Stability of steady rotations in the nonholonomic Routh problem

We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.      

The limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.     

The stability of generalized steady motion

The influence of dissipative forces on the stability of the generalized steady motion of a mechanical system with time-dependent constraints is investigated. As a preliminary, the problem of the limiting behaviour of solutions of a non-autonomous system is solved on the assumption that m first integrals and a function that decreases along every solution of the system are known. As an example, the motion of a gyroscope in gimbals.     

Stability of steady rotations of a heavy gyrostat about its principal axis

Stability of steady rotations of a gyrostat about its principal axis is investigated with the use of the Arnol'd —Moser theorem /1, 2/ extended to stationary motions /3, 4/. It is shown that steady rotations are stable for all parameter values that belong to the region where the necessary stability conditions are satisfied, except for some manifold of lesser dimension.     

The stability boundaries of the steady motion of a satellite with a gyroscope

Parts of the asymptotic stability boundaries of the uniform motion of the centre of mass of a system of bodies consisting of an asymmetrical satellite with a three-axis gyroscope in a circular orbit are investigated by the second Lyapunov method. Terms of the Lyapunov function that are higher than the second order are enlisted for the investigation. The sign-definiteness criterion of inhomogeneous forms is employed for the corresponding function. Parts of the stability boundaries in which the steady motion investigated is asymptotically stable are established using the Lyapunov asymptotic stability theorem. Application of the Barbashin and Krasovskii theorems reveals parts of the stability boundaries in which the steady motion is unstable. It is established that the asymptotic stability of the steady motion investigated is solved by expanding the Lyapunov function to sixth-order terms.     

The stability of the steady motions of a rigid body in a central field

The problem of the translational-rotational motion of a rigid body with a triaxial ellipsoid of inertia in a central gravitational field is considered. The body is modelled by a weightless sphere, at the ends of the three mutually perpendicular diameters of which there are point masses. It is shown that, unlike the cases when the approximate expression for the potential of the gravity forces is used, there are not only “trivial” steady motions of the body, for which the main central axes of inertia of the body coincide with the axes of the orbital system of coordinates, but also other classes of steady motions. In addition, the stability of these “trivial” steady motions is investigated, and the possibility of secular stability of the motions, unstable in the satellite approximation, is pointed out.     

The stability of the steady motion of a gyrostat with a liquid in a cavity

A symmetrical rigid body with a spherical base, carrying a rotor and having a cavity in the shape of an ellipsoid of revolution, completely filled with an ideal incompressible liquid in uniform vortex motion, is moving along an absolutely rough plane. It is shown that this system admits of an energy integral, Jellett's integral, the integral of constant vorticity and a geometric integral. The construction of a Lyapunov function as a linear combination of first integrals [1] yields the sufficient conditions for the rotation of the gyrostat about the vertically positioned axis of symmetry to be stable. The conditions for the gyrostat's rotation to be unstable are found. It is shown that the rotor may prove to have either a stabilizing or destabilizing effect on the system and that the gyrostat admits of motions of the type of regular precession. The sufficient conditions for the stability of these motions are obtained.     

On the stability of steady states in a granuloma model

We consider a free boundary problem for a system of two semilinear parabolic equations. The system represents a simple model of granuloma, a collection of immune cells and bacteria filling a 3-dimensional domain Ω(t)$\Omega (t)$ which varies in time. We prove the existence of stationary spherical solutions and study their linear asymptotic stability as time increases to infinity.     

The stability of the steady rotation of a system of three equidistant vortices outside a circle

A complete non-linear analysis of the stability of the steady rotation of three point vortices, placed in a plane at the vertices of a regular triangle outside a circular domain is carried out using the results of the Kolmogorov–Arnold–Moser theory. All the resonances of up to fourth order inclusive encountered here are listed and studied. The investigations of Havelock who solved this problem in a linear formulation are thereby completed.     

The von Neumann facet and a global asymptotic stability

We will study a multi-sector discrete-time optimal growth model with neoclassical non-joint technology and show that any path on ann-dimensional flat supported by the optimal steady state price will converge to the optimal steady state and is optimal. Burmeister and Graham have proved a similar result in a continuous-time setting. Although their result is limited, it is a first challenge to generalize the global stability result obtained by Uzawa and Srinivasan in a two-sector optimal growth model. One prominent advantage of our approach is that due to the discrete-time model setting, we can apply the duality approach and introduce the so called "von Neumann facet" intensively studied by McKenzie, which plays a very important role in proving the saddle point stability.     

The global asymptotic stability for F-M equation

In this paper, we consider a class of equations for the flow and magnetic field within the earth with initial-boundary value problem. We prove the existence of the stationary solution and show that it is global asymptotic stability.     

The local bifurcation and stability of nontrivial steady states of a logistic type of chemotaxis

Chemotaxis is a type of oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider local existence and stability of nontrivial steady states of a logistic type of chemotaxis. We carry out the bifurcation theory to obtain the local existence of the steady state and apply the expansion method on the chemotaxis to investigate the bifurcation direction. Moreover, by applying the bifurcation direction, we obtain the bifurcating steady state is stable when the bifurcation curve turns to right under certain conditions.     

The age-dependent population model: Numerical check of stability/instability of a steady state

An estimate for the boundaries of a rectangular region containing all (if any) roots with a positive real part of the characteristic equation of a given age-structured model is derived. A method for checking the stability of steady states of such equations based on the argument principle is then proposed.     

On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem

This paper is contributed to the Cauchy problem

     

On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is a contribution to the analysis of the Cauchy problem (0.1)$\{\begin{array}{c}\frac{\partial u}{\partial t}=\Delta u+\sum _{i=1}^k{c}_i{\left|x\right|}^{l_i}{u}^{p_i}\text{in}{\mathbb{R}}^n\times (0,T)\text{,}\hfill \\ u(x,0)=\phi \left(x\right)\text{in}{\mathbb{R}}^n\text{,}\hfill \end{array}$ with initial function $\phi \not\equiv 0$. The stability and asymptotic stability of the positive radial steady states, which are positive solutions of (0.2)$\Delta u+\sum _{i=1}^k{c}_i{\left|x\right|}^{l_i}{u}^{p_i}=0\text{,}$ have been discussed with $\phi$ in weighted Banach spaces.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.