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This paper is devoted to the asymptotic stability of diverging traveling waves for reaction–advection–diffusion equation u t - Δ u + α (t , y) u x = f (t , y , u) in cylinders. By the sliding method, we first establish a Liouville-type result. Then, using the Liouville-type result and truncation technique, we prove the asymptotic stability of the diverging traveling wave. [ABSTRACT FROM AUTHOR]

This paper presents various bifurcations of the McMillan map under perturbations of its coefficients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable fixed point is also demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on R × T L and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space. [ABSTRACT FROM AUTHOR]

In this paper, we first construct some explicit solutions to the b-family of equations, which will become unbounded in a finite time. Then, we investigate the asymptotic stability of the aforementioned singular solutions of the b-family of equations in the Sobolev space H s with s > 7 2 . It is also interesting to point out that this stability highly depends on the values of parameter b, that is, b ∈ (- 1 , 2 ] . The proof is based on the detailed analysis on the estimates of the perturbed solutions and the properties of the corresponding linear operators. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of solutions for a new class of generalized Halanay inequalities is studied via the fixed point method. This research provides a new approach to the study of the stability of Halanay inequality. To make the application of fixed point method in stability research more flexible and feasible, we introduce corresponding functions to construct an operator according to different characteristics of coefficients. The results obtained in this paper are applied to the stability study of a neural network system, which has high value in application. Moreover, three examples and simulations are given to illustrate the results. The conclusions in this paper greatly improve and generalize the relative results in the current literature. [ABSTRACT FROM AUTHOR]

The asymptotic stability of delayed reaction-diffusion neural networks with algebraic constraints, that is, singular delayed reaction-diffusion neural networks, is studied in this paper. In terms of Green's theorem, inequality technique and linear matrix inequalities (LMIs), two less conservative criterion for the asymptotic stability of singular delayed reaction-diffusion neural networks are given by endowing Lyapunov direct method and used to design an appropriate stabilizing feedback controllers. The results address both the effects of the delay and the algebraic constraints. In addition, these conditions have higher computational efficiency and can easily detect and stabilize the actual neural networks. Finally, the numerical simulations verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with the global asymptotic stability (GAS) of the fixed-point two-dimensional (2D) Lipschitz nonlinear digital filter (LNDF) in Fornasini–Marchesini second local state-space (FMSLSS) model with 2's complement overflow. In particular, the 2D system under study involves intrinsic system nonlinearities as well as nonlinearities due to overflow. New conditions for verifying the GAS of such LNDFs is developed in this paper. The criterion utilizes Lyapunov method and the properties of overflow arithmetic and Lipschitz nonlinearities. A criterion for the overflow stability of 2D systems (without intrinsic system nonlinearities) is also brought out. The obtained criteria are also applicable to digital filters with other frequently used overflow arithmetic (namely, saturation, zeroing, and triangular). The results presented in this paper can be used directly as criteria for elimination of overflow oscillations in 2D systems. The obtained results are compared with existing results. [ABSTRACT FROM AUTHOR]

This paper investigates the stability behaviour of externally interfered discrete-time system under the influence of various finite-register length nonlinearities. Such nonlinearities include quantization and overflow which are commonly originated during hardware implementation. The proposed criteria guarantee the realization of limit-cycle free discrete-time system in the presence of external disturbances and finite-register length nonlinearities. The stability conditions are fabricated by employing passivity-based approach under combinations of quantization and overflow nonlinearities. With the help of the appropriate Lyapunov function along with the passivity condition, the asymptotic stability for the discrete-time system is also ensured under zero disturbance. Relevant examples are presented to show the applicational scope of the work. [ABSTRACT FROM AUTHOR]

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill's equation based on L p -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill's equations of potential a (t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical stability of a family of explicit blowup solutions of the three-dimensional (3D) incompressible Navier-Stokes (NS) equations with smooth initial values, which is constructed in Guo et al. (2008). This family of solutions has finite energy in any bounded domain of ℝ3, but unbounded energy in ℝ3. Based on similarity coordinates, energy estimates and the Nash-Moser-Hörmander iteration scheme, we show that these solutions are asymptotically stable in the backward light-cone of the singularity. Furthermore, the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data. Finally, the result also shows that in the absence of physical boundaries, the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations. [ABSTRACT FROM AUTHOR]

Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x1, t ⩾ 0 where x1(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0) ≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥

The problem of global asymptotic stability (GAS) of fixed-point state-delayed uncertain discrete-time systems combined with a saturation operator in state-space is investigated in this paper. The uncertainties in the system are presumed to be norm-bounded, which has been frequently employed in robust control for uncertain systems. This paper proposes a new GAS criterion for the considered system. The saturation nonlinearities associated with the system, which operate exclusively in the linear region, are identified using a unique methodology. Such identification leads to an improved characterization of saturation nonlinearities involved in the system. The proposed criterion utilizes an upper bound of parameter uncertainties, an asymptotic bound on the system's states, and a precise characterization of saturation nonlinearities involving system parameters. The merit of the criterion is also exemplified through examples along with simulation results. [ABSTRACT FROM AUTHOR]

Many applications dealing with the mathematical formulation associated with real-order systems bring long memory complexity, and the subsequent system analysis seems very puzzling and challenging. The settling of random initial-time often limits simple mathematical analysis of many real-order nonlinear dynamic and control systems. This paper formulates some new theories of nonlinear non-autonomous real-order systems attached with random initial-time in the sense of Caputo derivative. A fundamental stability question often arises: can we find reasonable conditions that provide limiting behavior of non-trivial trajectories of such systems to some constant equilibrium vector? We introduce some new comparison results that deal with the construction of potential Metzler matrix, non-negative matrix and non-negative system that enables new order-dependent criteria for concluding non-trivial trajectory responses to constant zero equilibrium vector embedded within such systems. A novel general theorem is put forward which, in short, states that if a relative asymptotic stable system is constructed, it is possible to conclude the limiting behavior of the original system. In the end, some new emerging applications dealing with the problems of equilibrium analysis of dynamic models are discussed by using the proposed results. An engineering control problem demonstrating observer design to a new class of systems is proposed including the use of time-varying linear state feedback control law to illustrate the importance of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with an intelligent optimal control approach for position/force control of constrained reconfigurable manipulators in the presence of uncertainties. Reconfigurable manipulator has strong coupling, nonlinearity, uncertainties in the system and the environmental contact make the dynamical model much more complex. This necessitates the design of intelligent control scheme to achieve the control objectives as possible. Therefore to handle this complex system, firstly, the dynamic model is reduced into state space form describing the constrained and unconstrained state variables separately. Then the state-space form of error dynamics is utilized for quadratic optimization where the optimal control, the explicit solution of Hamilton Jacobi Bellman equation, is obtained by solving an algebraic Riccati equation. The proposed scheme integrates the linear optimal control, radial basis function (RBF) neural network and adaptive robust term to overcome the uncertainties of the complex system and to achieve the desired performance. The linear optimal controller optimizes the system whereas the nonlinearities and uncertainties of the system are compensated using RBF neural network and adaptive compensator. The asymptotic stability of the system is demonstrated using the optimal control theory and Lyapunov stability analysis. Finally, the proposed control approach is verified in comparative manner through simulation results with 2 DOF reconfigurable manipulators to perform various tasks. [ABSTRACT FROM AUTHOR]

This paper is concerned with the time-asymptotic stability of a nonlinear wave for the outflow problem of the isentropic compressible Navier–Stokes–Korteweg equations in the half space. Under some suitable assumptions on the spatial-asymptotic states and boundary data, the time-asymptotic profile is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L 2 -energy method and the decay (in both time and space variables) estimates of each component of the nonlinear wave, we prove that this nonlinear wave is time asymptotically stable under a small initial perturbation. [ABSTRACT FROM AUTHOR]

In this paper, we study the time-asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar Navier–Stokes–Poisson system in the half space. First, we make some suitable assumptions on the boundary data and space-asymptotic states such that the time-asymptotic state of the solution is a nonlinear wave, which is the superposition of the transonic stationary solution and the 2-rarefaction wave. Next, we show the stability of this nonlinear wave under a class of large initial perturbation, provided that the strength of the transonic stationary solution is small enough, while the amplitude of the 2-rarefaction wave can be arbitrarily large. The proof is completed by a delicate energy method and a continuation argument. The key point is to derive the positive upper and lower bounds of the particle densities. [ABSTRACT FROM AUTHOR]

This paper presents a state-dependent switching law to achieve asymptotic stability of a fourth-order switched system, in which each subsystem has two pairs of conjugate characteristic roots with positive real parts. Firstly, a linear vibration system of two-degrees-of-freedom is introduced to match the standard fourth-order linear subsystem. An energy function of each subsystem is then defined as its kinetic and potential energies. Secondly, an invertible transformation is introduced to convert a general fourth-order subsystem into a standard one, thereby constructing its energy function with definite physical meaning. Thirdly, in order to reasonably define an energy ratio function between two subsystems, an intermediate subsystem is introduced so that an identical invertible transformation can be used to convert the subsystem and the intermediate subsystem into standard fourth-order ones simultaneously. Based on the obtained energy ratio function, a state-dependent switching law is constructed to maximize the energy loss in a switching loop, thus stabilizing the fourth-order switched system at a fast speed. In the end, the effectiveness of the proposed method is illustrated by four numerical simulations. [ABSTRACT FROM AUTHOR]

The dynamics of quasilinear systems with regular moments of impulses is investigated when a perturbation generated by an unpredictable sequence is applied. The existence, uniqueness, as well as asymptotic stability of unpredictable solutions are proved making benefit of a novel definition regarding regular discontinuity moments. To achieve the main result a Gronwall type inequality for piecewise continuous functions is utilized. The theoretical results are supported with an illustrative example. This is the first time in the literature that the existence of unpredictable solutions is demonstrated for impulsive systems comprising periodic terms and regular moments of impulses. [ABSTRACT FROM AUTHOR]

This paper focuses on the robust control issue for interval type-2 Takagi-Sugeno (IT2 T-S) fuzzy discrete systems with input delays and cyber attacks. The lower and upper membership functions are first utilized to IT2 fuzzy discrete systems to capture parameter uncertainties. By considering the influences of input delays and stochastic cyber attacks, a newly fuzzy robust controller is established. Afterward, the asymptotic stability sufficient conditions in form of LMIs for the IT2 closed-loop systems are given via establishing a Lyapunov-Krasovskii functional. Afterward, a solving algorithm for obtaining the controller gains is given. Finally, the effectiveness of the developed IT2 fuzzy method is verified by a numerical example. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called "effective viscous flux", which enables us to obtain the desired energy estimates and regularity. The technique of the "effective viscous flux" turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models. [ABSTRACT FROM AUTHOR]

This article investigates the asymptotic stability of fractional-order (FO) systems with uncertainty and time-varying delay based on the sliding mode control (SMC) method. First, based on the SMC method, a suitable integral type fractional-order sliding mode surface (FOSMS) is designed and the dynamic equations of FO systems under SMC are obtained. Second, by inequality techniques, the condition for asymptotic stability of the FO system has been mathematically established. Then, a novel adaptive SMC law is introduced, which can make sure the accessibility of sliding mode surfaces (SMS). Finally, the feasibility of the results obtained in this paper is verified through a simulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of a modified chemostat model. We first demonstrate the existence of equilibria. Then, we present a mathematical analysis for the model, the invariance, the positivity, the persistence of the solutions, and the asymptotic global stability of the interior equilibrium. Some numerical simulations are carried out to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, the optimal formation control issue with collision avoidance and connectivity assurance is investigated for a class of second-order uncertain nonlinear multi-agent systems. First, the neural networks are employed in order to deal with the unknown nonlinear dynamics of the system. Then, an optimal formation control scheme is developed in the framework of the identifier–actor–critic. By constructing a new performance metric function containing collision avoidance and connectivity constraints, it is demonstrated that asymptotic convergence of the tracking error can be achieved under the proposed control scheme. Finally, the effectiveness of the proposed control method is validated by the numerical simulation example. [ABSTRACT FROM AUTHOR]

This paper aims to study the relationship between the timelike extremal hypersurfaces and the classical minimal surfaces. This target also gives the long time dynamics of timelike extremal hypersurfaces in Minkowski spacetime R 1 + M with the dimension M ≥ 2 . In this dimension, the stationary solution of timelike extremal hypersurface equation is the solution of classical minimal surface equation div ∇ u 1 + | ∇ u | 2 = 0 , ∀ x ∈ R M , which only admits the hyperplane solution by Bernstein theorem. We prove that this hyperplane solution as the stationary solution of timelike extremal hypersurface equation is asymptotic stablely by finding the hidden dissipative structure of linearized equation. Our result construct a unique global timelike non-small solution near the hyperplane. [ABSTRACT FROM AUTHOR]

This paper investigates a difficult problem of path-following control for parafoil dynamic systems in time-varying wind disturbance. A common representation of a parafoil dynamic system for following straight-line and orbit paths is introduced with the time-varying wind disturbance and modeling uncertainties simultaneously considered. Utilizing the barrier Lyapunov function and backstepping method, an adaptive path-following controller is then developed considering the yaw rate constraint. The closed-loop path-following control system is proved to be asymptotically stable. The key feature of the controller is that it not only ensures the superior capability to attenuate disturbances caused by the time-varying wind and the modeling uncertainties, but also avoids the yaw rate of the parafoil system violating the practical constraint. Experiment tests illustrate the parafoil performance obtained using the proposed path-following controller. [ABSTRACT FROM AUTHOR]

This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present the first and second order backward differentiation formulas (BDF1, vBDF1, BDF2 and vBDF2) with finite element method for a class of non-autonomous nonlocal partial differential equations with delay. First, the unique solvability of the numerical schemes is proved. The L 2 upper bound of the numerical solution is given by energy estimation. The convergence analysis is fulfilled for nonlinear delay term, with O (h q + Δ t) order for BDF1 and O (h q + Δ t 2) order for BDF2. The convergence order with respect to the spatial discretization is recovered to an optimal O (h q + 1) order for a kind of specific delay term ψ = ∇ · A (x , t) ∇ u . Asymptotic stability analysis is also given, and it is proved that the numerical solution will approach the stationary solution at an exponential speed. Finally, a few numerical experiments are presented to show the convergence order, asymptotic behavior, and multiple stationary solutions of the numerical solutions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to analyze the Hopf-like bifurcations in 3D piecewise linear systems. Such bifurcations are characterized by the birth of a piecewise smooth limit cycle that bifurcates from a singular point located at the discontinuity manifold. In particular, this paper concerns systems of the form x ˙ = A x + b ± which are ubiquitous in control theory. For this class of systems, we show the occurrence of two distinct types of Hopf-like bifurcations, each of which gives rise to a crossing limit cycle (CLC). Conditions on the system parameters for the coexistence of two CLCs and the occurrence of a saddle-node bifurcation of these CLCs are provided. Furthermore, the local asymptotic stability of the pseudo-equilibrium point is analyzed and applications in discontinuous control systems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study global stability dynamics for quasilinear damped Klein–Gordon equation with variable coefficients in R n for dimension n ≥ 1 , without assuming that the quasilinear term satisfies the null condition. Due to the influence of damping term, we are able to prove that if it admits a global smooth solution (including time quasi-periodic solution), then this quasilinear system must be asymptotic stable in Sobolev space under some assumptions on the variable coefficients of it. This result also implies that a smooth large solution can be constructed for such quasilinear system, which includes time quasi-periodic stable dynamics of Klein–Gordon equation. We note the global stability for a class of semilinear wave equation satisfying null condition has been shown by Zha (J Funct Anal 282:109219, 2022). [ABSTRACT FROM AUTHOR]

This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems. In particular, both Caputo definition and Riemann-Liouville definition are under consideration. With the convex assumption, several elementary fractional difference inequalities on Lyapunov functions are developed. According to the essential features of nabla fractional calculus, the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method. To substantiate the efficacy and effectiveness of the theoretical results, four examples are elaborated. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic stability of a new class of nonlinear time-varying real-order systems that involve multiple delays. First, we introduce a comparison principle and a new lemma that give an estimation for the bounds between the solutions to any two relative systems. Then, by using the inequalities and comparison methodology, we develop some new mathematical results for the asymptotic stability analysis of the zero solution to such a class of systems. We establish new fractional-order-dependent and delay-dependent, and fractional-order-dependent and delay-independent conditions for the analysis of such a class of systems. At the end, we present three examples and demonstrate that some established criteria are very effective for the analysis and control of such systems. [ABSTRACT FROM AUTHOR]

In this paper, we show that Landau solutions to the Navier–Stokes system are asymptotically stable under L 3 -perturbations. We give the local well-posedness of solutions to the perturbed system with the initial data in the L σ 3 space and the global well-posedness with the small initial data in the L σ 3 space, together with a study of the L q decay for all q > 3 . Moreover, we have also studied the local well-posedness, the global well-posedness and the stability in L p spaces for 3 < p < ∞ . [ABSTRACT FROM AUTHOR]

This work aims to develop an explicit approximation scheme for solving stochastic differential systems based on the Euler–Maruyama scheme. We demonstrate the strong consistency and convergence properties of the proposed scheme. In addition, in a mean-square sense, we perform linear and nonlinear asymptotic stability analysis for the multiplicative and additive cases. At the end of the paper, several numerical examples arising in biology, neurology, and population dynamics are carried out to illustrate the obtained convergence and stability properties. [ABSTRACT FROM AUTHOR]

The problem of l2-l∞ stability and disturbance attenuation performance analysis of fixed-point state-space digital filters with external disturbance and finite wordlength nonlinearities is studied in this paper. The finite wordlength nonlinearities are composite nonlinearities representing concatenations of quantization and overflow correction employed in practice. Using sector-based characterization of the finite wordlength nonlinearities, a new l2-l∞ stability criterion for the reduction of the effects of external disturbance to a given level and to confirm the nonexistence of limit cycles in the absence of external interference is established. Numerical examples are given to illustrate the efficacy of the proposed criterion. [ABSTRACT FROM AUTHOR]

In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. Moreover the strengths of two waves could be arbitrarily large. [ABSTRACT FROM AUTHOR]

To improve transportation capacity, dual overhead crane systems (DOCSs) are playing an increasingly important role in the transportation of large/heavy cargos and containers. Unfortunately, when trying to deal with the control problem, current methods fail to fully consider such factors as external disturbances, input dead zones, parameter uncertainties, and other unmodeled dynamics that DOCSs usually suffer from. As a result, dramatic degradation is caused in the control performance, which badly hinders the practical applications of DOCSs. Motivated by this fact, this paper designs a neural network-based adaptive sliding mode control (SMC) method for DOCS to solve the aforementioned issues, which achieves satisfactory control performance for both actuated and underactuated state variables, even in the presence of matched and mismatched disturbances. The asymptotic stability of the desired equilibrium point is proved with rigorous Lyapunov-based analysis. Finally, extensive hardware experimental results are collected to verify the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the authors apply steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data where . Based on the spectral analysis of the Lax pair, they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region with ∣ξ + 2∣ < 1, the long time asymptotic is given by in which the leading term is N(I) solitons, the second term is a residual error from a equation. For the region ∣ξ + 2∣ > 1, the long time asymptotic is given by in which the leading term is N(I) solitons, the second order term is soliton-radiation interactions and the third term is a residual error from a equation. These results are verification of the soliton resolution conjecture for the derivative Schrödinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of linearized stability for nonlinear fractional difference equations. Computational methods based on appropriate linearization theorem are standardly applied in bifurcation analysis of dynamical systems. However, in the case of fractional discrete systems, a theoretical background justifying its use is still missing. Therefore, the main goal of this paper is to fill in the gap. We consider a general autonomous system of fractional difference equations involving the backward Caputo fractional difference operator and prove that any equilibrium of this system is asymptotically stable if the zero solution of the corresponding linearized system is asymptotically stable. Moreover, these asymptotic stability conditions for equilibria of the system are described via location of all the characteristic roots in a specific area of the complex plane. In the planar case, these conditions are given even explicitly in terms of trace and determinant of the appropriate Jacobi matrix. The results are applied to a fractional predator-prey model and the fractional Lorenz model. Related experiments are supported by a numerical code that is appended as well [ABSTRACT FROM AUTHOR]

This paper investigates sampling dependent stability for aperiodic sampled-data systems by employing a Lyapunov-like functional that is time-dependent, and not imposed to be definite positive. Based on the system information on the sampling interval wholly rather than partly, a new Lyapunov-like functional is constructed, which extends existing ones by introducing the integral of the system state and the cross terms among this integral and the sampled state. To take advantage of the integral of the system state, integral equations of the sampled-data system are explored when estimating the derivative of the extended functional. By the Lyapunov-like functional theory, a new sampling dependent stability result is obtained for sampled-data systems without uncertainties. Then, the stability result is applied to sampled-data systems with polytopic uncertainties and a robust stability result is derived. At last, numerical examples are given to illustrate that the stability results improve over some existing ones. [ABSTRACT FROM AUTHOR]

This paper shows time-asymptotic nonlinear stability and existence of the viscous shock profiles to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which describes the motions of a conducting fluid in an electro-magnetohydrodynamics system. We can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the viscous shock profiles provided that the initial disturbance is small and of integral zero. [ABSTRACT FROM AUTHOR]

In this paper, we present a new solving approach for a class of multi-leader-follower games. For the problem studied, we firstly propose a neural network model. Then, based on Lyapunov and LaSalle theories, we prove that the trajectory of the neural network model can converge to the equilibrium point, which corresponds to the Nash equilibrium of the problem studied. The numerical results show that the proposed neural network approach is feasible to the problem studied. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of an electromechanical system is investigated. In this model, the phase of the system is a dynamical variable. The existence of four periodic orbits is proved. Two of them are asymptotically stable and the others are unstable. In the absence of viscous damping, when the parameters associated with the dimensionless voltages are adequately changed, each orbit is subject to a change of stability of the kind stable → unstable → stable with repetition of this pattern. This phenomenon is known as Sommerfeld effect. Moreover, the stability of these orbits depends on the zeros of the Bessel function J1. [ABSTRACT FROM AUTHOR]

This paper shows the existence and asymptotic Lyapunov stability of solutions with a moving inner layer (front) in a boundary value problem for a singularly perturbed parabolic reaction–advection–diffusion equation with the periodicity condition in time. In addition, the existence of solutions of this type for the corresponding initial boundary value problem is proved and a sufficient condition for their attraction to a periodic solution is proposed. For each problem, an asymptotic approximation of the solution is constructed and the existence and uniqueness theorems for such a solution with the constructed asymptotic behavior based on the asymptotic method of differential inequalities are proved. [ABSTRACT FROM AUTHOR]

In this paper we study stability and convergence of stage-based interpolation Runge–Kutta (SBIRK) methods in which the functional term is approximated by using the stage-values for nonlinear Volterra functional differential equations (VFDEs). This type of methods includes some popular Runge–Kutta methods, for example, some continuous Runge–Kutta methods, as special cases. The uniqueness of solution to discrete algebraic equation is first proved. Unconditional contractivity and asymptotic stability of this class of methods for nonlinear VFDEs are obtained under the most general framework. A global error bound on infinite integration interval for arbitrarily variable step-sizes is also derived for the first time. Some efficient SBIRK methods including Radau IA, Radau IIA and Lobatto IIIC Runge–Kutta methods with appropriate interpolation operator are provided. Numerical results illustrate the effectiveness of the proposed methods for VFDEs and verify our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper deals with a three-component parabolic system with density-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain. This system was used to describe the dynamics of a population of cells interacting with a chemoattractant and a nutrient. Based on the method of energy estimates and Moser iteration, we establish the global boundedness and the time-decay rates of the classical solutions under suitable conditions. This results perfect the corresponding results in Jin et al. (J Differ Equ 269:6758–6793, 2020). [ABSTRACT FROM AUTHOR]

In this paper, control design based on target tracking for a class of neutral systems is investigated by augmented error system approach. The dynamic equation of tracking error is derived by use of the desired output which is assumed to be known. Based on the main thought of preview control, the augmented error system is constructed. A criterion to guarantee the asymptotic stability of the corresponding nominal system is established. Through control design of the augmented error system, a delay-dependent control and a delay-independent control are respectively presented for the original neutral system. Control design in this paper contains integrators, which can effectively reduce static errors. Examples are given to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present a game theoretic framework for Cournot-Bertrand competition based on a nonlinear price function. The competition is between two firms and is assumed to take place in terms of pricing decision and quantity produced. However, the proposed objective function has not been used in literature before, yet the throughput obtained in this paper generalizes some of the existing results in literature. The competitive interaction between firms is described and analyzed using best-reply reaction, proposed adaptive adjustment and bounded rationality approach. The condition of stability of Nash equilibrium (NE) is induced by these approaches. Interestingly, we prove that there exists exactly a unique NE. Furthermore, it is noticed that when firms adopt best-reply and the proposed adaptive adjustment, the firms' strategic variables become asymptotically stable. On the contrary, when the bounded rationality is used both quantity and price behave chaotically due to bifurcation occurred. [ABSTRACT FROM AUTHOR]

The paper deals with the two-parameter matrix eigenvalue problems T j v j - λ 1 A j 1 v j - λ 2 A j 2 v j = 0 and T ~ j v ~ j - λ 1 A ~ j 1 v ~ j - λ 2 A ~ j 2 v ~ j = 0 , where λ j , λ ~ j ∈ C ; T j , A jk , T ~ j , A ~ jk (j , k = 1 , 2) are matrices. We derive the conditions, under which the problems have the same number of the eigenvalues in a given bounded domain. A two-parameter matrix eigenvalue problem is called a Schur–Cohn stable if its spectrum is inside the unit circle. It is said to be asymptotically stable if its spectrum is in the open left half-plane. As a consequence of the main result of the paper we obtain the tests for the Schur–Cohn and asymptotic stabilities. In addition, we suggest new localization results for the eigenvalues of the considered problems, which are sharp if the matrix coefficients are "close" to triangular ones. Our main tool is a new perturbation result for the determinants of linear matrix pencils. [ABSTRACT FROM AUTHOR]

This paper analyses the stability of cycles within a heteroclinic network formed by six cycles lying in a three-dimensional manifold, for a one-parameter model developed in the context of polymatrix replicator equations. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so-called projective map. The stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics. [ABSTRACT FROM AUTHOR]