Stability analysis of a rotating flexible system

We analyse here the equations of motion of a planar body consisting of a rigid body with attached flexible rod. These equations take the form of coupled ordinary and partial differential equations. We analyse the equations both with and without centrifugal stiffening effects. Using the energy-momentum method, we analyse nonlinear stability of the equilibria in each case. We also analyse the Hamiltonian and Poisson bracket structure of the system as well as the energy-momentum map and associated relative equilibria.Partly supported by NSF grant DMS-8701574 and AFOSR grant AFOSR-1SSA-87-0077 and by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Die Linienmethode bei nichtlinearen parabolischen Differentialgleichungen

The method of lines for parabolic differential equations consists in the discretization of the spatial variable only. In this way the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper it is proved that for the general nonlinear parabolic equation the solution of the discrete problem converges to the solution of the original problem, when the mesh size tends to zero. The principal tool in this investigation is the theory of ordinary differential inequalities and especially the concept of quasimonotonicity.     

Symmetry Analysis of Differential Equations Using MATHLIE

A general procedure for solving ordinary differential equations of arbitrary order is discussed. The method used is based on symmetries of a differential equation. The known symmetries allow the derivation of first integrals of the equation. The knowledge of at least r symmetries of an ordinary differential equation of order n with r n is the basis for deriving the solution. Our aim is to show that Lie's theory is a useful tool for solving ordinary differential equations of higher orders. Bibliography: 12 titles.     

Stability Analysis of Modified Multilag Methods for Volterra Integral Equations

Stability analysis of modified multilag methods for Volterraintegral equations of the second and first kind is presented,based respectively on the test equations This analysis reinforces the opinion that modified multilagmethods are advantageous over quadrature methods for Volterraequations. They allow us to combine the good stability propertiesof backward differentiation formulae and the efficiency of Adams-Moultonformulae for ordinary differential equations.     

Convergence properties of the Runge-Kutta-Chebyshev method

Summary The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length proportional tos ². The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of withs ². The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.Dedicated to Peter van der Houwen for his numerous contributions in the field of numerical integration of differential equations.Paper presented at the symposium Construction of Stable Numerical Methods for Differential and Integral Equations, held at CWI, March 29, 1989, in honor of Prof. Dr. P.J. van der Houwen to celebrate the twenty-fifth anniversary of his stay at CWI     

Consistency,convergence and stability of general discretizations of the initial value problem

Discretizations of nonlinear operators in Banach space are described and the concept of an inverse discretization introduced. In the main part of the paper, the very general formalism of BUTCHER for the initial value problem for ordinary differential equations is examined and the sufficiency of conditions for its stability and convergence is demonstrated. The order of convergence of these methods is discussed, and an example is given.     

A method for integration of unstable systems of ordinary differential equation subject to two-point boundary conditions

Instability problems in systems of differential equations are discussed. A matrix technique is given for producing numerical solutions to a system of ordinary differential equations with boundary conditions specified at each end of the interval when the system contains dominant solutions which give rise to numerical instability in conventional integration methods. A method of bringing up the initial conditions is described, whereby the two-point nature of the problem is made use of to stabilize the system. Three numerical examples are included.     

Multivariable Euler transform of systems of linear ordinary differential equations of Okubo normal form

The multivariable Euler transform of a solution of the system of linear ordinary differential equations of Okubo normal form is considered. The Pfaffian system satisfied by the transform is derived. Applications to the Appell hypergeometric functions \(F_{1}\), \(F_{3}\), and the Lauricella hypergeometric function \(F_{D}\) are given.     

Construction of a Class of Finite H ∞ Controllers for Infinite-Dimensional Systems

We consider the suppression of forced oscillations in distributed systems of a hyperbolic type by finite-dimensional controllers using an H objective. The system is split into a finite-dimensional and an infinite-dimensional subsystems. The controller receives a signal from the output of both systems. The class of controllers is described in the form of a system of ordinary differential equations.     

Stability in the Numerical Solution of Linear Parabolic Equations with a Delay Term

This paper is concerned with the stability of numerical processes that arise after semi-discretization of linear parabolic equations wit a delay term. These numerical processes are obtained by applying step-by-step methods to the resulting systems of ordinary delay differential equations. Under the assumption that the semi-discretization matrix is normal we establish upper bounds for the growth of errors in the numerical processes under consideration, and thus arrive at conclusions about their stability. More detailed upper bounds are obtained for -methods under the additional assumption that the eigenvalues of the semi-discretization matrix are real and negative. In particular, we derive contractivity properties in this case. Contractivity properties are also obtained for the -methods applied to the one-dimensional test equation with real coefficients and a delay term. Numerical experiments confirming the derived contractivity properties for parabolic equations with a delay term are presented.     

A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations

We investigate the stability properties of numerical methodsfor weakly singular Volterra integral equations of the secondkind. Our theory extends the stability theory of linear multistepmethods for ordinary differential equations. We introduce theconcepts of A-stability, A()-stability etc. for Abel-Volterraequations. The stability region is characterized in terms ofthe weights of the method. It is shown that the order of anA-stable convolution quadrature cannot exceed 2. Further westudy the stability properties of implicit Adam methods, withparticular emphasis on the question of A()-stability.     

Analyzing the stability behaviour of solutions and their approximations in case of index- differential-algebraic systems

When integrating regular ordinary differential equations numerically, one tries to match carefully the dynamics of the numerical algorithm with the dynamical behaviour of the true solution. The present paper deals with linear index- differential-algebraic systems. It is shown how knowledge pertaining to (numerical) regular ordinary differential equations applies provided a certain subspace which is closely related to the tangent space of the constraint manifold remains invariant.

     

On the approximation of nonlinear conflict-controlled systems of neutral type

We consider approximations of systems of nonlinear neutral-type equations in Hale’s form by systems of high-order ordinary differential equations. A procedure is given for the mutual feedback tracking between the motion of the original neutral-type conflict-controlled system and the motion of the approximating system of ordinary differential equations. The proposed mutual tracking procedure makes it possible to use approximating systems of ordinary differential equations as finite-dimensional modeling guides for neutral-type systems.     

Solvable models of layered neural networks based on their differential structure

We introduce the notion of solvable models of artificial neural networks, based on the theory of ordinary differential equations. It is shown that a solvable, three layer, neural network can be realized as a solution of an ordinary differential equation. Several neural networks in standard use are shown to be solvable. This leads to a new, two-step, non-recursive learning paradigm: estimate the differential equation which the target function satisfies approximately, and then approximate the target function in the solution space of that differential equation. It is shown experimentally that the proposed algorithm is useful for analyzing the generalization problem in artificial neural networks. Connections with wavelet analysis are also pointed out.     

On the existence of global attractors for one class of cascade systems

We investigate the qualitative behavior of solutions of cascade systems without uniqueness. We prove that solutions of a reaction-diffusion system perturbed by a system of ordinary differential equations and solutions of a system of equations of a viscous incompressible liquid with passive components form families of many-valued semiprocesses for which a compact global attractor exists in the phase space.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1287–1291, September, 2004.     

On Chebyshev Solution of Parabolic Partial Differential Equations

Now at Mathemarics Department, Assiut University Egypt A method is presented to transform parabolic equations to asystem of ordinary differential equations for the solution atthe Chebyshev points. The system may be solved analyticallyor by numerical methods and the Chebyshev coefficients are computed.We have the exact solution of a perturbed problem.     

The Inverse Problem of the Calculus of Variations: Separable Systems

This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the problem which Douglas, in his classification of pairs of such equations, called the separated case is generalized to arbitrary dimension. After identifying the conditions which should specify such a case for n equations in a coordinate-free way, two proofs of its variationality are presented. The first one follows the line of approach introduced by some of the authors in previous work, and is close in spirit, though being coordinate independent, to the Riquier analysis applied by Douglas for n = 2. The second proof is more direct and leads to the discovery that belonging to the separated case has an intrinsic meaning for the given second-order differential equations: the system is separable in the sense that it can be decoupled into n pairs of first-order equations.     

Transformations of linear connections

Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a geometric interpretation.     

Quasi-Stable Structures in Circular Gene Networks

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.