondary cells. This is accomplished with probability density functions and conditional sampling. In order to address this further, a base flow representing the streamwise rolls is introduced. The base flow intensity – based on a leading-order Galerkin projection – is allowed to vary in time through the introduction of time-dependent parameters. Upon substitution of the base flow into the streamwise momentum and streamwise vorticity transport equations, and via use of a vortex forcing model, we are able to assess the phase-space evolution (orbit) of the resulting system of ordinary differential equations. The system resembles the Lorenz system, but the forcing conditions differ intrinsically. Nevertheless, the system reveals that chaotic, non-periodic trajectories are possible for sufficient inertial conditions. Poincaré projection is used to assess the conditions needed for chaos, and to estimate the fractal dimension of the attractor. Its simplicity notwithstanding, the propensity for chaotic, non-periodic trajectories in the base flow model suggests similar dynamics is responsible for the large-scale reversals observed in the numerical and experimental datasets.

Constrained Model Predictive Control for a Hexapod Robot Walking on Irregular Terrain

It is usually a common task that walking on irregular terrain for a hexapod robot. This paper treats the problem of control of a hexapod robot that has been developed in a collaborative project of the Departments of Mechanical Engineering and Automation at Shanghai Jiao Tong University. We derive the hexapod robot’s kinematics and dynamics models, which are non-periodic, nonlinear, hybrid and open-loop unstable. In order to solve this challenge, we compute a transverse linearization about the desired motion which is a linear impulsive system. The real environment, where the robot performs tasks, are usually unstructured and dynamic, it is always difficult to know the size of obstacles the robot strikes on. In order to reduce its reliance on terrain information for locomotion control to the minimum, taking account of physical constraints, we propose a constrained model predictive controller for stabilization of non-periodic trajectories of the hexapod robot walking over irregular terrain. We prove the stability of the proposed controller theoretically. In the end we perform several simulations and experiments to verify the effectiveness of the control algorithm we proposed.

Tracking microparticle motions in three-dimensional flow around a microcubic array fabricated on the wall surface

This paper describes the application of a microscopic defocusing image system to track 3D particle motions in microflow around a microcubic array near a microchannel wall surface. The measurement principle and calibration method were evaluated to provide accurate 3D microparticle location. Particle trajectories were measured in two microchannels. Measured velocity profiles of Hele–Shaw flows for two Reynolds numbers agreed well with theoretical profiles. Three-dimensional particle tracking in fluid flow around a microcubic array exhibited quasi-periodic and non-periodic trajectory modes. The experimental results indicated that 3D particle motions are spatially and time dependent even when the flow rate is constant, and microparticle trajectories may deviate from steady flow streamlines.

Sensitivity Analysis of Chaotic Systems Using Unstable Periodic Orbits

A well-behaved adjoint sensitivity technique for chaotic dynamical systems is presented. The method arises from the specialisation of established variational techniques to the unstable periodic orbits of the system. On such trajectories, the adjoint problem becomes a time periodic boundary value problem. The adjoint solution remains bounded in time and does not exhibit the typical unbounded exponential growth observed using traditional methods over unstable non-periodic trajectories (Lea et al., Tellus 52 (2000)). This enables the sensitivity of period averaged quantities to be calculated exactly, regardless of the orbit period, because the stability of the tangent dynamics is decoupled effectively from the sensitivity calculations. We demonstrate the method on two prototypical systems, the Lorenz equations at standard parameters and the Kuramoto-Sivashinky equation, a one-dimensional partial differential equation with chaotic behaviour. We report a statistical analysis of the sensitivity of these two systems based on databases of unstable periodic orbits of size 10^5 and 4x10^4, respectively. The empirical observation is that most orbits predict approximately the same sensitivity. The effects of symmetries, bifurcations and intermittency are discussed and future work is outlined in the conclusions.

A comprehensive inversion approach for feedforward compensation of piezoactuator system at high frequency

Motion control of the piezoactuator system over broadband frequencies is limited due to its inherent hysteresis and system dynamics. One of the suggested ways is to use feedforward controller to linearize the input–output relationship of the piezoactuator system. Although there have been many feedforward approaches, it is still a challenge to develop feedforward controller for the piezoactuator system at high frequency. Hence, this paper presents a comprehensive inversion approach in consideration of the coupling of hysteresis and dynamics. In this work, the influence of dynamics compensation on the input–output relationship of the piezoactuator system is investigated first. With system dynamics compensation, the input–output relationship of the piezoactuator system will be further represented as rate-dependent nonlinearity due to the inevitable dynamics compensation error, especially at high frequency. Base on this result, the feedforward controller composed by a cascade of linear dynamics inversion and rate-dependent nonlinearity inversion is developed. Then, the system identification of the comprehensive inversion approach is proposed. Finally, experimental results show that the proposed approach can improve the performance on tracking of both periodic and non-periodic trajectories at medium and high frequency compared with the conventional feedforward approaches.

A sustainable running criterion for biped balance control

This paper aims at exploring a novel criterion for sustainable running of biped robots subject to external perturbations. A new notion of running-feasible state is introduced to ensure that a biped can sustain a running motion starting from this state by an allowable control. Then a running-feasible set is estimated by using the orbit energy and the symmetric property of the inverted pendulum model. Based on the developed sustainable running criterion, a non-periodic reference trajectory is generated within a running-feasible set for disturbance rejection in real time and a control law is then designed to track the reference trajectory. In simulation, a five-link, four-actuator biped transits its speed between 2.8 m/s and 4.8 m/s in responding to external disturbances of pushing forward and backward. The simulation results confirm the effectiveness of balance recovery in a running biped, resembling the reaction pattern of a human being.

Flexibility of bacterial flagella in external shear results in complex swimming trajectories.

Many bacteria use rotating helical flagella in swimming motility. In the search for food or migration towards a new habitat, bacteria occasionally unbundle their flagellar filaments and tumble, leading to an abrupt change in direction. Flexible flagella can also be easily deformed by external shear flow, leading to complex bacterial trajectories. Here, we examine the effects of flagella flexibility on the navigation of bacteria in two fundamental shear flows: planar shear and Poiseuille flow realized in long channels. On the basis of slender body elastodynamics and numerical analysis, we discovered a variety of non-trivial effects stemming from the interplay of self-propulsion, elasticity and shear-induced flagellar bending. We show that in planar shear flow the bacteria execute periodic motion, whereas in Poiseuille flow, they migrate towards the centre of the channel or converge towards a limit cycle. We also find that even a small amount of random reorientation can induce a strong response of bacteria, leading to overall non-periodic trajectories. Our findings exemplify the sensitive role of flagellar flexibility and shed new light on the navigation of bacteria in complex shear flows.

Effects of complex time on periodic and nonperiodic classical trajectories of one-dimensional Hamiltonian systems

In recent years, much research has been carried out on extending both quantum mechanics and classical mechanics into the complex domain by making parameters of real hermitian Hamiltonians or total energy of the system complex. In this paper we investigate the effects of complex time on periodic and nonperiodic trajectories of both hermitian and nonhermitian one-dimensional classical Hamiltonian systems. Most of the periodic classical trajectories of real hermitian systems turn into nonperiodic and open when the energy or the parameters of the potential become complex. We show that when time is taken as a complex quantity with a specific fixed phase angle or as a specific complex function, nonperiodic trajectories become periodic and closed. Furthermore, we show that real hermitian systems, such as H = p2/2m + x4 + bx3 + cx2 + dx (b, c, and d are real quantities) possess classical periodic trajectories for real energies even when time is complex (i.e., t = treiτ). It was found that there is a discrete set of τ values for which the trajectories of the preceding system are closed and periodic and periods associated with them form a countably infinite set.

Identification of regular and chaotic isothermal trajectories of a shape memory oscillator using the 0–1 test

Shape memory oscillators are thermomechanical hysteretic systems that, in a wide range of model parameters, can exhibit complex non-periodic non-linear dynamic responses under the excitation of a periodic force. In this study, the statistical 0–1 test based on the asymptotic properties of a Brownian motion chain is applied to periodic and non-periodic isothermal trajectories to examine the type of motion. The analysis is based on the computation of the control parameter K that approaches asymptotically 0 or 1 for regular and chaotic motions, respectively. The presented approach is independent of the integration procedure, being based on the characteristic sampling distance between the points of the analysed time series. The numerical results show that the test is able to unambiguously distinguish periodic from chaotic trajectories.

Stable dynamic walking over uneven terrain

We propose a constructive control design for stabilization of non-periodic trajectories of underactuated robots. An important example of such a system is an underactuated “dynamic walking” biped robot traversing rough or uneven terrain. The stabilization problem is inherently challenging due to the nonlinearity, open-loop instability, hybrid (impact) dynamics, and target motions which are not known in advance. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents “transversal” dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design, providing exponential orbital stability of the target trajectory of the original nonlinear system. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to a wide variety of hybrid nonlinear systems.

On the topological structure of singular attractors of nonlinear systems of differential equations

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a two-dimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and three-dimensional autonomous dissipative systems are generalized to autonomous multi- and infinite-dimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.