Stability Of Trivial Equilibrium Research Articles

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

Spatial Patterns and Bifurcation Analysis of a Diffusive Tumour-immune Model

In this paper, a diffusive tumour-immune model is presented. By comparing the effect of Neumann boundary conditions and Dirichlet boundary conditions on the stability of trivial equilibrium, we derive that the former can provide more mechanisms for spatial pattern formation of the model. By taking the diffusion rate of tumour cells as a parameter, we first give the local and global steady-state bifurcations emitting from the positive equilibrium of the model. Then the stability of the bifurcation solution is discussed by computing the second derivative of an appropriate function, which is different from the general case. Furthermore, numerical simulations provide an indication of the wealth of patterns that the system can exhibit. In particular, periodic oscillation and spot-like patterns can be observed in one-dimensional and two-dimensional simulations, respectively. All results obtained reveal the mechanism of interaction between tumour cells and immune system, which have profound significance for the development of tumour immunotherapy.

Potential forces and alternation of stability character in non-conservative systems

The influence of non-conservative forces (both positional and velocity-dependent) upon stability of equilibrium positions of mechanical systems has been intensely discussed for many years. In particular, it is well known that even small dissipative forces can cause instability. In the present work, the influence of potential forces upon stability of trivial equilibrium in 2DoF nonconservative systems is studied. Conditions are found when the increase in stiffness corresponding to one of generalized coordinates results in multiple (up to three) changes of stability character. This effect is illustrated by the example of an aeroelastic system with two translational degrees of freedom.

Change in the Character of Stability of Equilibrium in the Case of a Change in the Rigidity of a Generalize Coordinate

A linear mechanical system with n degrees of freedom is considered. The change in the character of stability of trivial equilibrium is studied in the case when the rigidity in one of the generalized coordinates changes. The maximum number of such alternations of a stability character is determined. An example of such a system is given in which the maximum amount of changes in the character of the stability occurs.

Global Stability of a Non-Smooth Predator–Prey System with Holling I Functional Response and Refuge Effect

This paper analyses the dynamics of a non-smooth predator-prey model with refuge effect, where the functional response is taken as Holling I type. To begin with, some preliminaries and the existence of regular, virtual, pseudo-equilibrium and tangent point are established. Then, the stability of trivial equilibrium and predator free equilibrium is discussed. Furthermore, it is shown that the regular equilibrium and the pseudo-equilibrium cannot coexist. Finally, the conclusion is given.

Asymptotic Stability of Impulsive Cellular Neural Networks with Infinite Delays via Fixed Point Theory

We employ the new method of fixed point theory to study the stability of a class of impulsive cellular neural networks with infinite delays. Some novel and concise sufficient conditions are presented ensuring the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. These conditions are easily checked and do not require the boundedness and differentiability of delays.

Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.

Delay-induced dynamics of an axially moving string with direct time-delayed velocity feedback

The local dynamics of an axially moving string under aerodynamic forces is investigated with a time-delayed velocity feedback controller. The retarded differential difference governing equation is obtained in modal coordinates of a two-degree-of-freedom system through Galerkin’s discretization procedure. The stability of trivial equilibrium is examined with the change of counting multiplicity of eigenvalue with positive real part. The Hopf bifurcation curves are determined in the controlling parameter spaces. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation of one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. The curves of excitation–response and frequency–response curves are shown with the effect of time delay. The stability analysis for steady-state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression. Moreover, the Poincaré–Bendixson theorem and energy considerations are used to investigate the existences and characteristics of quasi-periodic solutions when stability of the periodic solution is lost. Numerical results demonstrate the validity of the analytical prediction. Two different kinds of quasi-periodic solutions are found.

GLOBAL DYNAMIC PROPERTIES OF A SYNCHRONOUS MACHINE MODEL

In this paper, rich global dynamical properties are considered for a fundamental synchronous machine model: from global asymptotic stability of trivial equilibrium to chaotic oscillations. Based on Kalman–Yakubovich–Popov (KYP) lemma, static feedback controller is designed such that the synchronous machine model is global asymptotic stable which guarantees that the existence of limit cycles or chaotic oscillations is impossible.

ASYMPTOTIC SYNCHRONIZATION IN LATTICES OF COUPLED NONIDENTICAL LORENZ EQUATIONS

In this paper we study coupled nonidentical Lorenz equations with three different boundary conditions. Coupling rules and boundary conditions play essential roles in the qualitative analysis of solutions of coupled systems. By using Lyapunov stability theory, a sufficient condition is obtained for the global stability of trivial equilibrium of coupled system with Dirichlet condition. Then we restrict our attention on the dynamics of coupled nonidentical Lorenz equations with Neumann/periodic boundary condition and prove that the asymptotic synchronization occurs provided the coupling strengths are sufficiently large. That is, the difference between any two components of solution is bounded by the quantity O(ε/ max {c1, c2, c3}) as t → ∞, where ε is the maximal deviation of parameters of nonidentical Lorenz equations, and c1, c2 and c3 are the specified coupling strengths.