Bifurcations of completely integrable first-order ordinary differential equations

We consider an implicit first-order ordinary differential equation with complete integral. In [3], the authors give a generic classifications of first-order ordinary differential equations with complete integral with respect to the equivalence relation which is given by the group of point transformations. The classification problem is reduced to the classification of a certain class of divergent diagrams of mapping germs. In this paper, we give a generic classifications of bifurcations of such differential equations as an application of the Legendrian singularity theory. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Group classification of projective type second-order ordinary differential equations

Group classification with respect to admitted point transformation groups is carried out for second-order ordinary differential equations with cubic nonlinearity of the first-order derivative. The result is obtained with use of the invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of the system of geodesics to a two-dimensional surface.     

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.     

Parameter estimation of ordinary differential equations

This paper addresses the development of a new algorithm forparameter estimation of ordinary differential equations. Here,we show that (1) the simultaneous approach combined with orthogonalcyclic reduction can be used to reduce the estimation problemto an optimization problem subject to a fixed number of equalityconstraints without the need for structural information to devisea stable embedding in the case of non-trivial dichotomy and(2) the Newton approximation of the Hessian information of theLagrangian function of the estimation problem should be usedin cases where hypothesized models are incorrect or only a limitedamount of sample data is available. A new algorithm is proposedwhich includes the use of the sequential quadratic programming(SQP) Gauss–Newton approximation but also encompassesthe SQP Newton approximation along with tests of when to usethis approximation. This composite approach relaxes the restrictionson the SQP Gauss–Newton approximation that the hypothesizedmodel should be correct and the sample data set large enough.This new algorithm has been tested on two standard problems.     

Optimal solution of ordinary differential equations

We survey some recent optimality results for the numerical solution of initial value problems for ODE. We assume that information used by an algorithm about a right-hand-side function is partial. Two settings of information-based complexity are considered: the worst case and asymptotic. Upper and lower bounds on the error are presented for three types of information: standard, linear, and nonlinear continuous. In both settings, minimum error algorithms are exhibited.     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

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.     

One-step methods for ordinary differential equations

Summary It is proved that any consistent one-step method for solving the initial value problem for a first-order ordinary differential equation is convergent; no stability condition is required. An application is made to a similarly stated result, allowing part of the hypothesis in that case to be dropped.     

Nonuniqueness results for ordinary differential equations

In the present paper we give general nonuniqueness results which cover most of the known nonuniqueness criteria. In particular, we obtain a generalization of the nonuniqueness theorem of CHR. NOWAK, of SAMIMI's nonuniqueness theorem and of STETTNER's nonuniqueness criterion.     

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.