Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.     

Viscoelastic properties of a silica-filled styrene-butadiene rubber under uniaxial tension

A model composite — a silica-filled styrene-butadiene rubber with a various filler volume content — was tested for creep and creep recovery at different tensile load levels to evaluate the effect of viscoelasticity on the deformational properties of filled rubbers. A constitutive equation describing the diagram of equilibrium deformation of the composite in quasi-static loading was obtained from an analysis of creep test results. The equation was common for the filled rubber at different filler content. The existence of such a curve has been confirmed by experimental unloading diagrams registered in cyclic loading-unloading tests. It is shown that the phenomenological equations obtained from an analysis of creep recovery test results can be used successfully for describing the hysteresis loops of second and subsequent cycles for cyclic tests with a constant maximum stretch ratio.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

一类SIRS传染病模型

This paper considers an SIRS epidemic model that incorporates constant immigration rate, a general population-size dependent contact rate and proportional transfer rate from the infective class to susceptible class. A threshold parameter a is identified. If σ≤1, the disease-free equilibrium is globally stable. If σ＞1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence,global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.     

The stability of the relative equilibrium in an Earth–Moon orbital station system with a small reactive acceleration

The problem of stabilizing the relative equilibrium of an orbital station in an Earth–Moon system by imparting a small constant-modulus acceleration with constant orientation of its vector with respect to the body of the station, which is assumed to be a rigid body of variable mass, is considered. It is shown that, in the case of a small displacement of the centre of mass of the station (by means of a small reactive acceleration) with respect to the collinear libration point beyond the Moon, its relative equilibrium position can become stable by virtue of the equations of the first approximation.     

Weak equilibrium in a spatial model

Spatial models of two-player competition in spaces with more than one dimension almost never have pure-strategy Nash equilibria, and the study of the equilibrium positions, if they exist, yields a disappointing result: the two players must choose the same position to achieve equilibrium. In this work, a discrete game is proposed in which the existence of Nash equilibria is studied using a geometric argument. This includes a definition of equilibrium which is weaker than the classical one to avoid the uniqueness of the equilibrium position. As a result, a “region of equilibrium” appears, which can be located by geometric methods. In this area, the players can move around in an “almost-equilibrium” situation and do not necessarily have to adopt the same position.     

The Transient Motion of a Partially Immersed Rolling Strip in Water

An infinitely long thin strip is partly immersed in deep fluidand is fixed along its submerged end. It is given a small disturbancefrom its equilibrium position firstly by an applied rotationalforce and then by an initial angular displacement, the fluidcoming to rest before the strip, subject to a linear restoringforce, is released. Under the assumptions that viscosity andsurface tension may be neglected and the equations of motionlinearized, the ensuing two-dimensional displacement is describedby functions involving a Fourier integral. These are computedand the transient motion found. It is shown that after an initialstage the behaviour of the body is approximated by a dampedharmonic oscillatory motion which cannot be represented by asecond order differential equation with constant coefficients.This conflicts with the theory used by engineers and naval architectsin ship hydrodynamics. Ultimately the decay of motion is monotonic,decaying like t^–7 when an initial force is applied andt^–6 when there is an initial displacement. Comparisonis made with the corresponding behaviour of the undamped system.     

Stability of a Hamiltonian system in a limiting case

We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the 1: −1 resonance case where the linearized system has double pure imaginary eigenvalues ±iω, ω ≠ 0 and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.     

Optimal Impedance Load of a Bistable Energy Harvester

In this contribution we investigate a bistable energy harvester with regard to its optimal impedance load. A bistable energy harvester exhibits three different types of oscillation: Single-well (about a stable equilibrium), cross-well (between the wells) and inter-well (about the unstable equilibrium). The occurring oscillation type depends, for instance, on the excitation parameters and the initial conditions. It has already been observed ( [1]) that the optimal impedance, which allows to maximize the power output, varies for each oscillation type. In our investigations we complement these findings with analytical and numerical calculations. For our analysis we examine the non-dimensionalized coupled equations of a bistable energy harvester. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Bogdanov–Takens bifurcation in an oscillator with negative damping and delayed position feedback

In this paper, Bogdanov–Takens bifurcation occurring in an oscillator with negative damping and delayed position feedback is investigated. By using center manifold reduction and normal form theory, dynamical classification near Bogdanov–Takens point can be completely figured out in terms of the second and third derivatives of delayed feedback term evaluated at the zero equilibrium. The obtained normal form and numerical simulations show that multistability, heteroclinic orbits, stable double homoclinic orbits, large amplitude periodic oscillation, and subcritical Hopf bifurcation occur in an oscillator with negative damping and delayed position feedback. The results indicate that negative damping and delayed position feedback can make the system produce more complicated dynamics.     

Optimization incentive and relative riskiness in experimental stag-hunt games

We compare the experimental results of three stag-hunt games. In contrast to Battalio et al. (Econometrica 69:749–764, 2001), our design keeps the riskiness ratio of the two strategies at a constant level as the optimization premium is increased. We define the riskiness ratio as the relative payoff range of the two strategies. We find that decreasing the riskiness ratio while keeping the optimization premium constant decreases sharply the frequency of the payoff-dominant equilibrium strategy. On the other hand an increase of the optimization premium with a constant riskiness ratio has no effect on the choice frequencies. Finally, we confirm the dynamic properties found by Battalio et al. that increasing the optimization premium favours best-response and sensitivity to the history of play.     

TWO DIFFERENTIAL INFECTIVITY EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.     

The existence theorem and exact solutions of a variational problem on high-temperature phase transition with zero surface tension

A model of a two-phase elastic medium with classical energy density is considered. In the case under consideration the energy functional depends on the temperature, which is assumed to be constant along the whole body. The question on the equilibrium state for σ=0 is studied, and a family of exact solutions to the problem is constructed. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 201–212.     

On improving the Hashin-Shtrikman bounds for the effective properties of three-phase composite media

For a composite comprising an isotropic mixture of two isotropicdielectric materials, the Hashin-Shtrikman bounds for the overalldielectric constant tensor are attainable and hence are thebest possible. Considering instead a three-phase composite,the Hashin-Shtrikman bounds are the best that are known in termsof volume fractions alone, and yet, in the limit of vanishingvolume fraction of the material of greatest dielectric constant,the three-phase upper bound remains strictly greater than thetwo-phase bound. A similar comment applies to the lower bound,in relation to a small volume fraction of the material withthe smallest dielectric constant. Although this phenomenon mayreflect a limitation of the Hashin-Shtrikman methodology, itremains conceivable that some microgeometries exist for whichall the ‘third’ phase is positioned in regions ofhigh field concentration, so that it always has a large effect.This paper resolves this problem to some extent, by generatinga new upper bound that ranges continuously from the Hashin-Shtrikmantwo-phase bound to the Hashin-Shtrikman three-phase bound asthe volume fraction c₃ of the ‘third’ material increasesfrom zero. The Hashin-Shtrikman three-phase bound thus cannotbe optimal, at least when c₃ is small. The method of derivationof the new bound relies on an application of the theory of functionsof bounded mean oscillation, recently developed in the contextof bounding the behaviour of nonlinear composites.     

Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method

This paper deals with the study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. The paper considers a micro-beam suspended between two conductive micro-plates, subjected to a same actuation voltage. The nonlinear governing differential equation of motion about static equilibrium position using calculus of variation theory and Taylor series expansion has been linearized and implementing a Galerkin based reduced order model a Mathieu type equation has been obtained. By improving variational iteration method combining with method of strained parameters transition curves, separating stable from unstable regions have been obtained. The results of variational iteration method, perturbation and direct numerical integration methods for some cases selected from different regions (stable and unstable regions) have been compared.     

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.