On integrability in elementary functions of certain classes of nonconservative dynamical systems

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries. As is known, the concept of integrability is sufficiently broad and undeterminate in general. In its construction, it is necessary to take into account in what sense it is understood (it is meant that a certain criterion according to which one makes a conclusion that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function classes the first integrals are sought for, etc. (see also [1, 4, 14, 17, 20–22, 35, 40–42, 47, 83–85, 117, 120]).     

On stability of certain key types of rigid body motion in a nonconservative field

In this activity the qualitative analysis of spatial problems of the real rigid body motions in a resistant medium is fulfilled. A nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack is constructed. An analysis of plane and spatial models (in the presence or absence of an additional tracking force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Qualitative stability of certain nonlinear systems

To certain nonlinear dynamical systems naturally correspond simplicial complexes. This correspondence is a generalization of the familiar relationship between the interaction matrix of a linear dynamical system and the signed digraph of that matrix. By defining stability in terms of attractor regions (as opposed to attractor trajectories), we can specify qualitative linear algebraic conditions involving the simplicial complex which insure stability of the nonlinear system. The analysis uses only signs of coefficients in the dynamical system. A model of E. Lorenz [3] is an example of a three-dimensional system which fulfills the stability conditions and which is known to have a strange attractor within the attractor region. In summary, the linear qualitative tests described (Theorems 2 and 3) can be applied to certain nonlinear dynamical systems to yield preliminary information about the global stability of such systems.     

Gyroscopic stabilization of nonconservative systems

The problem is considered of the stabilization of a mechanical system having only nonconservative positional forces by adding gyroscopic forces. The gyroscopic stabilization is proved to be always realizable in the case of a degenerate matrix of nonconservative forces and even number of coordinates. If the matrix of nonconservative forces is nonsingular then a possibility of the gyroscopic stabilization is established for all systems whose number of coordinates is divisible by four. For a nonautonomous systemwith nonconservative positional forces and dissipative forces with complete dissipation, some sufficient conditions are obtained for stabilization up to the exponential stability by addition of gyroscopic forces.     

Stability of nonconservative systems and applications

The paper studies the stability of mechanical systems subjected to dissipative, gyroscopic, and nonconservative positional forces.     

Role of mass intensity on the stability of elastic systems

Summary A theorem on the role of mass intensity on the stability of undamped, linearly elastic systems subjected to stationary loads (conservative or nonconservative) is derived.

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Zusammenfassung Es wird ein Satz über die Rolle der Massenverteilung bei der Stabilität ungedämpfter linearelastischer Systeme unter stationärer Belastung (konservative oder nichtkonservative) hergeleitet.

     

Variational method in the stability analysis of nonconservative problems

Zusammenfassung Das Stabilitätsproblem eines Voigt-Kelvin-Körpers, welcher nichtkonservativen und gyroskopischen Kräften unterworfen ist, wird — ähnlich der bekannten klassischen Methode — als Variationsaufgabe formuliert. Zwei Variationsprinzipien, basierend auf beschränkten zulässigen Verschiebungen, werden aufgestellt und in einem Näherungsverfahren verwendet, das als Verallgemeinerung des vielverwendeten Ritzschen Verfahrens als konservatives Problem gelten kann. Zur Instruktion wird ein Beispiel gegeben.     

A study of the stability of the solutions of certain nonlinear systems

We use the method of Lyapunov functions to study the stability of the solutions of certain nonlinear systems close to Bazhevskii systems, and also stability in the cone x≥0, x∈Rⁿ, of a system of differential equations with a polynomial right-hand side. We consider an application to the method of Lyapunov vector functions in the case when the system of equations is essentially nonlinear. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 6–12.     

Integrable nonconservative dynamical systems on the tangent bundle of the multidimensional sphere

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.     

Note on exponential stability of certain nonlinear differential systems with time-varying delays

One of the main results in a recent paper [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213] is extended to a more general nonlinear differential system with time-varying delays. A restrictive condition for global exponential stability in the above paper is also removed.     

On the stability of solutions of certain systems of ordinary differential equations

Summary The object of the paper is to give n-dimensional extensions of some stability results for certain Aizerman-type systems of differential equations of order two.     

Global exponential stability of certain switched systems with time-varying delays

This paper is focused on global exponential stability of certain switched systems with time-varying delays. By using an average dwell time (ADT) approach that is different from the method in [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213], we establish a new global exponential stability criterion for the switched linear time-delay system under the ADT switching. We also apply this method to a general switched nonlinear time-delay system. A numerical example is given to show the effectiveness of our results.     

Creep stability of polymer rods

The author examines the problem of the buckling of a hinger rod of rigid homogeneous polymer material under constant load. The results of an experimental investigation are presented. The theoretical calculations are based on the nonlinear generalized Maxwell equation. A numerical solution has been obtained on a computer. The results of this solution are compared with the experimental data. An analytic solution that makes it possible to estimate certain limiting values is obtained for the linearized equation.Mekhanika Polimerov, Vol. 4, No. 1, pp. 145–150, 1968     

Optimization of a nonconservative elastic system with stability constraint

Journal of Optimization Theory and Applications - The stability of an elastic system may, under certain circumstances, be affected by the distribution of the mass. By a suitable mass distribution,...