Liapunov functions and error bounds for approximate solutions of ordinary differential equations

Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.     

On stability and L p solutions of ordinary differential equations

Summary The usual definition of the stability of a solution of a system of ordinary differential equations is extended by introducing two positive control functions. These functions are used to control the rate of growth of the in?tial position of the solution and the rate of growth of the solution. Definitions and results are also given for the corresponding analogues of boundedness, weak boundedness, and uniform properties of the sotions of differential equations. The problem of determining when solutions of certain linear and weakly nonlinear differential equations lie in a modified Lp-space is also considered. This research was supported by the National Science Foundation under grant GP-8921. Entrata in Redazione il 13 maggio 1969.     

On the stability of semi-implicit methods for ordinary differential equations

The aim of this paper is to analyze the stability properties of semi-implicit methods (such as Rosenbrock methods,W-methods, and semi-implicit extrapolation methods) for nonlinear stiff systems of differential equations. First it is shown that the numerical solution satisfies y ₁ (h)y ₀, if the method is applied with stepsizeh to the systemy =Ay ( denotes the logarithmic norm ofA). Properties of the function(x) are studied. Further, conditions for the parameters of a semi-implicit method are given, which imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh. The restriction on the stepsize, however, does not depend on the stiffness of the differential equation. Finally, the presented theory is applied to the extrapolation method based on the semi-implicit mid-point rule.     

A note on exponential stability of quasi-linear ordinary differential equations

It is shown that the exponential stability of the zero solution of a quasi-linear ordinary differential equation can be characterized completely by its linear part.     

A note on exponential stability of quasi-linear ordinary differential equations

It is shown that the exponential stability of the zero solution of a quasi-linear ordinary differential equation can be characterized completely by its linear part.     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.     

Two-sided stability and convergence of multistage and multistep methods for ordinary differential equations

A norm is introduced which allows the extension of bistability and biconvergence results of Stummel (“Topics in Numerical Analysis, II,” Academic Press, New York, 1975; “Approximation Methods in Analysis,” Aarhus Universiteit, 1973) (which apply to one-step methods) to the case of multistage and multistep methods.     

Flag systems and ordinary differential equations

We discuss the Monge problem in the theory of ordinary differential equations and prove the Cartan criterion for first order Monge systems with the added hypothesis of homogeneity. Next, we examine Hilbert's counter-example and finally give a brief account on the two special cases involving flag systems of length two and three. Entrata in Redazione il 1 agosto 1998.     

Projective sets and ordinary differential equations

We prove that for the set of Cauchy problems of dimension which have a global solution is -complete and that the set of ordinary differential equations which have a global solution for every initial condition is -complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for the set of Cauchy problems of dimension which have a global solution even if we perturb a bit the initial condition is -complete.

     

Some third-order ordinary differential equations

The paper deals with periodic orbits in three systems of ordinarydifferential equations. Two of the systems, the Falkner–Skanequations and the Nosé equations, do not possess fixedpoints, and yet interesting dynamics can be found. Here, periodicorbits emerge in bifurcations from heteroclinic cycles, connectingfixed points at infinity. We present existence results for suchperiodic orbits and discuss their properties using careful asymptoticarguments. In the final part results about the Nosé equationsare used to explain the dynamics in a dissipative perturbation,related to a system of dynamo equations.     

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.     

On structural stability of ordinary differential equations with respect to discretization methods

Summary. On compact -dimensional discs, Morse-Smale differential systems having no periodic orbits are considered. The main result is that they are correctly reproduced by one-step discretization methods. For methods of order and stepsize sufficiently small, the time--map of the induced local flow and the -discretized system are joined by a conjugacy -near to the identity. The paper fits well in the rapidly growing list of results stating that hyperbolic/transversal structures are preserved by discretization. The proof relies heavily on techniques elaborated by Robbin (1971) in establishing his structural stability theorem on self-diffeomorphisms of compact manifolds. Received February 24, 1994 / Revised version received February 20, 1995