Existence Of Stable Limit Cycle Research Articles

Dynamics of coupled thermoacoustic modes: Noise and frequency spacing effects

Practical combustion systems consist of multiple thermoacoustic modes that are linearly stable or unstable at a wide range of frequencies. When coupled, the modal interactions are a strong function of the frequency spacing between them – having very significant implications on interpretation of linear stability or instability of modes from experimental data. Frequency spacing between the modes particularly influences the stability and existence of potential limit cycle solutions. In this paper, we extend earlier deterministic studies to include the effects of noise, studying how deterministic dynamics change with the addition of noise and the simultaneous impact of frequency spacing (i.e., closely, or widely spaced). Specifically, this paper characterizes three key effects of noise, which are (1) distribution of limit cycle amplitudes (“diffusion”), (2) shift in the average limit cycle amplitudes (“noise induced drift”), and (3) introduction of new bifurcations that are not present in the deterministic system. Key conclusions from this work are that in highly noisy practical environments, a deterministic/ low noise analysis will not be sufficient to explain even critical qualitative features of the system – such as existence of stable limit cycles or their amplitudes. Moreover, the nature of noise-induced effects on nonlinear modal interactions can be fundamentally different for degenerate, closely spaced, and widely spaced modes.

Study of internal oscillations in a dynamic system through an external signal implemented in circadian rhythms

Through the analysis carried out on a dynamic model that is represented as a system of ordinary differential equations that describes the behavior of the circadian cycles; we will show and analyze in the next document what are the conditions that allow the synchronization of the circadian clock oscillator with the external modification oscillator. The implementation of this type of techniques in anatomical problems is highlighted, which are rare in the literature. The implementations will be carried out through different simulations using numerical techniques and the way in which we will determine the coupling conditions of an internal cycle of the system versus external cycles will be detailed. In the final development of this work, we will be able to see in the model without an external modification signal the existence of stable limit cycles and discover the moment in which the synchronization of the internal oscillator and the external modification signal occurs. These types of problems are common when making biological models that are described by a physical analysis.

Transient behavior towards the stable limit cycle in the Sel’kov model of Glycolysis: A physiological disorder

A simplified model for the complex glycolytic process was historically proposed by Sel’kov. It showed the existence of stable limit cycle as an example of Poincare’-Bendixson theorem. This limit cycle is nothing but the time eliminated Lissajous plot of the concentrations of Adenosine-diphosphate (ADP) and Fructose-6-phosphate (F6P) of a normal/healthy human. Deviation from this limit cycle is equivalent to the deviation of normal physiological behavior. It is very important to know how long a human body will take to reach the glycolytic stable limit cycle, if deviated from it. However, till now the convergence time, depending upon different initial parameter values, was not studied in detail. This may have great importance in understanding the recovery time for a diseased individual deviated from normal cycle. Here the convergence time for different initial conditions has been calculated in original Sel’kov model. It is observed that convergence time, as a function of the distance from the limit cycle, gets saturated away from the cycle. This result seems to be a physiological disorder. A possible mathematical way to incorporate this in the Selkov model, has been proposed.

Autonomous relay control system for satellite formation operating in low Earth orbit

A relay control algorithm is proposed for the satellite formation consisting of a «passive» (virtual) spacecraft moving in an unperturbed circular orbit, and several “active” spacecraft maneuvering relative to it. The purpose of control is to keep each active spacecraft on its finite path relative to the virtual spacecraft defining the center of formation with as low as possible propellant consumption. To describe the motion of maneuvering spacecraft relative to the center of formation, modified Hill–Clohessy–Wiltshire equations are used accounting for the earth's flattening and aerodynamic drag. The main emphasis of this paper is on the study of the existence of stable limit cycles and determination of their domains of attraction, as well as the search for values of the relay control system parameters ensuring a minimum propellant consumption for maintaining a proper dynamic formation behavior. Key words: relay control, point transformation method, formation flying, limit cycle, phase plane.

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus‐focus” and “focus‐center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus‐center,” we use a pure geometric approach. In the case “focus‐focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two‐dimensional system describing the qualitative behavior of a two‐dimensional excitable membrane model.

Limit Cycles in Replicator-Mutator Dynamics with Game-Environment Feedback

This paper considers the coevolutionary game and environment dynamics under mutations of strategies. Individuals’ game play affects the dynamics of changing environments while the environment in turn affects the decision-making dynamics of individuals through modulating game payoffs. For some such closed-loop systems, we prove that limit cycles will never appear; however, in sharp contrast, after allowing mutations of strategies in these systems, the resulting replicator-mutator dynamics under environmental feedback may well exhibit Hopf bifurcation and limit cycles. We prove conditions for the Hopf bifurcation and thus the existence of stable limit cycles, and also illustrate these results using simulations. For the coevolutionary game and environment system, these stable limit cycles correspond to sustained oscillations of population’s decisions and richness of the environmental resource.

Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

An averaging-based Lyapunov technique to design thermal oscillators: A case study on free piston Stirling engine

This article introduces a novel averaging-based Lyapunov approach to design complex oscillators such as free-piston Stirling engines (FPSEs). First, a class of thermal oscillators namely the FPSE is introduced. Then, the existence of limit cycle in dynamic response of the oscillator, which is a required criterion for steady operation, is investigated via the proposed technique. Accordingly, a systematic way containing two conditions is presented to justify the performance of the oscillator system. As the preliminary condition, instability of the equilibrium state of the oscillator is studied through the proposed averaging-based Lyapunov theorem (necessary condition). Subsequently, as the second condition, the existence of stable limit cycle in the oscillator response is investigated through the mentioned method (sufficient condition). Next, some useful equations for effective prediction of the oscillator performance are presented using the proposed averaging-based Lyapunov method. Finally, the simulation outcomes, as well as the experimental data, clearly reveal the effectiveness of the proposed methodology in designing the free piston Stirling oscillator, which can be further extended to other oscillators with periodic response such as engines, energy harvesters, vibration/acoustic isolators, and resonant absorbers as well.

Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts

Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current $ I^{\text{ext}} $ and the derivative order $ \gamma $, on top of the other seven usual parameters $ a, b, c, d, \nu_1, \nu_2 $ and $ x_{\text{rest}}. $ Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as $ \gamma $ decreases. This inverse proportionability reveals a system likely to be $ \gamma- $controlled with $ \gamma $ varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as $ \gamma $ changes from 1 down to 0.

TCP With Virtual Queue Management Policies: Stability and Bifurcation Analysis

In this paper, we analyze a model for transport control protocol (TCP) along with a non-adaptive virtual queue (VQ) and an adaptive virtual queue (AVQ) management policy. In the class of transport protocols, we focus on compound TCP as it is the default protocol in the Windows operating system. We start by conducting a local stability analysis for the underlying fluid models. For the VQ policy, we show that small virtual buffers play an important role in ensuring stability, whereas the AVQ policy could readily lose local stability as the link capacity, the feedback delay, or the link’s damping factor gets large. With both the queue policies, the protocol parameters of compound TCP also influence stability. Furthermore, in both the models, we explicitly show that as parameters vary the loss of local stability would occur via a Hopf bifurcation. For the AVQ policy, we are also able to analytically verify if the Hopf bifurcation is super-critical, and determine the stability of the bifurcating limit cycles. Packet-level simulations, conducted over two topologies, using the network simulator (NS2) confirm the existence of stable limit cycles in the queue size.

Globally stable control of a dynamic bipedal walker using adaptive frequency oscillators

SUMMARYWe present a control method for a simple limit-cycle bipedal walker that uses adaptive frequency oscillators (AFOs) to generate stable gaits. Existence of stable limit cycles is demonstrated with an inverted-pendulum model. This model predicts a proportional relationship between hip torque amplitude and stride frequency. The closed-loop walking control incorporates adaptive Fourier analysis to generate a uniform oscillator phase. Gait solutions (fixed points) are predicted via linearization of the walker model, and employed as initial conditions to generate exact solutions via simulation. Global stability is determined via a recursive algorithm that generates the approximate basin of attraction of a fixed point. We also present an initial study on the implementation of AFO-based control on a bipedal walker with realistic mass distribution and articulated knee joints.

Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics

The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multiagent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices. In the noncirculant case we illustrate how stable limit cycles of the dynamics are coupled to embedded directed cycles in the payoff graph. The limit cycles correspond to oscillations of grammar dominance in language evolution and to oscillations in behavioral preferences in social networks; for decision-making systems, the limit cycles correspond to sustained oscillati...

Dynamic analysis of unstable Hopfield networks

In this paper, the dynamic behaviors of unstable Hopfield neural networks (HNNs) with asymmetric connections are studied. It is found that the solution of the HNN is bounded and the HNN is a dissipative system. In addition, sufficient conditions for the instability of the equilibrium point and the existence of stable limit cycles are proposed. Some numerical simulations are given to illustrate the effectiveness of the proposed results. It is shown that some HNNs exhibit two independent limit cycles or chaotic attractors which are symmetric to each other with respect to the origin.

2A2-E17 バウンシング・ロッド・ダイナミクスに基づく受動走行の安定化に関する実験的研究

In our earlier work, we have demonstrated a passive running based on bouncing rod dynamics (hinged-movement) by experiment and simulation. However, stability of the passive running is unstable. In this paper, we first analyze the causes of running failure. Secondly, we stabilize the passive running by foot. Finally, we demonstrate a possibility of existence of stable limit cycle.

CONVERGENCE PROPERTIES OF PREDATOR-PREY SYSTEMS WITH FUNCTIONAL RESPONSE

In the field of population dynamics and chemical reaction the possibility or the existence of spatially and temporally nonhomogeneous solutions is a very important problem. For last 50 years or so there have been many results on the pattern formation of chemical reaction systems studying reaction systems with or without diffusions to explain instabilities and nonhomogeneous states arising in biological situations. In this paper we study time-dependent properties of a predator-prey system with functional response and give sufficient conditions that guarantee the existence of stable limit cycles.

Dynamics of modified Leslie–Gower-type prey–predator model with seasonally varying parameters

Abstract A modified Leslie–Gower-type prey–predator model composed of a logistic prey with Holling’s type II functional response is studied. The axial point (1, 0) is found to be globally asymptotically stable in a domain. Condition for stability of the non-trivial equilibrium point is obtained. The existence of stable limit cycle of the system is also established. The analysis for Hopf bifurcation is carried out. The numerical simulations are carried out to study the effects of seasonally varying parameters of the model. The system shows the rich dynamic behavior including bifurcation and chaos.

Controller Design in Presence of Non-Smooth Nonlinearity: A Flight Control Example

The problem of output regulation with guaranteed transient performances in the presence of non-smooth nonlinearity is discussed. The fast dynamical controller with the relative highest derivative of output signal in feedback loop is used. Consequently, two-time-scale motions are induced in the closed-loop system. Stability conditions imposed on the fast and slow modes and sufficiently large mode separation rate can ensure that the full-order closed-loop system achieves the desired properties in such a way that the output transient performances are desired and insensitive to external disturbances and parameter variations in the system. The existence of stable limit cycle in the fast motion subsystem gives the robustness of the output transient performances in the presence of non-smooth nonlinearity and model uncertainty.

Limit cycle prediction for ballooning strings

This paper uses the Hopf bifurcation and center manifold theorems to predict the limit cycles of ballooning strings observed in many textile manufacturing processes. The steady state and linearized solutions of the non-linear governing equations are reviewed. The non-linear dynamic equations are discretized and projected to a finite-dimensional linear normal mode space, resulting in a set of non-linearly coupled but linearly uncoupled ordinary differential equations. The first two equations, corresponding to the lowest normal mode, are analyzed using the Hopf bifurcation theorem. The first Lyapunov coefficient is calculated to prove the existence of stable limit cycles for double-loop balloons with small string length. Analysis of the first two modes using the center manifold method verifies the effectiveness of the two-dimensional approximation. The bifurcation theorem, however, fails to apply to the large string length Hopf bifurcation point because the first Lyapunov coefficient is indeterminable. Numerical simulation of the non-linear two-dimensional equations agrees with experimental results quantitatively for small string length and qualitatively for large string length.

Autotune identification for systems with right-half-plane poles and zeros

Abstract Autotuning based on relay feedback tests have received a great deal of attention recently. The relay feedback tests fail to generate sustained oscillation in some cases. For systems with RHP poles and/or zeros, incorrect autotune identification procedure may give erroneous information and unstable limit cycle. Frequency domain analyses provide a criteria for prediction of the existence of stable limit cycle. Furthermore, a procedure is given to ensure the success of a relay feedback test. The analyses and procedure are illustrated through open loop stable and unstable systems. Results show that, except for some rare occasion, successful relay feedback experiments can be obtained following the proposed procedure.

About Deterministic Extinction in Ratio-dependent Predator–Prey Models

Ratio-dependent predator–prey models set up a challenging issue regarding their dynamics near the origin. This is due to the fact that such models are undefined at (0,0). We study the analytical behavior at (0,0) for a common ratio-dependent model and demonstrate that this equilibrium can be either a saddle point or an attractor for certain trajectories. This fact has important implications concerning the global behavior of the model, for example regarding the existence of stable limit cycles. Then, we prove formally, for a general class of ratio-dependent models, that (0,0) has its own basin of attraction in phase space, even when there exists a non-trivial stable or unstable equilibrium. Therefore, these models have no pathological dynamics on the axes and at the origin, contrary to what has been stated by some authors. Finally, we relate these findings to some published empirical results.