Second-order Differential Systems Research Articles

Second-order difference systems in unbounded domains: Existence conditions and construction of exact and approximate bounded solutions

In this paper existence conditions and explicit expressions of bounded solutions for nonhomogeneous second-order difference systems defined in unbounded domains are given. Bounded solutions are expressed in terms of series involving a bounded fundamental matrix solution. Computable approximate solutions are constructed and error bounds in terms of the data are given.

Noncollision Periodic Solutions to Some Singular Dynamical Systems with Very Weak Forces

We prove a double variational characterization of the set of all the periodic solutions of the Kepler problem. We provide a sufficient condition for the existence of a periodic solution with prescribed period or energy for autonomous second-order differential systems with Keplerian-type potentials. The proofs are based on linking arguments.

Kamenev type theorems for second-order matrix differential systems

We consider the second order matrix differential systems (1) (P(t)Y')' + Q(t)Y = 0 and (2) Y″ + Q(t)Y = 0 where Y, P, and Q are n × n real continuous matrix functions with P(t), Q(t) symmetric and P(t) positive definite for t ∈ [t 0 , ∞) (P(t) > 0, t ≥ t 0 ). We establish sufficient conditions in order that all prepared solutions Y(t) of (1) and (2) are oscillatory. The results obtained can be regarded as generalizing well-known results of Kamenev in the scalar case

Oscillation with Respect to Partial Variables of Linear Second-Order Differential Systems

Oscillation with Respect to Partial Variables of Linear Second-Order Differential Systems

Oscillation with respect to partial variables of linear second-order differential systems

Consider the second-order vector differential system (1) x″(t)+Q(t)x(t)=0 and matrix differential system (2) X″(t)+Q(t)X(t)=0, where x(t) is an n-dimensional vector function and X(t) and Q(t) are n×n continuous matrix functions. In this article, we establish the concept that systems (1) and (2) are oscillatory with respect to partial variables. Some sufficient conditions are obtained; several examples are given to illustrate the results

Design of optimal second-order state estimators

The present consideration of the design of online computation-saving second-order state estimators for second-order vector-matrix differential systems proposes a class of such estimators which is proven to possess guaranteed convergence. A class of optimal second-order estimators is then obtained, and the conditions required for optimality are identified. The estimator proposed offers high performance in conjunction with online computation reductions sufficiently great to allow the estimation of the large number of state variables associated with control of large, flexible space structures represented by high-dimensional second-order systems.

Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes

An isoparametric finite element formulation based on a higher order displacement model for dynamic analysis of multilayer unsymmetric composite plates is presented. Newmark and Wilson-θ schemes are used for time integration of the discrete coupled second-order ordinary differential equation system of dynamic equilibrium and results are compared. A special mass diagonalization scheme is used which conserves total mass of the element and also includes the effects of rotary inertia terms. Effects of consistent/diagonal mass matrix, time step, and finite element mesh are investigated. Several numerical examples are presented and results are compared with those available in the literature.

Asymptotics of solutions of a second-order linear differential system with a special right-hand side

Asymptotics of solutions of a second-order linear differential system with a special right-hand side

Positive solutions of elliptic systems with bounded nonlinearities

SynopsisSemilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

A class of second-order differential equations and related first-order systems

A class of second-order nonlinear differential equations which arises in several branches of mathematical physics is considered. It is shown that equations of this class may be factorised into first-order equations of 'Riccati type'. Conditions are obtained, on the coefficient functions of the second-order equations, for the first-order equations to be of matrix Riccati form, whose solutions have a finite superposition property. The factorisation into first-order equations is then not unique, and there is an alternative first-order set of equations whose solutions do not have this superposition property. A second-order equation arising in the theory of pellet fusion processes is investigated in detail. Solutions are obtained when the corresponding first-order equations are of matrix Riccati from and shown to be equivalent to solutions derived by alternative methods. Lagrangian systems giving rise to equations of the class are also considered.

Boundary value problems for second-order differential systems

Sufficient conditions for the existence of a solution to the BVP x″ = f( t, x, x′), x(0) = Q 0 x(1), x′(0) = Q 1 x′(1) are obtained. Here Q 0, Q 1 are nonsingular n × n matrices, with Q 0 orthogonal. These extend results for the periodic case and are proved via a modification of a degree-theoretic approach.

A reduction of order two for infinite-order Lagrangians.

Given a Lagrangian system depending on the position derivatives of any order, and assuming that certain conditions are satisfied, a second-order differential system is obtained such that its solutions also satisfy the Euler equations derived from the original Lagrangian. A generalization of the singular Lagrangian formalism permits a reduction of order keeping the canonical formalism in sight. Finally, the general results obtained in the first part of the paper are applied to Wheeler-Feynman electrodynamics for two charged point particles up to order 1/${c}^{4}$.

Oscillation results for selfadjoint differential systems

We consider the second-order differential system, (1) ( R( t) Y′)′ + Q( t) Y = 0, where R, Q, Y are n × n matrices with R( t), Q( t) symmetric and R( t) positive definite for t ϵ [ a, + ∞) ( R( t) > 0, t ⩾ a). We establish sufficient conditions in order that all prepared solutions Y( t) of (1) are oscillatory; that is, det Y( t) vanishes infinitely often on [ a, + ∞). The conditions involve the smallest and largest eigen-values λ n ( R −1( t)) and λ 1(∝ a t Q( s) ds), respectively. The results obtained can be regarded as generalizing well-known results of Leighton and others in the scalar case.

An Asymptotic Method for Second-order Critically Damped Nonlinear Equations

A second-order nonlinear differential system modeling nonoscillatory processes and characterized by critical damping is considered. A new perturbation technique, based on the work of Krylov, Bogoliubov and Mitropolskii, is developed to find the solution of the system. The method is illustrated by an example.

Realization of Poincaré group induced by a second-order ordinary differential system. Noninteraction theorem

A generalization of the predictive relativistic mechanics is studied where the initial conditions are taken on a general hypersurface of M4. The induced realizations of the Poincaré group are obtained. The same procedure is used for the Galileo group. Noninteraction theorems are derived for both groups.

Composite variational principles, added variables, and constants of motion

It is shown that any second-order differential system admits a variational formulation via the introduction of suitable additional variables. The new variables are related to the existence of invariant 1-forms and to solutions for the adjoint of the equations of variation of the given system. The connections among invariant forms, constants of motion, and infinitesimal invariance transformations are then discussed in some detail.

An Oscillation Criterion for Linear Second-Order Differential Systems

An Oscillation Criterion for Linear Second-Order Differential Systems

Oscillation of two-dimensional linear second-order differential systems

This article is concerned with the oscillatory behavior at infinity of the solution y:[ a, ∞) → R 2 of a system of two second-order differential equations, y″( t) + Q( t) y( t) = 0, tϵ[ a, ∞); Q is a continuous matrix-valued function on [ a, ∞) whose values are real symmetric matrices of order 2. It is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix ∝ a t Q( s) ds tends to infinity as t → ∞. This proves a conjecture of D. Hinton and R. T. Lewis ( Rocky Mountain J. Math. 10 (1980), 751–766) for the two-dimensional case. Furthermore, it is shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity.

An oscillation criterion for linear second-order differential systems

This article is concerned with the oscillatory behavior at infinity of the solution y: (a, infinity) ..-->.. R/sup n/ of a system of n second-order differential equations, y''(t) + Q(t)y(t) = 0, t epsilon (a, infinity); Q is a continuous matrix-valued function on (a, infinity) whose values are real symmetric matrices of order n. It is shown that the solution is oscillatory at infinity if (at least) n - 1 eigenvalues of the matrix integral/sub a//sup t/Q(t) dt tend to infinity as t ..-->.. infinity. 7 references.

Stability analysis and stability conditions in the delay intervals for second-order delay systems

Abstract In tills paper the stability and the stability conditions of a number of second-order retarded delay linear differential systems are discussed. The paper shows especially that the system which has stationary oscillation when without delay can become asymptotically stable in the large in certain suitable delay intervals.