Liapounov Stability Research Articles

Stability and dynamics of a plastic softening oscillator

This paper deals with the stability of a single-degree-of-freedom plastic softening oscillator. Understanding such an elementary model concerns, for instance, the seismic behaviour of concrete or steel structures. The associated dynamic system is a complex hysteretic system. Using appropriate internal variables, it can be written as a singular autonomous system. Liapounov stability of the solutions is then studied. A domain of perturbations associated with a stable solution is exhibited. This domain looks like a truncated cone in the three-dimensional phase space. It can be read as a critical displacement or energy that the oscillator can support during a seismic excitation. The difference with the “equivalent” linearized elastic system is highlighted. The unloading part of the response of the inelastic system has a stabilising effect.

Stabilité et dynamique d'un oscillateur endommageable

ABSTRACT This paper deals with stability of a simple damage oscillator, loaded by a constant traction force. Understanding such an elementary model concerns the seismic behavior of concrete structures. The associated dynamic system is a complex hysteretic system: very few results are available in the literature. Using appropriate internal variables, the dynamic system can be written as a singular autonomous system. Liapounov stability of the numerous solutions is then studied. The differences with the “equivalent” non-linear elastic system are highlighted. The unloading part of the anelastic system has a stabilizing effect.

Stabilité et dynamique d'un oscillateur endommageable

ABSTRACT This paper deals with stability of a simple damage oscillator, loaded by a constant traction force. Understanding such an elementary model concerns the seismic behavior of concrete structures. The associated dynamic system is a complex hysteretic system: very few results are available in the literature. Using appropriate internal variables, the dynamic system can be written as a singular autonomous system. Liapounov stability of the numerous solutions is then studied. The differences with the “equivalent” non-linear elastic system are highlighted. The unloading part of the anelastic system has a stabilizing effect.

A New Characterization of M-Matrices and H-Matrices

This paper deals with the characterization of M-matrices and H-matrices, with positive diagonal entries, in term of strongly accretive matrices. The applications concern on the one hand the convergence analysis of parallel asynchronous Schwarz alternating method, applied to the solution of nonlinear boundary value problems and on the other hand the Liapounov stability analysis for large scale evolution systems.

On the Dual Stability of a Von Neumann Facet and the Inefficacy of Temporary Fiscal Policy

This study establishes two main results in a dynamic general equilibrium model. The first is to demonstrate the dual Liapounov stability of a von Neumann facet without the restrictive assumptions on the structure of underlying technologies that are commonly adopted in the optimal growth literature. The second result is to demonstrate that a temporary change in fiscal policy has almost no effect on present and future consumption. While such inefficacy of temporary fiscal policy has been discussed in the context of the permanent income hypothesis, in this study, it is proved in the dynamic general equilibrium framework under a set of basic assumptions of general equilibrium theory.

Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on [formula omitted]N—III. Center curves and liapounov stability

Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on [formula omitted]N—III. Center curves and liapounov stability

On nonlinear stability of stationary Euler flows on a rotating sphere

We consider Euler fluids on a rotating sphere and prove the Liapounov stability for a class of stationary, eventually nonsmooth flows. In particular, in the smooth case, we adapt to our context the classical Arnold method.

Variational version of thermal conduction fluctuations in the context of the most probable route

An expression for the square-law fluctuation of internal energy is obtained in the framework of the local equilibrium approach with the aid of the non-equilibrium distribution function presented by Zubarev (1974). It is shown that this fluctuation satisfies the necessary conditions for its use as Liapounov's function. It is demonstrated that Liapounov's stability requirement leads to a fluctuating functional which has a minimum for the hydrodynamic stage. The infinitesimality estimates of the right parts are found for two balance equations: energy and entropy. It is shown that the local potential of Prigogine and Glansdorff (1971) can be derived from perturbed fluctuating functional obtained here. With the aid of the well known formula of the Onsager-Machlup postulate for the density probability transition of a thermodynamic system from some non-equilibrium state, the path integrals are obtained for the evolution of the system in the hydrodynamic stage.

A primal route to the Turnpike and Liapounov stability

A primal route to the Turnpike and Liapounov stability

Nonlinear Oscillations

Part 1 The mathematical pendulum as an illustration of linear and non-linear oscillations - systems which are similar to a simple linear oscillator: Undamped free oscillations of the pendulum damped free oscillations forced oscillations. Part 2 Liapounov stability theory and bifurcations: The concept of Liapounov stability the direct method of Liapounov stability by the first approximation the Poincare map the critical case of a conjugate pair of eigenvalues simple bifurcation of equilibria and the Hopf bifurcation. Part 3: Self-excited oscillations in mechanical and electrical systems analytical approximation methods for the computation of self-excited oscillations analytical criteria for the existence of limit cycles forced oscillations in self-excited systems self-excited oscillations in systems with several degrees of freedom Part 4 Hamiltonian systems: Hamiltonian differential equations in mechanics canonical transformations the Hamilton-Jacobi differential equation canonical transformations and the motion perturbation theory Part 5 Introduction to the theory of optimal control: Control problems, controllability the Pontryagin maximum principle transversality conditions and problems with target sets canonical perturbation theory in optimal control.

Detailed balance, Liapounov stability, and entropy in resonance fluorescence

Resonance fluorescence from a two-level atom is investigated as a problem in nonequilibrium statistical mechanics. The dynamical properties of the atomic steady state are investigated through the principle of detailed balance for open quantum Markovian systems. The saturated atom is shown to be described by a set of equations which obey detailed balance. A Liapounov function is found which guarantees the stability of the atomic steady state for arbitrary values of the driving and damping parameters. Two other Liapounov functions are introduced which are similar in form to the Boltzmann-Gibbs and Prigogine et al. entropy functionals and which reduce to these entropies in the saturation limit. Thus, it is shown that either entropy definition yields an $H$ theorem for the open atomic system where detailed balance obtains.

Conservation laws and Liapounov stability of the free rotation of a rigid body

The paper derives the well known stabilities of free rotation of a rigid body about its principal axes of least and greatest moments of inertia directly from the constancy of the kinetic energy and of the square of the angular momentum. The resulting proof of Liapounov stability yields new quantitative measures of this stability. Involving only simple algebra, it depends on satisfying a weak sufficient condition that insures an unchanging sign of the main component of the angular velocity ω. The method cannot be used, however, to prove the well known instability of rotation about the intermediate axis.

Stability of the traction boundary value problem in linear elastodynamics

It is shown that, provided certain rigid body motions are excluded, a positive definite strain energy is sufficient for the Liapounov stability of the solution to the title problem.