We consider the stability of synchronized states (including equilibrium point, periodic orbit, or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on the master stability function and Gershgörin disk theory, to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics. S...

For some one-dimensional discrete-time autonomous population models, local stability implies global stability of the positive equilibrium point. One of the known techniques is the enveloping method. In this paper, we present a survey on the enveloping method to study global stability of single periodic population models. In the other words, we present the conditions in which “individual envelop...

In this paper, we consider the global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new exponential stability criteria of the system are derived by using the Lyapunov functional method and the linear matrix inequality approach for estimating the upper bound of the derivative of Lyapunov functional. Finally, we illustrate ...

Liquid metal wall concept has received a great deal of attention recently because of its perceived advantages in addressing high heat flux, magnetohydrodynamic (MHD) stability, and other related issues in advanced confinement schemes. Despite its inherent and clear benefits, this concept also poses potentially serious problems from the MHD equilibrium and stability point of view. In particular,...

In this paper, a fractional-order model for the spread of computer viruses with saturated treatment rate is presented. This model consists of three components: Susceptible, Infectious, and Recovered. The SIR epidemic model with saturated treatment rate is considered. The stability of the disease free and endemic equilibrium points are studied. The global stability of the disease free equilibriu...

In this paper we focus on a generalized Hénon–Heiles system in a rotating reference frame, in such a way that Lagrangian-like equilibrium points appear. Our goal is to study their nonlinear stability properties to better understand the dynamics around these points. We show the conditions on the free parameters to have stability and we prove the superstable character of the origin for the classi...

In this paper, a discrete-time food chain characterized by three species is modeled by a system of three nonlinear difference equations. The existence and local stability of the equilibrium points of the discrete dynamical system are analyzed. It is shown that for some parameter values the interior equilibrium point loses its stability through a discrete Hopf bifurcation. Basic properties of th...

In this lecture we introduce a useful tool to study the robustness against disturbances of nonlinear systems. Our starting point is the observation that, given a globally asymptotically stable equilibrium, a small enough perturbation of the vector field should not change the phase portrait much, and hence should not destroy the stability properties. However, how large can the perturbations be b...

The majority of slope stability analyses performed in practice still use traditional limit equilibrium approaches involving methods of slices that have remained essentially unchanged for decades. This was not the outcome envisaged when Whitman & Bailey (1967) set criteria for the then emerging methods to become readily accessible to all engineers. The ®nite element method represents a powerful ...

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium ...