Systems Of Second-order ODE Research Articles

Conjugate points of dynamic pairs and control systems

We study the geometry of dynamic pairs $(X,\VV)$ on a manifold $M$, where $X$ is a vector field and $\VV$ is a distribution on $M$, both satisfying a regularity condition. Special cases are pairs defined by systems of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian geometries, semi-Hamiltonian systems and control-affine systems. Analogs of conjugate points from the calculus of variations are defined for the pair $(X,\VV)$. The main results give estimates for the position of conjugate points in terms of a curvature operator, analogously to the Cartan--Hadamard and Bonet--Myers theorems. Contrary to classical cases, no metric is given a priori, the distribution $\VV$ may be nonintegrable and the curvature operator is defined in terms of $(X,\VV)$.

Qualitative analysis of a mechanical system of coupled nonlinear oscillators

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to + ∞ , provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations.

Inertias of matrices and sign patterns related to a system of second order ODEs

The inertia of real matrices of the form C=[ADIO] and of the sign pattern matrix C of C are considered, where A is irreducible and D is a positive diagonal matrix. Such matrices C occur in applications that give rise to systems of linear second order differential equations. If D=dI is a positive scalar matrix, then the eigenvalues of C are determined by the magnitude of d and the eigenvalues of A, giving results for the inertia of C when A has certain specified forms. Results for the inertia of C are also given when D is an arbitrary positive diagonal matrix. The set of possible refined inertias of general sign pattern matrices C of the above form are determined, and are completely specified when A is a spectrally arbitrary sign pattern of order 2 or 3.

Hermite Fitted Block Integrator for Solving Second-Order Anisotropic Elliptic Type PDEs

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves.

Stability of sign patterns from a system of second order ODEs

The stability and inertia of sign pattern matrices with entries in {+,−,0} associated with dynamical systems of second-order ordinary differential equations x¨=Ax˙+Bx are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C=[ABIO] of order 2n, and eigenvalue properties are considered for sign patterns C=[ABDO] of order 2n, where A,B are the sign patterns of A,B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C, as well as the refined inertia of C. Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.

Particle Dynamics with Elastic Collision at the Boundary: Existence and Partial Uniqueness of Solutions

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $\Omega $ . Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.

Green operators for low regularity spacetimes

In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to well-posedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H1 and a suitable Green matrix that solves a system of second order ODEs.

Bounded solutions to a system of second order ODEs and the Whitney pendulum

Bounded solutions to a system of second order ODEs and the Whitney pendulum

A note on the existence of multiple solutions for a class of systems of second order ODEs

We prove the existence of multiple solutions for some systems of second order ODEs with Dirichlet boundary conditions. Such systems are obtained by coupling scalar ODEs with different growth conditions. The proof relies on a global continuation technique.

A Neumann problem for a system depending on the unknown boundary values of the solution

A semilinear system of second order ODEs under Neumann conditions is studied. The system has the particularity that its nonlinear term depends on the (unknown) Dirichlet values y(0) and y(1) of the solution. Asymptotic and non-asymptotic sufficient conditions of Landesman-Lazer type for existence of solutions are given. We generalize our previous results for a scalar equation, and a well known result by Nirenberg for a standard nonlinearity independent of y(0) and y(1).

Conservation laws for the geodesic equations of the canonical connection on Lie groups in dimensions two and three

In order to characterize the systems of second-order ODEs which admit a regular Lagrangian function, Noether symmetries for the geodesic equations of the canonical linear connection on Lie groups of dimension three or less are obtained, so the character- ization of these geodesic equations through their Noether's symmetries Lie Algebras is investigated. The corresponding conservation laws and the first integral for each geodesics are constructed.

Periodic solutions of singular radially symmetric systems with superlinear growth

We prove the existence of infinitely many periodic solutions for periodically forced radially symmetric systems of second-order ODE’s, with a singularity of repulsive type, where the nonlinearity has a superlinear growth at infinity. These solutions have periods, which are large integer multiples of the period of the forcing, and rotate exactly once around the origin in their period time, while having a fast oscillating radial component. Analogous results hold in the case of an annular potential well.

Branches of index-preserving solutions to systems of second order ODEs

We investigate the existence of a continuum of index-preserving solutions to a Dirichlet problem associated with a parameter-dependent system of second order ordinary differential equations, developing a detailed analysis on the behaviour of the branches of nontrivial solutions. Our approach is based on the Rabinowitz global bifurcation Theorem combined with the notion of index and nullity of suitable linear boundary value problems. An application of the result to the study of branches of odd, periodic solutions for suitable systems of two linearly coupled pendulums of lenghts variables is also analyzed.

Élie Cartan’s geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan’s method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sense, using our implementation of Cartan’s method, we establish two new equivalence results. We establish when a system of second order ODEs is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDEs with two independent variables and one dependent variable is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.

Nonlinear Systems of Second-Order ODEs

We study existence of positive solutions of the nonlinear system in ; in ; , where and . Here, it is assumed that , are nonnegative continuous functions, , are positive continuous functions, , , and that the nonlinearities satisfy superlinear hypotheses at zero and . The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of and .

Branches of forced oscillations in degenerate systems of second-order ODEs

We use fixed point index methods to study the set of forced oscillations in periodically perturbed systems of ODEs on manifolds. We prove the existence of branches of periodic solutions for a particular class of system where, contrary to the usual ‘nondegeneracy’ assumption, the leading vector field is neither trivial nor has a set of compact zeros.

Superlinear systems of second-order ODE’s

We discuss the existence of positive solutions of the system − u ″ = f ( t , u , v , u ′ , v ′ ) in ( 0 , 1 ) , − v ″ = g ( t , u , v , u ′ , v ′ ) in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = v ( 0 ) = v ( 1 ) = 0 where the nonlinearities f and g satisfy a superlinearity condition at both 0 and ∞ . Our main result is the proof of a priori bounds for the eventual solutions. As an application, we consider the Dirichlet problem in an annulus for systems of semilinear elliptic equations with nonlinearities depending on the gradient as well. As a second application, we consider fourth-order elastic beam equations with dependence also on the derivatives u ′ , u ″ , u ‴ .

Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics.

This paper analyzes the problem of persistence of homoclinic solutions to perturbed systems of second order ODE's. These systems arise from PDE's, when considering solutions in the form of travelling waves. It is shown that homoclinic solutions persist in the presence of dissipation. Dissipation can be balanced by nonautonomous terms of compact support, which are controlled by a single parameter. This result is applied to prove the existence of torsional pulse-like travelling waves propagating along a nonelastic DNA molecule. In this case the energy is added to system by advancing the RNA polymerase.

Jacobi fields and linear connections for arbitrary second-order ODEs

We discuss generalisations of Jacobi fields and Raychaudhuri’s equation from the geodesic case to that of an arbitrary system of second-order ODEs. Our results are obtained using a natural choice of linear connection on evolution space.

Automatic Integration of Systems of Second-Order ODE’s Separated into Subsystems

This paper continues to lay the foundation for Runge–Kutta type methods for integrating systems of ODE’S in which some of the variables are more slowly varying than others within the specified error bounds. Second- rather than first-order systems are considered because they permit the development of more efficient algorithms. It is assumed that the system has been separated into several subsystems requiring different stepsizes. After the presentation of general theorems and a stepsize control analysis, a particular Runge–Kutta type method is developed and applied to two example problems including a perturbation problem. It is shown that a wide class of perturbation problems may be solved in a like manner.