Relative Equilibria in the Spherical,Finite Density Three-Body Problem

The relative equilibria for the spherical, finite density three-body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical five relative equilibria for the point-mass three-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways of these relative equilibria are mapped out as the angular momentum of the system is increased. This is done under the assumption that they have equal and constant densities and that the entire system rotates about its maximum moment of inertia. The transition to finite density greatly increases the number of relative equilibria in the three-body problem and ensures that minimum energy configurations exist for all values of angular momentum.     

Existence of the Broucke periodic orbit and its linear stability

In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Roberts? symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable.     

Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors

This paper considers the bilinear minimax control problem of an important class of parabolic systems with Robin boundary conditions. Such systems are linear on state variables when the control and disturbance are fixed, and linear on the control or disturbance when the state variables are fixed. The objective is to maintain target state variables by taking account the influence of noises in data, while a desired power level and adjustment costs are taken into consideration. Firstly we introduce some classes of bilinear systems and obtain the existence and the uniqueness of the solution, as well as stability under mild assumptions. Afterwards the minimax control problem is formulated. We show the existence of an optimal solution, and we also find necessary optimality conditions. Finally, to illustrate the abstract results, we present two examples of neutron fission systems.     

Stability diagram for 4D linear periodic systems with applications to homographic solutions

We consider a family of 4-dimensional Hamiltonian time-periodic linear systems depending on three parameters, λ₁, λ₂ and ε such that for ε=0 the system becomes autonomous. Using normal form techniques we study stability and bifurcations for ε>0 small enough. We pay special attention to the d'Alembert case. The results are applied to the study of the linear stability of homographic solutions of the planar three-body problem, for some homogeneous potential of degree −α, 0<α<2, including the Newtonian case.     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.     

Dynamics of product inhibition on lactic acid fermentation

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincare-Bendixson theorem, we obtain the condition for the globally asymptotical stability of the equilibria. Secondly, using the Floquet’s theorem and small-amplitude perturbation method, we obtain the biomass-free periodic solution is locally stable if R₂ < 1. The permanent conditions of the system are also given. Finally, our findings are confirmed by means of numerical simulations.     

The number and stability of limit cycles for planar piecewise linear systems of node–saddle type

The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles.     

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

This paper is concerned with the stabilization problem of Timoshenko beam in the presence of linear dissipative boundary feedback controls. Using C₀-semigroups theory we establish the existence and the uniqueness of solution of the proposed closed loop system. In order to consider the asymptotic behavior of the closed loop system, we first discuss the existence of nonzero solution of a closely related boundary value problem. Then we derive various necessary and sufficient conditions for the system to be asymptotically stable. Finally, we prove the equivalence between the exponential stability and the asymptotic stability for the closed loop system.     

Asymptotic behaviour of the stability parameter for a family of singular-limit Hill's equation

Under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular-limit Hill's equation which depends on three parameters. We use the blow-up techniques introduced in [R. Martínez, A. Samà, C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in R⁴. Applications, J. Differential Equations 226 (2006) 652-686]. The main contribution of this paper concerns the study of the nondegeneracy conditions. We give a geometrical interpretation of them, in terms of heteroclinic orbits for some related systems. In this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied. As a motivation and application we consider the vertical stability of homographic solutions in the three-body problem.     

On the number of limit cycles in general planar piecewise linear systems of node–node types

We investigate the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with node–node dynamics. Using the Liénard-like canonical form with seven parameters, some sufficient and necessary conditions for the existence of limit cycles are given by studying the fixed points of proper Poincaré maps. In particular, we prove the existence of at least two nested limit cycles and describe some parameter regions where two limit cycles exist. The main results are applied to the PWL Morris–Lecar neural model to determine the existence and stability of the limit cycles.     

Aspects of the planetary Birkhoff normal form

The discovery of the Birkhoff normal form around circular, co-planar motions for the planetary system opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a direct proof (after the proof in [18]) of the celebrated Arnold’s Theorem [5] on the stability of planetary motions. In this paper, after reviewing the story of the proof of this theorem, we focus on technical aspects of this normal form. We develop an asymptotic formula for it that may turn to be useful in applications. Then we provide two simple applications to the three-body problem: we prove that the “density” of the Kolmogorov set of the spatial three-body problem does not depend on eccentricities and the mutual inclination but depends only on the planets’ masses and the separation among semi-axes (going in the direction of an assertion by V. I.Arnold [5]) and, using Nehoro?ev Theory [33], we prove, in the planar case, stability of all planetary actions over exponentiallylong times, provided mean-motion resonances are excluded. We also briefly discuss difficulties for full generalization of the results in the paper.     

Sur les tissus plans de rang maximal et le problème de Chern

We explain how it is possible to obtain the necessary conditions on functions defining a germ of planar web in order that it be of maximal rank. Then we apply this method to the study of maximal rank 5-webs. We show the existence of many exceptional 5-webs non-equivalent to Bol's web, thus giving an answer to Chern's problem. To cite this article: L. Pirio, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

捕食者与食饵都染病的捕食-被捕食模型分析

建立并分析了一个捕食者和食饵都染病的捕食-被捕食模型,求得了它的非负平衡点.利用Hurwitz判据,用特征根的方法得到了边界平衡点局部渐近稳定的充分条件.进一步利用LaSalle不变性原理获得了正平衡点全局渐近稳定的充分条件.     

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the Lp-Lq estimates.     

Generalizations and applications of the nowhere zero linear mappings in network coding

In [2], Alon and Tarsi proposed a conjecture about the nowhere-zero point in linear mappings. In this paper, we first study some generalizations of this problem, and obtain necessary and sufficient conditions for the existence of nowhere point in these generalized problems under the assumption |F|?n+2, where n is the number of rows of the matrix A. Then we apply the results in these generalizations to give a polynomial time algebraic construction of the acyclic network codings.     

Equilibria and global dynamics of a problem with bifurcation from infinity

We consider a parabolic equation ut−Δu+u=0 with nonlinear boundary conditions , where as |s|→∞. In [J.M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity, Proc. Roy. Soc. Edinburgh Sect. A 137 (2) (2007) 225-252] the authors proved the existence of unbounded branches of equilibria for λ close to a Steklov eigenvalue of odd multiplicity. In this work, we characterize the stability of such equilibria and analyze several features of the bifurcating branches. We also investigate several question related to the global dynamical properties of the system for different values of the parameter, including the behavior of the attractor of the system when the parameter crosses the first Steklov eigenvalue and the existence of extremal equilibria. We include Appendix A where we prove a uniform antimaximum principle and several results related to the spectral behavior when the potential at the boundary is perturbed.     

Complex cauchy problem in the class of analytic functions with integral metric

We study the complex Cauchy problem for a system of linear differential equations in the class of analytic functions with integral metric. In the case of a Hardy-Lebesgue type weighted L _p -space, we obtain necessary and sufficient conditions for the local solvability of the problem.     

Some homoclinic phenomena in the three-body problem

We prove the existence of transversal homoclinic points in the collinear three-body problem, restricted and general, and in the planar circular restricted three-body problem. As a consequence the shift of Bernoulli is proved to be included as a subsystem of a suitable section of the flow for the three cases studied. Then the existence of all the possible types of final evolution follows.     

A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

     

具有食饵避难的Leslie-Gower捕食系统最优收获分析

研究了一类具有食饵避难的Leslie-Gower捕食与被捕食系统收获模型,利用Hurwitz判据,得到了正平衡点局部渐近稳定,进一步构造了适当的Lyapunov函数,证明了正平衡点的全局渐近稳定性.并且在捕获努力量假说下,对发生食饵避难的两种群同时捕获,考虑了生态经济平衡点的存在性和利用Pontryagin最大值原理对两种群进行最优收获,得到当贴现率为零时,既保持了生态平衡,又使得在渔业开发过程中取得最大经济利益.