Stability Of Periodic Motions Research Articles

The dynamics of ensemble of neuron-like elements with excitatory couplings

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three parameters that define: a) the beginning of activation of an element α, b) the duration of the activation δ and c) the strength of the coupling g. We observed a rich diversity of different types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed the canard in-phase solutions and the chaotic anti-phase activity, which corresponds to a strange attractor that appears via the cascade of period doubling bifurcations of limit cycles.In addition, we investigated an interesting phenomenon when two different chaotic attractive regimes corresponding for two different types of chaotic anti-phase activity merge in a single strange attractor. As a result, a new type of chaotic anti-phase regime appears by explosion from the collision of these two strange attractors.We also provided the detailed study of bifurcations which lead to the transitions between all these regimes. We detected on the (α, δ) parameter plane regions that correspond to the above-mentioned regimes. We also showed numerically the existence of bistability regions where various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specified regions corresponding to coexisting various types of sequential activity.

An efficient method for simulating the dynamic behavior of periodic structures with piecewise linearity

An efficient method based on the parametric variational principle (PVP) is proposed for simulating the dynamic behavior of periodic structures with large number of piecewise linearity. The formulation for the gap-activated springs is proposed based on PVP, which can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method, an efficient numerical time-integration method is developed to obtain the dynamic responses of the system. For this method, the matrix exponential of only one unit cell of the system is computed, which greatly improves the computational efficiency. Dynamic responses of a 3 degrees of freedom (DOF) piecewise linear system under harmonic excitations are given to demonstrate the validation of the proposed method. The piecewise linear dynamic system can exhibit very complex vibrational behavior, such as stable periodic motion, multi-periodic motion, quasi-periodic motion and chaotic motion, which can be successfully predicted by using bifurcation theory. Moreover, it is demonstrated that the proposed method can be used to efficiently determine dynamic responses of a periodic piecewise linear system with large number of DOFs and large number of gap-activated springs under harmonic excitations.

Sequent period-(2m − 1) motions to chaos in the van der Pol oscillator

In this paper, independent, symmetric, periodic motions in a van der Pol oscillator are predicted through a semi-analytical method. This semi-analytic method is based on the discretization of the corresponding continuous nonlinear system for an implicit mapping. Through the implicit mapping structures, stable and unstable periodic motions are obtained analytically. A sequence of periodic motions to chaos via 1(S) ◁ 3(S) ◁ ··· ◁ (2m − 1)(S) ◁ ··· is discovered. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The frequency–amplitude characteristics of periodic motions are discussed. Numerical simulations of the periodic motions are carried out for comparison of numerical and analytical results. Such a periodic motion sequence is for a better understanding of dynamics of the van der Pol oscillator.

Analyzing Bifurcation, Stability and Chaos for a Passive Walking Biped Model with a Sole Foot

Human walking is an action with low energy consumption. Passive walking models (PWMs) can present this intrinsic characteristic. Simplicity in the biped helps to decrease the energy loss of the system. On the other hand, sufficient parts should be considered to increase the similarity of the model’s behavior to the original action. In this paper, the dynamic model for passive walking biped with unidirectional fixed flat soles of the feet is presented, which consists of two inverted pendulums with L-shaped bodies. This model can capture the effects of sole foot in walking. By adding the sole foot, the number of phases of a gait increases to two. The nonlinear dynamic models for each phase and the transition rules are determined, and the stable and unstable periodic motions are calculated. The stability situations are obtained for different conditions of walking. Finally, the bifurcation diagrams are presented for studying the effects of the sole foot. Poincaré section, Lyapunov exponents, and bifurcation diagrams are used to analyze stability and chaotic behavior. Simulation results indicate that the sole foot has such a significant impression on the dynamic behavior of the system that it should be considered in the simple PWMs.

A nonlinear model of bicycle shimmy

ABSTRACTThis paper presents a nonlinear model accurately describing, both qualitatively and quantitatively, the onset and dynamics of bicycle shimmy. Methods of nonlinear dynamics, such as numerical continuation and bifurcation analysis, show that the model exhibits two stable periodic motions found experimentally in on-road tests: the weave and wobble (or shimmy) mode. The modelling results are compared with experimental data collected by riding a racing bicycle downhill at high speeds with hands on the handlebar. The model predicts with surprising accuracy the amplitudes and frequencies of the oscillations, the longitudinal velocity at which they occur, as well as the substantial independence of wobble frequency and amplitude from the forward speed. The lateral acceleration of the upper tube of the frame near the steering axis reaches 5–10 g, both in the model and in the data. The analysis shows that wobble onset and amplitude is particularly sensitive to changes in the torsional stiffness of the frame and strongly depends on tyre lateral force and aligning torque at the wheel–road contact point. It also allows to quantify the additional viscous rotary damping that should be added to the steering assembly to prevent wobble.

Three-dimensional vibration of cantilevered fluid-conveying micropipes—Types of periodic motions and small-scale effect

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of fluid-conveying cantilevered micropipes. Particular attention is given on the derivation and analysis of the reduced equations, and the small-scale effect on the periodic motions. Based on the modified couple stress theory (MCST), the governing equations are derived by using Hamilton’s principle. The material length scale parameter and large-deflection-induced geometric nonlinearities given by the Lagrangian strain tensor are incorporated into the governing equations. Utilizing the center manifold theory, normal form method and O(2) symmetry, the original governing equations can be rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system. Then two possible types of periodic motions, i.e. planar periodic and spatial periodic motions, together with their stabilities are investigated by means of averaging methods and numerical simulations. Results show that the larger the dimensionless material length scale parameter is, the wider the region of mass ratio for stable planar periodic motion is. Particularly, the presence of small length scale parameter makes micropipes be more likely to oscillate in a plane. It is also shown that for mass ratio corresponding to the hysteresis of the curves of critical flow velocity versus mass ratio, the stabilities for bifurcating periodic motions at lower, moderate and higher critical flow velocities may be different.

Quasi-periodic dynamics in system with multilevel pulse modulated control

In this paper, the authors describe the transitions from the regular periodic mode to quasiperiodicity that can be observed in a multilevel pulse-width modulated control system for a high-power heating unit. The behavior of such system can be described by a set of two coupled non-autonomous differential equations with discontinuous right-hand sides. The authors reduce the investigation of this system to the studying of a two-dimensional piecewise-smooth map. The authors demonstrate how a closed invariant curve associated with quasiperiodic dynamics can arise from a stable periodic motion through a border-collision bifurcation. The paper also considers a variety of interesting nonlinear phenomena, including phase-locking modes, the coexistence of several stable closed invariant curves, embedded one into the other and with their basins of attraction separated by intervening repelling closed curves.

Significantly stable mode of the ultracold atomic wave packet in amplitude-modulated parabolic optical lattices

We show that a conspicuous wave packet of ultracold noninteracting Bosonic atoms emerges in a 1-dimensional parabolic optical lattice as in the setup of the Aarhus experiment [P. L. Pedersen ${\it et}$ ${\it al.}$, Phys. Rev. A ${\bf 88}$, 023620 (2013)], given the lattice height is harmonically modulated with a particular amplitude at a resonant frequency. We show that this wave packet, coined "${\it 4bandPWP}$" here, executes stable time-wise periodic motion for infinitely long time. We apply the Floquet theory to analyze the parameter dependence of ${\it 4bandPWP}$ in detail. Our analysis shows that it consists mainly of two principal Floquet eigenstates of the periodically driven Hamiltonian. The informative Husimi representation yields temporal slices of the phase space of ${\it 4bandPWP}$, visually identifying moments where the inter-band transitions take place. The provided data should aid the experiment in locating ${\it 4bandPWP}$.

Simplest mechanical model of stable hopping with inelastic ground-foot impact

Abstract The dynamic analysis of legged locomotion is challenging because of many reasons, such as the possibly high degrees of freedom of the model, the alternating topology in certain phases of walking or running, the presence of under-actuation and over-actuation, the geometric nonlinearities and ground-foot impact induced non-smoothness. Control issues makes models even more complex, although having models which includes the control strategy of the brain is very helpful in biomechanics. Such models helps to understand how the brain keeps the body in balance, how the energy level is maintained and how the motion patterns are generated. Before developing a fully extended, very complex model of walking or running, we focus on the dynamic analysis of a piecewise smooth model of hopping, which possesses some fundamental characteristics of locomotion systems: 1) different topology in ground and flight-phase; 2) energy absorption due to partially/fully inelastic ground-foot collision; 3) active control strategy for maintaining a certain energy level; 4) different control strategies in ground and flight-phases. We prove that our two degrees of freedom model provides stable periodic motion, i.e. vertical hopping in wide range of parameters. We present the application of stability analysis methods that can be adapted for more complex models of legged locomotion.

求解周期性分段线性系统动态响应的高效数值方法

An efficient method based on the parametric variational principle (PVP) was proposed for computing the dynamic responses of periodic piecewise linear systems with multiple gap-activated springs. Through description of gap-activated springs with the PVP, the complex nonlinear dynamic problem was transformed to a standard linear complementary problem. This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method (PIM), an efficient numerical time-integration method was developed to obtain the dynamic responses of the system. This method indicates that there are a large number of identical elements and zero elements in the matrix exponents of a periodic structure, and saves computation load and computer storage by avoiding repeated calculation and storage of these elements. Numerical results validate the proposed method. The dynamic behaviors of a 5-DOF piecewise linear system under harmonic excitations were analyzed, including the stable periodic motion, the quasi-periodic motion and the chaotic motion. In comparison with the Runge-Kutta method, the proposed method has satisfactory correctness and efficiency.

On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

The problem of orbital stability of a periodic motion of an autonomous two-degreeof- freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion. Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.

A pendulum on a rotating and vibrating foundation

The motion of a pendulum is investigated in a uniform gravity field. The foundation of the pendulum is assumed to rotate around the vertical simultaneously accomplishing harmonic vibrations along it with a high frequency and small amplitude. With use of the classical perturbation theory and modern methods of analysis of nonlinear dynamical sets, the problem on the existence and stability of periodic motions of a pendulum with the period equal to that of the vibrations of the foundation is solved.

Traveling-wave solutions in continuous chains of unidirectionally coupled oscillators

Proposed is a mathematical model of a continuous annular chain of unidirectionally coupled generators given by certain nonlinear advection-type hyperbolic boundary value problem. Such problems are constructed by a limit transition from annular chains of unidirectionally coupled ordinary differential equations with an unbounded increase in the number of links. It is shown that any preassigned finite number of stable periodic motions of the traveling-wave type can coexist in the model.

The dynamics of eccentric vibration mechanism (Part 1)

This paper presents a mathematical model of a new multi-striker eccentric shock-vibration mechanism (ESVM) with a crank-sliding bar vibration exciter (CSVE) and an arbitrary number of pistons. Analytical solutions for the parameters of the model are obtained to determine the regions of existence of stable periodic motions. Under the assumption of an absolutely inelastic collision of the piston, we derive equations that single out a bifurcational unattainable boundary in the parameter space, which has a countable number of arbitrarily complex stable periodic motions in its neighborhood. We present results of numerical simulations, which illustrate the existence of periodic and stochastic motions. The methods proposed in this paper for investigating the dynamical characteristics of the new crank-type conrod mechanisms allow practitioners to indicate regions in the parameter space, which allow tuning these mechanisms into the most efficient periodic mode of operation, and to effectively analyze the main changes in their operational regimes when the system parameters are changed.

On the theory of multi-pulse vibro-impact mechanisms

This paper presents a mathematical model of a new multi-striker eccentric shock-vibration mechanism with a crank-sliding bar vibration exciter and an arbitrary number of pistons. Analytical solutions for the parameters of the model are obtained to determine the regions of existence of stable periodic motions. Under the assumption of an absolutely inelastic collision of the piston, we derive equations that single out a bifurcational unattainable boundary in the parameter space, which has a countable number of arbitrarily complex stable periodic motions in its neighbourhood. We present results of numerical simulations, which illustrate the existence of periodic and stochastic motions. The methods proposed in this paper for investigating the dynamical characteristics of the new crank-type conrod mechanisms allow practitioners to indicate regions in the parameter space, which allow tuning these mechanisms into the most efficient periodic mode of operation, and to effectively analyze the main changes in their operational regimes when the system parameters are changed.

Global Behavior of a Vibro-Impact System With Multiple Nonsmooth Mechanical Factors

A nonlinear mechanical model of a vibro-impact system influenced by double nonsmooth mechanical factors that combine elastic and rigid impact is described. The theoretical solutions to judge the periodic motion stability of the system are presented, and three different “gazing” motions and the corresponding conditions are described. The transition and coupling of periodic motions by the nonsmooth mechanical factors are demonstrated. The formation mechanism of sticking motion, chattering motion, and the periodic cavity by the influence of gazing bifurcation are analyzed. The coexistence of periodic motions and the extreme sensitivity of the initial value within the high frequency region are studied. The distribution of the attractor and the corresponding attracting domain corresponding to different periodic motions are also studied.

Turning motion analysis for the finite-width rimless wheel as a passive walker

The focus of studies in the field of passive walking has often been on straight walking, while less attention has been paid to the field of turning motions. The purpose of this paper is to investigate the passive motions with a focus on turning of a finite width rimless wheel as the simplest 3D model of the passive biped walkers. This study is divided into two main sections: Firstly, the limited passive turning is considered; secondly, the infinite passive turning is analyzed. A Poincare map corresponding to a step is one of the common methods used for the determination of the periodic motions (limit cycles) and their specifications. It is emphasized that the Poincare map has only one fixed point, indicating only one stable periodic motion that it is parallel to the steepest descent slope surface. In addition, the effect of variation among some parameters on rotational behavior and its convergence are investigated. The results of simulation are also verified via ADAMS software for 50 steps. In second part, we apply a novel surface profile namely “helical slope” for producing the continuous passive turning. More precisely, the wheel can turn stably on a helical circular path. Similarly, the stability of corresponding limit cycle for different values of revolving velocities and slopes of the helical surface are studied. The results indicate that the turning motion have the stronger stability than the straight walking.

Kinematics and dynamics of a particle on a non-simple harmonic vibrating screen

The motion of a particle on a screen is directly affected by the motion of the screen if airflow and intergranular friction are ignored. To study this effect, a mathematical model was established to analyze the motion of a planar reciprocating vibrating screen, and a matrix method was employed to derive its equation of motion. The motion of the screen was simulated numerically and analyzed using MATLAB. The results show that the screen undergoes non-simple harmonic motion and the law of motion of each point in the screen is different. The tilt angle of the screen during screening is not constant but varies according to a specific periodic function. The results of numerical simulations were verified through experiments. A high-speed camera was used to track the motion of three points in the longitudinal direction of the screen. The balance equation for forces acting on a single particle on the screen was derived based on the non-simple harmonic motion of the screen. These forces were simulated using MATLAB. Different types of particle motion like slipping forward, moving backward, and being tossed to different parts of the screen were analyzed. A vibro-impact motion model for a particle on the non-simple harmonic vibrating screen was established based on the nonlinear law of motion of the particle. The stability of fixed points of the map is discussed. Regimes of different particle behaviors such as stable periodic motion, period-doubling bifurcation motion, Hopf bifurcation motion, and chaotic motion were obtained. With the actual law of motion of the screen and the behavior of a particle on the screen, a theoretical basis for design optimization of the screen is provided.

Dynamic behavior analysis and time delay feedback control of gear pair system with backlash non-smooth characteristic

The present work investigates the non-smooth vibration characteristic and time delay feedback control of a gear pair system involving backlash and time-varying mesh stiffness. Firstly, a gear pair model with backlash non-smooth characteristic is established. Then in combination with the discontinuity mapping method, Floquet theory is presented to determine the stability and bifurcation of periodic response, and the period doubling bifurcation has been accurately predicted. Moreover, the maximal Lyapunov exponent is obtained to determine the chaos state in gear pair system which is conform to the bifurcation diagram and Poincare section. Finally, a time delay feedback is introduced to control the dynamic behaviors of the system, and numerical simulation results show that the system can be effectively controlled from chaotic motion into stable periodic motion by increasing the delay feedback gain or delay time.

Stability, Bifurcation and Chaos of a Traveling Viscoelastic Beam Tuned to 3:1 Internal Resonance and Subjected to Parametric Excitation

Analytical-numerical approach has been adopted to investigate the stability, bifurcation and dynamic behavior (including chaotic behavior) of axially moving viscoelastic beam subjected to parametric excitation resulting from speed variation in the presence of 3:1 internal resonance between the first two modes of vibration. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent coefficients. The direct method of multiple scales is employed to analyze the joint influence of the combination of parametric resonance and internal resonance with the focus on steady state responses. Equilibrium solutions along with their stability and bifurcations are determined by continuation algorithm while direct time integration is used for dynamic behavior for various system parameters. The results are compared with the previous works depicting the principal parametric resonances of the first and second modes. Significant comparative analysis results are reported in the stability and bifurcation of frequency response analysis. The dynamic responses show a range of behavior viz. stable periodic, mixed mode, quasiperiodic and unstable chaotic motion of the system. Numerical results illustrate various typical and interesting nonlinear phenomena of the traveling system which are not found in the existent literature.