Liapunov Function Research Articles

Global Asymptotic Stability Analysis of Fixed Points for a Density-Dependent Single-Species Population Growth Model

In a density-dependent single-species population growth model, a simple method is proposed to explicitly and directly derive the analytic expressions of reliable regions for local and global asymptotic stability. Specifically, first, a reliable region ΛLAS is explicitly represented by solving the fixed point and utilizing the asymptotic stability criterion, over which the fixed point is locally asymptotically stable. Then, two types of auxiliary Liapunov functions are constructed, where the variation of the Liapunov function is decomposed into the product of two functions and is always negative at the non-equilibrium state. Finally, based on the Liapunov stability theorem, a closed-form expression of reliable region ΛGAS is obtained, where the fixed point is globally asymptotically stable in the sense that all the solutions tend to fixed point. Numerical results show that our analytic expressions of reliable regions are accurate for both local and global asymptotic stability.

Prescribed performance LOS guidance-based dynamic surface path following control of unmanned sailboats

The unmanned sailboat path following control problem with model uncertainties and unknown external disturbances is addressed. Firstly, different sailing strategies are set for the sailboat under three different wind fields in nature for its vulnerability to the true wind field so that the sailboat can complete the path-following task under different wind directions. Secondly, a barrier Lyapunov function with prescribed performance is used to limit position errors and heading errors of the unmanned sailboat within prescribed constraints. A radial-based function neural network technique is then introduced to approximate model uncertainties. Then an adaptive law is introduced to estimate an upper bound on the sum of the external environmental disturbances and neural network approximation errors. A path-following control scheme for the unmanned sailboat is proposed by the backstepping method and combined with the dynamic surface control technique. Finally, stability analysis is performed by Liapunov functions, demonstrating that the proposed control method can guarantee that all error signals in the closed-loop system are semi-global uniform ultimate bounded and that the prescribed performance constraints are not violated. Finally, the effectiveness of the scheme is demonstrated by simulation.

A Novel MPPT Algorithm for Photovoltaic Systems Based on Improved Sliding Mode Control

Due to the poor tracking performance and significant chattering of traditional sliding mode control in the maximum power point tracking (MPPT) algorithm, a novel MPPT algorithm based on sliding mode control for photovoltaic systems is proposed in this paper. The sliding mode control structure and new sliding mode surface of the multi-power reaching law are designed with the boost converter as the carrier of the photovoltaic system, and the sigmoid function is proposed to replace the symbolic function and saturation function in the power reaching law to improve the reaching rate and control quality of the traditional sliding mode control. Furthermore, the Liapunov function is employed to analyze the accessibility, existence and stability of the improved sliding mode control. Simulation results under dynamic and partial shading conditions show that compared with exponential sliding mode and constant speed sliding mode, the improved sliding mode control strategy can quickly track the maximum power point of photovoltaic systems under various atmospheric conditions. The proposed MPPT algorithm has stronger robustness and universality. Additionally, the efficiency of the proposed algorithm is improved by 2.3% and 5.6% as compared to the exponential sliding model control algorithm and constant velocity sliding model control algorithm. In addition, the experimental platform is constructed to further validate the feasibility and effectiveness of the proposed algorithm.

Hydrodynamic Impacts of Short Laser Pulses on Plasmas

We determine conditions allowing for simplification of the description of the impact of a short and arbitrarily intense laser pulse onto a cold plasma at rest. If both the initial plasma density and pulse profile have plane symmetry, then suitable matched upper bounds on the maximum and the relative variations of the initial density, as well as on the intensity and duration of the pulse, ensure a strictly hydrodynamic evolution of the electron fluid without wave-breaking or vacuum-heating during its whole interaction with the pulse, while ions can be regarded as immobile. We use a recently developed fully relativistic plane model whereby the system of the Lorentz–Maxwell and continuity PDEs is reduced into a family of highly nonlinear but decoupled systems of non-autonomous Hamilton equations with one degree of freedom, the light-like coordinate ξ=ct−z instead of time t as an independent variable, and new a priori estimates (eased by use of a Liapunov function) of the solutions in terms of the input data (i.e., the initial density and pulse profile). If the laser spot radius R is finite and is not too small, the same conclusions hold for the part of the plasma close to the axis z→ of cylindrical symmetry. These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons.

Asymptotic and transient dynamics of SEIR epidemic models on weighted networks

We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.

Stability of a class of nonlinear hierarchical size-structured population model.

This paper investigates the existence of positive equilibrium as well as the stability of positive equilibrium and zero equilibrium in a nonlinear size-structured hierarchical population model. Under the condition that larger individuals are more competitive advantages than smaller ones, a non-zero fixed point theorem is used to show that there is at lest one positive equilibrium in the system. Moreover, we obtain the stability results of positive equilibrium and zero equilibrium by deriving characteristic equations and establishing Liapunov function. Finally, some numerical experiments are presented.

Analysis of an epidemic model with latent infection

An infectious disease model with latent infection is studied, and the basic reproductio n number of the model is obtained When the basic reproducing number is less than 1, the mode l has only one disease-free equilibrium. When the basic reproducing number is greater than 1, t he disease-free equilibrium is unstable. The model also has a unique positive equilibrium. The global asymptotic stability of the two equilibria is analyzed by linearization method, Liapunov function method and geometric method.

Analysis of a contact problem for a viscoelastic Bresse system

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an adequate Liapunov function. Then, using the finite element method and the implicit Euler scheme, a finite element approximation is introduced. A discrete stability property and a priori error estimates are proved. Finally, some numerical experiments are performed to demonstrate the decay of the discrete energy and the numerical convergence.

Dynamical Study of a Vector Host Epidemic Model With Non-Monotone Incidence

Epidemics of vector borne disease studied in this paper when disease incidence rate is non- monotone. Non monotonic behaviors of infected hosts as well as infected vectors are studied for influence of increase in disease on susceptible host population and effect of repellents and insecticides on vectors, respectively. Suitable Liapunov functions are constructed to discuss the global asymptotic stabilities at the equilibrium points. Results are verified by conducting numerical simulation. Mechanism of disease spread by vectors and hosts during the epidemic can be better understood with the help of this model.

Dynamics of a non-autonomous predator-prey system with Hassell-Varley-Holling Ⅱ function response and mutual interference

In this paper, we establish a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. We construct some sufficient conditions for the permanence, extinction and globally asymptotic stability of system by use of the comparison theorem and an appropriate Liapunov function. Then the sufficient and necessary conditions for a periodic solution of the system are obtained via coincidence degree theorem. Finally, the correctness of the previous conclusions are demonstrated by some numerical cases.

Liapunov functions for neural network models

Liapunov functions for neural network models

Construction of Liapunov functions for highly nonlinear dynamic systems with feedback

The paper provides stability and qualitative research of oscillations of a highly nonlinear dynamic system with feedback. For a system satisfying Liapunov theorem, definite-positive recurrent functions of order 1 with negative derivatives are constructed. Sufficient stability conditions are established for the considered cases of power nonlinearities. Surface diagrams of Liapunov functions and their derivatives are constructed for various values of the feedback parameter. Using the Liapunov criterion, the behavior of the trajectories of dynamic system on the state planes and near the singular points is investigated. Possible limit cycles are determined based on the Poincaré method of contact curves. Integral curves and phase trajectory diagrams are constructed numerically using the Mathcad 13 software package. A transition process from an unstable focus to self-oscillating and relaxation vibration modes is established, as well as the corresponding limit cycles consistent with analytical definition of rings containing the indicated limit cycles.

Differential systems stability analysis based on matrix multiplicative criteria

This article proposes a method of stability analysis in the sense of Liapunov for systems of ordinary differential equations. The method is based on matrix multiplicative stability criteria obtained from transformations of numerical integration difference schemes. Criteria are formed in the form of necessary and sufficient conditions. A distinctive feature of the criteria is that they do not employ the methods of the qualitative theory of differential equations. Specifically, for the case of linear systems, the evaluation of characteristic numbers and characteristic indicators is not necessary. When analyzing the stability of nonlinear systems, the construction of Liapunov functions is not required. Matrix multiplicative form of the criteria allows computerization of stability analysis. The software implementation of the criteria is performed in real time and, based on its results, allows to make an unambiguous conclusion about the nature of stability of the system under study.

Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

Information-theory based variational principles have proven effective at providing scalable uncertainty quantification (i.e., robustness) bounds for quantities of interest in the presence of nonparametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincaré, $\log$-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds for time-averaged observables, comparing a Markov process to a second (not necessarily Markov) process. These bounds are well behaved in the infinite-time limit and apply to steady-states of both discrete and continuous-time Markov processes.

A new SOR-type iteration method for solving linear systems

Many discretization methods for continuous dynamical systems have the iterative nature and therefore can provide iterative techniques for solving problems in numerical linear algebra. In this paper, based on the discrete gradient and the variation-of-constants formula for ordinary differential equations, a new SOR-type iteration method is proposed for solving the linear system Ax=b. The convergence of the new method is guaranteed by the decay of the Liapunov function. Numerical experiments are carried out to show the effectiveness of the new method.

Global Stability of a Predator-Prey Model with the Logistic Term

This paper is mainly to analyze stability of equilibria in a new predator-prey model, Logistic-Volterra model. It exists three different equilibria for the model. The stability of these equilibria for the model are discussed by Liapunov function or other principle. Through theoretical analysis, (0,0) is unstable and existent in any situations. (N1, 0) is existent and global asymptotic stability if δ2 < k2 / N1, and (x1*, x2*) is existent and uniformly and globally asymptotic stability if δ2 > k2 / N1. Then, some simulation are given and verify the proof of the result. At last, some conclusions about biology were raised.

Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times.

We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” (), evolving through the Liouville equation for the non-equilibrium classical distribution , with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution , as weight function to generate orthogonal polynomials (’s) in momenta. The moments of , implied by the ’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.

Simulation of Optimal Harvesting of Three Species Ecological Model with Closed Interval of Biological Parameter Using by Liapunov Function

The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords: Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/IEL/9-1-04

Simulation of Optimal Harvesting of Three Species Ecological Model with Closed Interval of Biological Parameter using by using First Order Nonlinear Ordinary Differential Equations

The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords : Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/JAAS/52-02

Simple and Effective Theory of Movement Steadiness

It is proposed the very simple and quick method for estimation of the asymptotic stability of any nonlinear dynamic systems, in particular, of the high-dimensional systems for which Tailor series of the right-hand sides of the differential equations converge very slowly. In such problems, the sum of terms of the order of smallness higher than two can substantially exceed the value of any term of second order. In this case, Lyapunov’s methods cannot guarantee correct stability estimate at all. The new method does not use the notion of Liapunov function and, therefore, one has no numerous shortcomings of all Liapunov methods. In this paper, it is proposed to replace the very complex problem of the searching for Liapunov function with a very simple problem of the searching maximum of the function of n coordinates (that is of the velocity of variation in metrics of the perturbed state space). However, one is not intended for the linear systems.