Asymptotic Orbital Stability Research Articles

Stability of solitary wave solutions in the Lugiato–Lefever equation

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato–Lefever equation on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation. These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies θ∈(0,π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (0,\\pi )$$\\end{document}, while unstable waves are found for angles θ∈(π,2π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (\\pi ,2\\pi )$$\\end{document}. Furthermore, we establish asymptotic orbital stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov–Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates.

Control systems with Thyristor converters

A system is considered, in the control loop of which there are thyristor converters or a special type of indirect control system. An example of constructing self-oscillations in a system describing the behavior of electrical devices using such elements in the control loop is given. Solutions for self-oscillations of a given period are constructed. Conditions for the existence of orbital asymptotic stability and Lyapunov stability for periodic solutions with a given period are presented, and a control loop is synthesized that ensures the existence of such solutions.

Uniform asymptotic stability for convection–reaction–diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter \varepsilon . The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale \varepsilon -dependent topology reduces to the classical W^{1, \infty} -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations.

Longitudinal shock waves in a class of semi-infinite stretch-limited elastic strings

In this paper, we initiate the study of wave propagation in a recently proposed mathematical model for stretch-limited elastic strings. We consider the longitudinal motion of a simple class of uniform, semi-infinite, stretch-limited strings under no external force with finite end held fixed and prescribed tension at the infinite end. We study a class of motions such that the string has one inextensible segment, where the local stretch is maximized, and one extensible segment. The equations of motion reduce to a simple and novel shock front problem in one spatial variable for which we prove existence and uniqueness of local-in-time solutions for appropriate initial data. We then prove the orbital asymptotic stability of an explicit two-parameter family of piece-wise constant stretched motions. If the prescribed tension at the infinite end is increasing in time, our results provide an open set of initial data launching motions resulting in the string becoming fully inextensible and tension blowing up in finite time.

Self-excited periodic motion in underactuated mechanical systems using two-fuzzy inference system

Dynamic systems with self-excited periodic motion have a wide range of applications in mechanical systems. For instance, a class of non-minimum phase underactuated and nonprehensile systems require self-oscillations instead of tracking an external reference signal. To solve this problem, we proposed a two fuzzy inference system, which is based on fuzzy logic theory, in order to generate a periodic output. The idea is to enforce the conditions that ensure the orbital asymptotic stability of the periodic solution of the closed-loop system. In this study, we considered Euler-Lagrange systems, with emphasis in underactuated systems. The describing function method was used to design the fuzzy controller and set the desired frequency and amplitude of the periodic output. Moreover, in accordance with Loeb's criteria, we established sufficient conditions for orbital stability. Finally, we tested and validated our proposal, via simulation and experiments on two laboratory platforms: a single-link pendulum and an underactuated non-minimum-phase rotational inverted pendulum.

Exciting extreme events in the damped and AC-driven NLS equation through plane-wave initial conditions.

We investigate, by direct numerical simulations and for certain parametric regimes, the dynamics of the damped and forced nonlinear Schrödinger (NLS) equation in the presence of a time-periodic forcing. It is thus revealed that the wave number of a plane-wave initial condition dictates the number of emerged Peregrine-type rogue waves at the early stages of modulation instability. The formation of these events gives rise to the same number of transient "triangular" spatiotemporal patterns, each of which is reminiscent of the one emerging in the dynamics of the integrable NLS in its semiclassical limit, when supplemented with vanishing initial conditions. We find that the L2-norm of the spatial derivative and the L4-norm detect the appearance of rogue waves as local extrema in their evolution. The impact of the various parameters and noisy perturbations of the initial condition in affecting the above behavior is also discussed. The long-time behavior, in the parametric regimes where the extreme wave events are observable, is explained in terms of the global attractor possessed by the system and the asymptotic orbital stability of spatially uniform continuous wave solutions.

Dynamics of bilateral control system with state feedback for price adjustment strategy

The price fluctuation of commodities affects their demand, and it strongly influences the development of industry and the stability of the international economy. A price-output model with two price-dependent impulses is presented in this paper to describe their mutual restrictions and the price adjustment strategies. The equilibria and dynamic properties of the impulsive-free price-output model are qualitatively analyzed, and the results show the hazard of free price fluctuation. For the price-output model with two price-dependent impulses, conditions for the existence of the order-2 periodic solution are obtained, and the orbital asymptotic stability is proved. Those results reveal the positive role of price adjustment strategies. Finally, some numerical simulations are carried out to verify the theoretical results.

Asymptotic behavior in the dynamics of a smooth body in an ideal fluid

This paper addresses a conservative system describing the motion of a smooth body in an ideal fluid under the action of an external periodic torque with nonzero mean and of an external periodic force. It is shown that, in the case where the body is circular in shape, the angular velocity of the body increases indefinitely (linearly in time), and the projection of the phase trajectory onto the plane of translational velocities is attracted to a circle. Asymptotic orbital stability (or asymptotic stability with respect to part of variables) exists in the system. It is shown numerically that, in the case of an elliptic body, the projection of the phase trajectory onto the plane of translational velocities is attracted to an annular region.

Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws

We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first-order scalar hyperbolic balance laws under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover, the convergence toward a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.

The entry-exit theorem and relaxation oscillations in slow-fast planar systems

The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel's geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two time-scale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper.

Stability of Periodic Motions and Synthesis of Relay Sampled Data Control Systems

This article is devoted to research and design of relay systems with control of data sampling. It is shown that the time sample has a significant effect on the parameters of periodic oscillations. We propose an exact method for analyzing periodic modes in digital self-oscillatory control systems with a two-position relay element and a linear piecewise-linear part is proposed. The proposed approach extends the phase hodograph method to the class of systems operating in discrete time. Two approaches have been developed to assess the stability of periodic motions in such systems. In the first approach, a discrete representation of a plant is considered and areas of stability are defined for each possible limit cycle. The sampling of the control system causes a delay in the switching of the relay in a batch mode in comparison with the continuous case. The second approach assumes the replacement of a discrete system by an equivalent continuous system with a time delay. Further, the asymptotic orbital stability of self-oscillations in a relay control system (RCS) with a delay is estimated. We consider the linearization of relay systems with digital control of the input signal. It is also shown that when linearizing a relay element in a digital RCS using a useful signal, the relay transfer ratio will belong to a certain range of values. Synthesis of corrective devices for relay control systems with regard to digital implementation has been reviewed. At the stage of optimization of parameters of the relay control system, the sample is taken into account. The model example demonstrates an advantage in the synthesis of digital technologies. It is shown that when optimizing the controller parameters with regard to time discretization, it was possible to provide the desired frequency of self-oscillations, which ensures the best accuracy of the tracking mode.

Stability, convergence and Hopf bifurcation analyses of the classical car-following model

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles traversing a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Classical Car-Following Model (CCFM). Specifically, we analyze the CCFM in no delay, small delay and arbitrary delay regimes. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using. Next, we derive the necessary and sufficient condition for local stability of the CCFM for an arbitrary delay. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation, thus resulting in limit cycles in system dynamics. Physically, these limit cycles manifest as back-propagating congestion waves on highways. In the context of human-driven vehicles, our work provides phenomenological insight into the impact of reaction delays on the emergence and evolution of traffic congestion. In the context of self-driven vehicles, our work has the potential to provide design guidelines for control algorithms running in self-driven cars to avoid undesirable phenomena. Specifically, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the CCFM. Next, we characterize the rate of convergence of the CCFM, and bring forth the interplay between local stability, non-oscillatory convergence and the rate of convergence of the CCFM. Further, to better understand the oscillations in the system dynamics, we characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincare normal forms and the center manifold theory. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations.

Local Stability and Hopf Bifurcation Analysis for Compound TCP

We conduct a local stability and Hopf bifurcation analysis for Compound transmission control protocol (TCP), with small Drop-Tail buffers, in three topologies. The first topology consists of two sets of TCP flows having different round-trip times and feeding into a core router. The second topology consists of two distinct sets of TCP flows, regulated by a single-edge router and feeding into a core router. The third topology comprises two distinct sets of TCP flows, regulated by two separate edge routers and feeding into a core router. In each case, we conduct a local stability analysis and obtain conditions on the network and protocol parameters to ensure stability. If these conditions get marginally violated, we show that the underlying systems lose local stability via a Hopf bifurcation. After exhibiting a Hopf, a key concern is to determine the asymptotic orbital stability of the bifurcating limit cycles. We then present a detailed analytical framework to address the stability of the limit cycles and the type of the Hopf bifurcation by invoking Poincare normal forms and the center manifold theory. We finally conduct packet-level simulations to corroborate our analytical insights.

A Lycaon pictus impulsive state feedback control model with Allee effect and continuous time delay

Allee effect (i.e. sparse effect) is active when the population density is small. Our purpose is to study such an effect of this phenomenon on population dynamics. We investigate an impulsive state feedback control single-population model with Allee effect and continuous delay. We first qualitatively analyze the singularity of this model. Then we obtain sufficient conditions for the existence of an order-one periodic orbit by the geometric theory of impulsive differential equations for the survival of endangered populations and obtain the uniqueness of an order-one periodic orbit by the monotonicity of the subsequent function. Furthermore, we prove the orbital asymptotic stability of an order-one periodic orbit using the geometric properties of successor functions to confirm the robustness of this control. Finally, we verify the correctness of our theoretical results by using some numerical simulations. Our results show that the release of artificial captive African wild dog (Lycaon pictus) can effectively protect the African wild dog population with Allee effect.

Nonlinear aeroelastic response of wind turbines using Simo-Vu-Quoc rods

• Combination of GEBT-based Simo-Vu-Quoc rods and BEMT for large wind turbines. • Lagrangian description obviates the need to explicitly define centrifugal/coriolis forces. • Aeroelastic coupling includes the effects of both structural deformations and vibration velocities. • Priming of aeroelastic solver by moderating structural response in the acceleration phase. • Asymptotic stability and orbital stability (limit-cycles) observed with and without gravity. A methodology is presented to obtain the nonlinear aeroelastic response of horizontal axis wind turbines using accurate and computationally efficient aerodynamic and structural descriptions. For computing the geometrically nonlinear deformations of the rotor blades , we employ Simo-Vu-Quoc’s geometrically exact finite-strain spatial rod model, which treats the displacements of the beam reference line and the rotations of the cross-section as configuration variables, without any restrictions on their magnitudes or those of the resulting 1-D strain measures. The structural description is Lagrangian, obviating the need for an explicit definition of centrifugal/coriolis forces. Towards the aerodynamic side, Blade Element Momentum Theory enhanced with several important corrections is employed, along with a dynamic stall model that accounts for the unsteady effects. A comprehensive treatment of the aeroelastic computations within the restrictions imposed by the aerodynamic theory is presented with the aid of a ‘shadow’ frame of reference rotating rigidly with the deforming blades. A novel strategy ‘primes’ the aeroelastic solver by moderating the response of rotor in its acceleration phase. An integrated MATLAB tool WindGEAR incorporating the two coupled codes is developed to simulate the fully-coupled aeroelastic response of the NREL 5 MW wind turbine. The results, when compared with published data, establish WindGEAR firmly as a robust wind turbine nonlinear aeroelastic analysis tool in time-domain.

The modified optimal velocity model: stability analyses and design guidelines

Reaction delays are important in determining the qualitative dynamical properties of a platoon of vehicles traveling on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Modified Optimal Velocity Model (MOVM). Specifically, we analyze the MOVM in three regimes – no delay, small delay and arbitrary delay. In the absence of reaction delays, we show that the MOVM is locally stable. For small delays, we then derive a sufficient condition for the MOVM to be locally stable. Next, for an arbitrary delay, we derive the necessary and sufficient condition for the local stability of the MOVM. We show that the traffic flow transits from the locally stable to the locally unstable regime via a Hopf bifurcation. We also derive the necessary and sufficient condition for non-oscillatory convergence and characterize the rate of convergence of the MOVM. These conditions help ensure smooth traffic flow, good ride quality and quick equilibration to the uniform flow. Further, since a Hopf bifurcation results in the emergence of limit cycles, we provide an analytical framework to characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the resulting non-linear oscillations. Finally, we corroborate our analyses using stability charts, bifurcation diagrams, numerical computations and simulations conducted using MATLAB.

Current State and Prospects for Development of the Theory of Relay Feedback Systems

This paper presents a brief review of the theory of the relay feedback systems. Analysis and synthesis of the relay self-oscillating systems is currently a classical topic in the control theory. The main problems of the theory are analysis of the possible self-oscillations, stability, estimation of robustness, and regulator design. The first works devoted to the analysis of the relay feedback systems appeared in 1950s. However, many problems concerning this theory are still unresolved. Systematization of the theory of the relay control systems and the definition of possible directions for research in future are presented in this article. The main attention is paid to the method of studying of the self-oscillating systems for the phase locus method. This method was developed in the Department of Automatic Control Systems of Tula State University. Introduction contains a brief historical review of the problems and methods for their solution within the framework of the theory of the relay systems of automatic control. The second section provides a brief review of the methods for studying of the self-oscillations in the control systems in the frequency domain. The main attention in this section is given to the methods, based on the Tsypkin locus. The third section deals with the methods of analysis and synthesis of the relay systems of automatic control in the state space. The method of the phase locus is considered. This method is applicable to the study of the systems with a nonlinear control object. Within the framework of this method, a synthesis algorithm was generated. The recent work within the framework of this theory was devoted to the synthesis of stable self-oscillating control systems. An analysis of the global asymptotic orbital stability of the self-oscillations in the systems with a linear control object in the form of linear matrix inequalities is considered, as well as the problems of designing of the digital relay controllers.

On Orbital Stabilization for Industrial Manipulators: Case Study in Evaluating Performances of Modified PD+ and Inverse Dynamics Controllers

Orbital stabilization is one of the available alternatives to the classical asymptotic stabilization, known as the reference tracking control, which is typically considered and implemented for controlling motions of industrial robot manipulators. Since asymptotic orbital stability means convergence of solutions of a closed-loop system to an orbit of a reference trajectory, instead of tracking it as a function of time, new feedback designs can potentially improve performance with respect to several key criteria for industrial manipulators such as absolute path accuracy for tool’s motions and robustness to uncertainties in the model. The main outcomes of this paper are a new class of controllers that achieve asymptotic orbital stabilization of motions and a novel analytical method for analysis and redesign of system’s dynamics using an excessive set of easy-to-compute transverse coordinates. The contributions have been validated in a series of experimental studies performed on a standard industrial robot ABB IRB 140 with the IRC5-system extended with an open control interface. The outcomes of the tests show that the proposed redesign allows achieving significantly reduced deviations of the actual trajectories from the desired ones at different ranges of speeds and for several different paths, often outperforming the state-of-the-art commercial implementations. A comprehensive discussion of one of such experiments is given.

On the Emergence and Orbital Stability of Phase-Locked States for the Lohe Model

We study the emergence and orbital stability of phase-locked states of the Lohe model, which was proposed as a non-abelian generalization of the Kuramoto phase model for synchronization. Lohe introduced a first-order system of matrix-valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase-locked states of the Lohe model. In this paper, we provide an analytical framework to confirm Lohe’s observations of emergent phase-locked states. This extends earlier special results on lower dimensions to any finite dimension. For the construction and orbital stability of phase-locked states, we introduce Lyapunov functions to measure the ensemble diameter and dissimilarity between two Lohe flows, and using the time-evolution estimates of these Lyapunov functions, we present an admissible set of initial states, and show that an admissible initial state leads to a unique phase-locked asymptotic state.

Study on a giant panda reintroduction state feedback control pulse model with diffusion between two patches

We establish a pulse model for giant pandas reintroduction with state feedback control and diffusion between two patches. A unique order one periodic solution of the system with orbital asymptotic stability are obtained by using the geometry theory of semi-continuous dynamic systems. Especially, the order one periodic solution is bigger than the number of minimum viable population. Besides, we also prove that the system does not have order k (k = 2, 3,...) periodic solutions, which means the order one periodic solution is the unique periodic solution. It is essential for design and development of the reintroduction of captive giant pandas work in the future.