On implicit systems of differential equations

This article considers implicit systems of differential equations. The implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit systems of differential equations, we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation. Our approach is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach of this article is different, we have no linearity requirements at all, instead we rely on algebraic geometry. In particular, we use birational mappings to produce an explicit system. The methods developed in this article are easily implemented using various computer algebra systems.     

Generic singularities of implicit systems of first order differential equations on the plane

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.     

On certain nonlinear systems of partial differential equations of the second order

We consider nonlinear systems of five partial differential equations of the second order with one and two unknown functions. We obtain explicit conditions of compatibility which provide the unique solvability of problems with initial data.     

On monotonic solutions of systems of nonlinear second order differential equations

Bounded, monotonic, and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed. Some necessary and sufficient conditions for boundedness of all solutions are established. Moreover, all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals. The obtained results extend and improve many known results of various authors.     

Matrix-second order differential equations and chaotic Hamiltonian systems

We consider matrix-second order differential equations which are perturbations of the harmonic flow on the space of matrices. Experimental evidence of the non integrability of the two degrees of freedom Hamiltonian system provides an indication of the non existence of a Lax pair with commuting eigenvalues for perturbations of order six. This shows the specificity of quartic perturbations for which such a Lax pair was precedently obtained.Supported by Istituto Nazionale di Alta Matematica F. Severi. This article has been written while the second author was visiting ETH.     

On the convergence of solutions of certain systems of second order differential equations

Summary The object of this paper is to furnish an n-dimensional analogue of a convergence result obtained in [3] by Loud for the equation (1.4).     

On solving implicit differential equations on parallel computers

We construct and analyse integration methods for solving initial value problems for implicit differential equations (IDEs) that can be efficiently used on parallel computer systems. We construct an IDE method for general IDEs of arbitrarily high index, and two methods that can be applied to partitioned IDEs. The partitioned IDE methods both exploit the special form of the problem and converge faster than the general IDE method. The first partitioned IDE method is suitable for higher-index problems, the second partitioned IDE method only applies to index 1 problems, but is considerably less expensive on parallel computers. This paper presents the results presented in June 1995 at the Seminario Matematico e Fisico organized by the Mathematics Department of the Polytechnics University of Milano. Conferenza tenuta da P.J. van der Houwen il 5 giugno 1995     

Efficient higher order implicit one-step methods for integration of stiff differential equations

A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.     

Stability of certain systems of second order differential equations

Summary Conditions are obtained which are sufficient for a system of ordinary second order differential equations to have an autonomous quadratic Lyapunov function. From these are derived conditions which ensure that every pair of solutions converges as t_?+∞ Entrata in Redazione il 9 novembre 1968.