Multiple duffing problem in a folding structure with hill-top bifurcation

This paper reviews the theoretical basis and its application for a multiple type of Duffing oscillation. This paper uses a suitable theoretical model to examine the structural instability of a folding truss which is limited so that only vertical displacements are possible for each nodal point supported by both sides. The equilibrium path in this ideal model has been found to have a type of “hill-top bifurcation” from the theoretical work of bifurcation analysis. Dynamic analysis allows for geometrical non-linearity based upon static bifurcation theory. We have found that a simple folding structure based on Multi-Folding-Microstructures theory is more interesting when there is a strange trajectory in multiple homo/hetero-clinic orbits than a well-known ordinary homoclinic orbit, as a model of an extended multiple degrees-of-freedom Duffing oscillation. We found that there are both globally and locally dynamic behaviours for a folding multi-layered truss which corresponds to the structure of the multiple homo/hetero-clinic orbits. This means the numerical solution depends on the dynamic behaviour of the system subjected to the forced cyclic loading such as folding or expanding action. The author suggests simplified theoretical models for hill-top bifurcation that help us to understand globally and locally dynamic behaviours, which depends on the static bifurcation problem. Such models are very useful for forecasting simulations of the extended Duffing oscillation model as essential and invariant nonlinear phenomena.     

Statistical cluster points and turnpike *

In this paper we study an asymptotic behaviour of optimal paths of a difference inclusion. The turnpike property in some wording [5,8, and so on] provided that there is a certain stationary point and optimal paths converge to that point. In this case only a finite number terms of the path (sequence) remain on the outside of every neighbourhood of that point

In the present paper a statistical cluster point introduced in [1] instead of the usual concept of limit point is considered and tue turnpiKe tueorem is proved, Mere it is es-ablished that there exists a stationary point which is a statistical cluster point for the all optimal paths. In this case not only a finite number but also infinite number terms of the path may remain on the outside of every small neighbourhood of the stationary point, but the number of these terms in comparison with the number of terms in the neighbourhood is so small that we can say:the path “almost” remains in this neighbourhood

Note that the main results are obtained under certain assumptions which are essentially weaker than the usual convexity assumption. These assumptions first were introduced for continuous systems in [6]     

小波Galerkin法在非线性分岔问题求解中的应用

通过一个典型的Bratu问题,研究了小波Galerkin法(WGM)在非线性分岔问题求解方面的应用.首先,利用基于Coiflet的小波Galerkin法,对一维和二维Bratu方程进行离散;然后针对单参数问题,推导了追踪解曲线的伪弧长格式和直接计算极值型分岔点的扩展方程;针对双参数问题,推导了追踪稳定边界的伪弧长格式和...     

Flow topology in an L-shaped cavity with lids moving in the same directions

Flow development and eddy structure in an L-shaped cavity with lids moving in the same directions have been investigated using both tools from topological and numerical methods. In particular, structural bifurcation near a nonsimple degenerate point is investigated by making a local analysis of the velocity field based on a Taylor series expansion. The streamlines of a Hamiltonian vector field system are simplified by using the homotopy invariance of the index theory. A series of bifurcation curves are constructed to determine the sequence of flow structures by which eddies are generated in the L-shaped cavity.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Bifurcation of periodic solutions of a system close to a Lyapunov system

Bifurcation of 2π-periodic solutions (2π-ps) of a system of second-order differential equations close to a Lyapunov system is investigated. The case of principal resonance, when an eigenfrequency of the linear oscillations of the unperturbed system is close to the frequency of the perturbing impulse, is considered. It is shown that, at certain values of the problem parameters, bifurcation of the 2π-ps that are generated from an equilibrium position, occurs. A constructive method is proposed for finding the bifurcation curve, as well as 2π-ps on it. The examples considered are bifurcation of 2π-ps in the problem of the oscillations of a mathematical pendulum with a horizontally vibrating suspension point, and in the problem of the planar oscillations of an artificial satellite in a weakly elliptical orbit. The bifurcation curves for these examples are constructed and the corresponding 2π-ps are found.     

Bifurcation in the neighbourhood of a non-isolated singular point

The Lyapunov-Schmidt method for bifurcation problems has, until recently, been applied only to operator equations whose singular points are isolated in the solution set of the equation. For bifurcation at a multiple eigenvalue involving several parameters, however, singular points are often non-isolated. In this paper, the case of intersecting curves of singular points is considered. Under natural hypotheses on these curves, and assuming suitable transversality conditions on the first order nonlinearity of the operator, it is shown that the solution set of the equation may be completely determined locally in terms of the solutions of associated finite dimensional polynomial equations.     

Bifurcations on contact manifolds

Consider a 1-parameter compactly supported family of Legendrian submanifolds of the 1-jet bundle of a compact manifold with its natural contact structure and a path of intersection points of the Legendrian family with the 1-jet of a constant function. Since the contact distribution is a symplectic vector bundle, it is possible to assign a Maslov-type index to the intersection path. We show that the non-vanishing of the Maslov intersection index implies that there exists at least one point of bifurcation from the given path of intersection points. This result can be viewed as a kind of analogue in bifurcation theory of the Arnold-Sandon conjecture on intersections of Legendrian submanifols. The proof is based on the technique of generating functions that relates the properties of Hamiltonian diffeomorphisms to the Morse theory of the associated functions.     

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.     

NONLINEAR MATRIX MODELS AND POPULATION DYNAMICS

Nonlinear matrix difference equations are studied as models for the discrete time dynamics of a population whose individual members have been categorized into a finite number of classes. The equations are treated with sufficient generality so as to include virtually any type of structuring of the population (the sole constraint is that all newborns lie in the same class) and any types of nonlinearities which arise from the density dependence of fertility rates, survival rates and transition probabilities between classes. The existence and stability of equilibrium class distribution vectors are studied by means of bifurcation theory techniques using a single composite, biologically meaningful quantity as a bifurcation parameter, namely the inherent net reproductive rate r. It is shown that, just as in the case of linear matrix equations, a global continuum of positive equilibria exists which bifurcates as a function of r from the zero equilibrium state at and only at r = 1. Furthermore the zero equilibrium loses stability as r is increased through 1. Unlike the linear case however, for which the bifurcation is “vertical” (i.e., equilibria exist only for r = 1), the nonlinear equation in general has positive equilibria for an interval of r values. Methods for studying the geometry of the continuum based upon the density dependence of the net reproductive rate at equilibrium are developed. With regard to stability, it is shown that in general the positive equilibria near the bifurcation point are stable if the bifurcation is to the right and unstable if it is to the left. Some further results and conjectures concerning stability are also given. The methods are illustrated by several examples involving nonlinear models of various types taken from the literature.     

Class field theory for curves over local fields

The purpose of this paper is to generalize some results of Bloch [3] concerning class field theory for curves over local fields. Bloch developed his theory mainly under the assumption that curves have good reduction. By using a different method, we shall develope a theory including the bad reduction case.     

Stability and Hopf bifurcation of a delayed predator-prey system with nonlocal competition and herd behaviour

In this paper, we investigate the stability and Hopf bifurcation of a diffusive predator-prey system with herd behaviour. The model is described by introducing both time delay and nonlocal prey intraspecific competition. Compared to the model without time delay, or without nonlocal competition, thanks to the together action of time delay and nonlocal competition, we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous. We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves. Furthermore, by the computation of normal forms for the system near equilibria, we investigate the stability and direction of Hopf bifurcation. Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.     

Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Bifurcations of fuzzy nonlinear dynamical systems

We present some recent developments of the fuzzy generalized cell mapping method (FGCM) in this paper. The topological property of the FGCM and its finite convergence of membership distribution vector are discussed. Powerful algorithms of digraphs are adopted for the analysis of topological properties of the FGCM systems. Bifurcations of fuzzy nonlinear dynamical systems are studied by using the FGCM method. A backward algorithm is introduced to study the unstable equilibrium solutions and their bifurcation. We have found that near the deterministic bifurcation point, the fuzzy system undergoes a complex transition as the control parameter varies. In this transition region, the steady state membership distribution is dependent on the initial condition. If we use the measure and topology of the α-cut (α = 1) of the steady state membership function of the persistent group representing the stable fuzzy equilibrium solution to characterize the fuzzy bifurcation, assuming the uniform initial condition within the persistent group, the bifurcation of the fuzzy dynamical system is then completed within an interval of the control parameter, rather than at a point as is the case of deterministic systems.     

Testing separability in marked multidimensional point processes with covariates

In modeling marked point processes, it is convenient to assume a separable or multiplicative form for the conditional intensity, as this assumption typically allows one to estimate each component of the model individually. Tests have been proposed in the simple marked point process case, to investigate whether the mark distribution is separable from the spatial–temporal characteristics of the point process. Here, we extend these tests to the case of a marked point process with covariates, and where one is interested in testing the separability of each of the covariates, as well as the mark and the coordinates of the point process. The extension is not at all trivial, and covariates must be treated in a fundamentally different way than marks and coordinates of the process, especially when the covariates are not uniformly distributed. An application is given to point process models for forecasting wildfire hazard in Los Angeles County, California, and solutions are proposed to the problem of how to proceed when the separability hypothesis is rejected.     

Collapse of Keldysh chains and stability boundaries of non‐conservative systems

Eigenvalue problems for non‐selfadjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of a multiple eigenvalue with the Keldysh chain of arbitrary length is investigated. Explicit expressions describing bifurcation of eigenvalues are found. The obtained formulae use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stability theory and sensitivity analysis of non‐conservative systems. Mechanical examples are considered and discussed in detail.     

A remark on multiobjective stochastic optimization via strongly convex functions

Many economic and financial applications lead (from the mathematical point of view) to deterministic optimization problems depending on a probability measure. These problems can be static (one stage), dynamic with finite (multistage) or infinite horizon, single objective or multiobjective. We focus on one-stage case in multiobjective setting. Evidently, well known results from the deterministic optimization theory can be employed in the case when the “underlying” probability measure is completely known. The assumption of a complete knowledge of the probability measure is fulfilled very seldom. Consequently, we have mostly to analyze the mathematical models on the data base to obtain a stochastic estimate of the corresponding “theoretical” characteristics. However, the investigation of these estimates has been done mostly in one-objective case. In this paper we focus on the investigation of the relationship between “characteristics” obtained on the base of complete knowledge of the probability measure and estimates obtained on the (above mentioned) data base, mostly in the multiobjective case. Consequently we obtain also the relationship between analysis (based on the data) of the economic process characteristics and “real” economic process. To this end the results of the deterministic multiobjective optimization theory and the results obtained for stochastic one objective problems will be employed.     

Adjoint lifts and modular endomorphism algebras

We prove that the ramification of the endomorphism algebra of the Grothendieck motive attached to a non-CM cuspform of weight two or more is completely determined by the slopes of the adjoint lift of this form, when the slopes are finite. We treat all places of good and bad reduction, answering a question of Ribet about the Brauer class of the endomorphism algebra in the finite slope case.     

An internal-variable theory of thermo-viscoelastic constitutive relations at finite strain

Based on the nonequilibrium thermodynamic theory, a new thermo-viscoelastic relation at finite strain is proposed. Under the assumption that the specific heat at a fixed strain and fixed internal variables can be regarded as a constant, a new expression for the free energy which decouples the mechanical and the thermal effects is derived. Through an analysis of the mesoscopic deformation mechanism of solid polymers, a set of internal variables is introduced, and an internal-variable constitutive theory in thermo-viscoelasticity at finite strain is formulated. An explicit expression of a thermoviscoelastic constitutive relation is obtained for solid polymers in the case where their molecular network has a randomly oriented distribution function at reference configuration. Moreover, the relationship between the relaxation time and the temperature is also discussed. The viscoelastic constitutive theory proposed in reference is only a linear approximation of the present theory.