The derivatives of equilibrium states

We find some estimates for the derivatives of equilibrium states of subshifts of finite type. We prove the differentiability (with respect to the potential) of integrals of certain discontinuous functions for the equilibrium state of a potential.research supported by CNPq, Brazil     

Principal eigenvalues,topological pressure,and stochastic stability of equilibrium states

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ^ε (V) for the operatorL ^ε = ɛL + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 0 from Donsker-Varadhan’s variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems. This work was supported by US-Israel Binational Science Foundation.     

Existence,uniqueness and stability of equilibrium states for non-uniformly expanding maps

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact metric spaces and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.     

Convergence to equilibrium states for fluid models of many-server queues with abandonment

Fluid models, in particular their equilibrium states, have become an important tool for the study of many-server queues with general service and patience time distributions. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under some conditions. Our method builds on the framework of measure-valued processes, which keeps track of the remaining patience and service times.     

The stability of equilibrium in systems with friction

Mechanical systems with non-ideal geometrical constraints are considered. The possible lack of uniqueness of the solution of the problem of determining the generalized accelerations and reactions with respect to specified coordinates and velocities is taken into account in solving the problem of the stability of an equilibrium state. A number of necessary and sufficient conditions of stability are obtained. It is shown that the results are also applicable in the case of unilateral constraints subject to the condition that a specific hypothesis concerning the character of the impacts on the constraints is adopted. A problem on the stability of a rigid body on a rough plane in the two-dimensional case is solved as an example.     

The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ. The research of M. Urbański was supported in part by the NSF Grant DMS 0400481. H. Sumi thanks University of North Texas for kind hospitality, during his stay there.     

带有分段常数变量和避难所的天敌一害虫模型的稳定性和分支行为

研究一类带有分段常数变量和避难所的天敌-害虫模型的稳定性和分支行为.首先通过计算转化得到天敌-害虫模型对应的差分模型,利用线性稳定性理论讨论了正平衡态局部渐近稳定的充分条件.其次以害虫种群的内禀增长率或逃脱率为分支参数,利用分支理论研究了模型正平衡态处产生翻转分支周期解和Neimark-Sacker分支周期解的充分条件;并且使用正规形理论和中心流形定理构造了判断分支周期解稳定性的阈值.最后数值模拟验证了理论分析的正确性,并展示了该模型复杂的动力学行为.     

The stability of the equilibrium of two-phase elastic solids

The distinctive features of the loss of stability of elastic solids which undergo phase transitions are investigated for the case of small deformations. The non-uniqueness of the solution of the boundary-value problem for the describing of the thermodynamic equilibrium of a two-phase body is caused by the non-linearity associated with the unknown interface. The solution can be chosen by comparing the potential energies of the body in the two-phase and single phase states and by analysing of the local stability of the two-phase states. A linearized boundary-value problem is formulated which describes infinitesimal small perturbations of an initial two-phase state which is in thermodynamic equilibrium. Analysis of the stability of the two-phase state reduces to an investigation of the bifurcation points and the behaviour of the small solutions of the system of integrodifferential equations in terms of functions describing the perturbations of the interface. The problem of the non-uniqueness and loss of stability of centrisymmetric equilibrium two-phase deformations is investigated as an example. A theorem concerning the number of centrisymmetric solutions is proved. The energy changes accompanying the formation and development of two-phase states and the stability of the solutions obtained are investigated. The concept of topological instability as a bifurcation is introduced, as a result of which the type of geometry of a solution of the boundary-value problem changes and surfaces of separation of the phases actually appear and disappear. Macrodiagrams of the deformational are constructed which demonstrate the effect of deformation softening in the path of a phase transition.     

The stability of the equilibrium of holonomic conservative systems

The problem of the stability of the equilibrium positions and steady motions of holonomic conservative systems has been fairly completely treated in a number of reviews [44,58,9]. However, investigations are continuing in this field and a number of new important results have recently been obtained (in 1982–1992). This review analyses these results and compares them with previous ones.     

Nonuniqueness of equilibrium states for bars in tension

This paper presents an infinite linear program with the optimal value of the maximizing problem strictly greater than the optimal value of the minimizing problem.     

The stability of the equilibrium position of scleronomous mechanical systems

The problem of the stability of the equilibrium position of a scleronomic mechanical system is considered. The comparison method enables this problem to be reduced to the problem of the stability of scalar differential equations. The stability conditions are found for certain types of scalar comparison equations (Sections 1–4), and the sufficient conditions for the stability of the equilibrium positions of various scleronomous mechanical systems are determined from these (Sections 5–9).     

The stability of the equilibrium ellipsoids of a rotating liquid

The stability of the Riemann equilibrium ellipsoids, in the class of perturbations which satisfy the Dirichlet assumptions, is investigated using Rumyantsev's method [1]. It is shown that the equations of motion of the Dirichlet liquid ellipsoid are Hamiltonian on each combined level of the moment and circulation integrals, which corresponds to well-known results [2], although it is not a consequence of them. This fact provides, generally speaking, additional possibilities for solving the problem of determining the instability region. In the parameter space of each of the two families [3, 4] of Riemann ellipsoids, the region U for which almost all the equilibrium ellipsoids belonging to it are unstable is determined in explicit analytical form. It is shown that the stability region can be specified in explicit analytical form in both cases.     

On stability of perfect equilibrium points

We consider stability of Selten's perfect equilibrium point against slight imperfections of rationality of players. As its stability is not sufficient, we strengthen the perfectness concept and define astrictly perfect equilibrium point. We provide sufficient conditions for this equilibrium point.