On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat

We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.     

The equilibrium positions of an ellipsoid with a cavity containing a liquid on a horizontal plane

The equilibrium positions of an ellipsoid with an ellipsoidal cavity, partially filled with an ideal incompressible liquid, on a horizontal plane in a uniform gravitational field are considered. All trivial and non-trivial equilibrium positions are found and the conditions for their stability are obtained. The results are presented in the form of bifurcation diagrams.     

Enhanced and reduced natural convection using magnetic fluid in a square cavity

Natural convection using a magnetic fluid was studied in a square cavity under the influence of a permanent magnet. The aim was to explore the degree by which heat transfer may be controlled, enhanced or reduced, by investigating a set of different distances of a permanent magnet to the cavity. These distances of the magnet were set such that the cavity was in some cases fully dominated by buoyancy or by the magnetic body force and in other cases partly dominated by either of both body forces in different parts of the fluid. The effect on heat transfer was characterised by an averaged Nusselt number, Rayleigh and magnetic Rayleigh number. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a Generalized Sobolev System

The linear stability problem of the rotational motion of a rigid body around a fixed point containing an inner cavity filled up with an ideal fluid is considered. In this paper, we also assume that the fluid is rotating. The effect of the angular velocities of the rigid body and the fluid in the stability problem is studied. The case of a cavity ellipsoidal is presented in detail.     

The steady motions of a system of two elastically coupled bodies

The problem of the existence, branching and stability of the steady motions of a system of two elastically coupled bodies in a central gravitational field is considered. Each body is simulated by a weightless rod with point masses at opposite ends. It is assumed that the rods are essentially attached at their mass centres, and the composite body is moving in a plane containing the attracting centre. Both trivial and non-trivial steady motions are studied, on the assumption that none of the principal axes of inertia of the body coincides with the radius vector of the centre of mass or with a tangent to the orbit; it is also assumed that the rods are not orthogonal to one another. The stability of all steady motions is fully investigated and an atlas of bifurcation diagrams presented.     

The stability of the rotation of a heavy body with a viscous filling

The stability of the permanent rotation of a symmetrical heavy body with a viscous filling is investigated. A finite-dimensional phenomenological model of the “internal friction” with which the filling acts on the wall of the cavity is constructed based on the Helmholtz equations for a vortex. The boundaries of the stability limit are constructed and the interaction between the internal friction and the external damping is tracked. It is shown that the cases of a cavity that is oblate and prolate along the axis of rotation lead to the existence of different forms of stability regions.     

Analysis of Dynamics in a Complex Food Chain with Ratio-Dependent Functional Response

In this paper, we study a new model obtained as an extension of athree-species food chain model with ratio-dependent functional response. We provide non-persistence and permanence results and investigate the stability of all possible equilibria in relation tothe ecological parameters. Results are obtained for the trivial andprey-only equilibria where the singularity of the model prevents linearization, and the remaining semi-trivial equilibria are studiedusing linearization. We provide a detailed analysis of conditionsfor existence, uniqueness, and multiplicity of coexistencee quilibria, as well as permanent effect for all species. The complexity of the dynamics in this model is theoretically discussedand graphically demonstrated through various examples and numerical simulations.     

Dynamic behaviors of the periodic predator–prey system with distributed time delays and impulsive effect

In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].     

Invariant sets in the Goryachev–Chaplygin problem: existence,stability and branching

The existence, stability and branching of invariant sets in the problem of the motion of a heavy rigid body with a fixed point, which satisfies the Goryachev–Chaplygin conditions, are discussed. Both trivial invariant sets, in which the pendulum-like motions of a Goryachev–Chaplygin spinning top lie, as well as non-trivial invariant sets, in which the motion of the top is described by elliptic functions of time, are investigated.     

The permanent rotations of a balanced non-autonomous gyrostat

The permanent rotations of a gyrostat about its fixed centre of gravity are investigated. It is assumed that the lines of action of the time-dependent gyrostatic momentum vector maintain a constant position in a reference system attached to the carrier body. It is shown that, if the total angular momentum of the gyrostat is non-zero, permanent rotations can only occur about its principal axes of inertia. In that case the gyrostatic momentum vector must be collinear with one of the principal axes of inertia of the gyrostat.     

Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect

In this paper, a predator–prey system which based on a modified version of the Leslie–Gower scheme and Holling-type II scheme with impulsive effect are investigated, where all the parameters of the system are time-dependent periodic functions. By using Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We use standard bifurcation theory to show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. As an application, we also examine some special case of the system to confirm our main results.     

System of Laplace equations in a half-space with a free interface,capillary-gravitational waves,bifurcation, and symmetry

We study the stability of branching solutions of a system of two nonlinearly perturbed Laplace equations in a half-space with two differential relations on the interface. This system describes the motion of a two-layer fluid. To construct and study the related branching systems, methods of group analysis of differential equations (RZhMat 1978 11B883K, RZhMat 1983 11A813K) and the S. Lie-L.V. Ovsyannikov technique of invariants and invariant manifolds are used.     

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.     

Linear problems of the stability of a type of rotation of a satellite about the centre of mass

The stability in the first approximation of the rotation of a satellite about a centre of mass is investigated. In the unperturbed motion the satellite performs, in absolute space, three rotations around the normal to the orbital plane in a time equal to two periods of rotation of its centre of mass in the orbit (Mercury-type rotation). Three cases of such rotations are considered: the rotations of a dynamically symmetrical satellite and a satellite, the central ellipsoid of inertia of which is close to a sphere, in an elliptic orbit of arbitrary eccentricity, and the rotation of a satellite with three different principal central moments of inertia in a circular orbit.     

Periodic Perturbations of a Nonconservative Center

Small time-periodic perturbations of an oscillator whose restoring force has a conservative as well as a dissipative component are studied. The stability of the equilibrium and the bifurcation of an invariant two-dimensional torus from the equilibrium are considered. The focus quantity and the bifurcation equation determining the stability and branching character of the equilibrium are constructed.     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

On conditions of existence of permanent rotations of connected rigid bodies system about vertical vector

In contrast to the classical problem of motion of a heavy rigid body about a fixed point where the permanent rotations are well known and completely investigated [7, 3] as the most simple and good visually demonstrated type of motions, in multibody mechanics under an increasing of quantity of the system bodies, mechanical parameters and the order of differential motion equations the study of such motions is more complicated problem. The problem on permanent rotations of two connected rigid bodies under influence of gravity force was investigated in [2, 4]. In this paper a system consisting of arbitrary constant quantity, n, of heavy rigid bodies which are sequentially jointed in a chain is considered. The conditions of existence of motions when each body permanently rotates about the vertical vector are determined. These conditions are analyzed in a general case when the bodies angular velocities are different.     

Global dynamics of a delayed predator–prey model with stage structure for the predator and the prey

A delayed predator–prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory for infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using suitable Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator–extinction equilibrium and the coexistence equilibrium do not exist, and that the predator–extinction equilibrium is globally stable when the coexistence equilibrium does not exist. Further, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE)

It is known that a branching process in a random environment (BPRE) which is subcritical or critical either dies with probability one or, in the trivial case, corresponds to an immortal sterile population. In the supercritical case, various conditions are known to be necessary for noncertain extinction while other conditions are known to be sufficient. In this paper, a necessary and sufficient condition for noncertain extinction of a supercritical BPRE is given. In particular, it is shown that a supercritical BPRE has noncertain extinction if and only if there exists a random truncation, depending only on the environmental sequence, such that the truncated BPRE is supercritical and such that the sequence of truncation points grows more slowly than any exponential sequence.