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.     

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.     

Modelling homeorhesis by ordinary differential equations

Homeorhesis is a necessary feature of any living system. If a system does not perform homeorhesis, it is nonliving. The present work develops the sufficient conditions for the ODE model to describe homeorhesis and suggests the structure of the model. The proposed homeorhesis model is fairly general. It treats homeorhesis as piecewise homeostasis. The model can be specified in different ways depending on the specific system and specific purposes of this analysis. An example of the specification is the PhasTraM model, the homeorhesis-aware nonlinear reaction–diffusion model for hyperplastic oncogeny in the previous works of the author. The qualitative agreement of the developed homeorhesis model with the living-system experimental results is noted. The work also shows that the basic mathematical models (such as the active-particle generalized kinetic theory) are substantially more important for the living-matter studies than in the case of nonliving matter. A few directions for future research are suggested as well.     

Limit processes in ordinary differential equations

Summary Differential equations with right-hand sides including terms oscillating with high frequency and large amplitude are studied. The limit equation deduced involves resonance terms in the form of Lie brackets.Dedicated to the 60th birthday of Prof. Hans Wilhelm Knobloch     

Computing with polynomial ordinary differential equations

In 1941, Claude Shannon introduced the General Purpose Analog Computer (GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later on electronic) machines of that time.Following Shannon’s arguments, functions generated by the GPAC must satisfy a polynomial differential algebraic equation (DAE). As it is known that some computable functions like Euler’s $\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt$ or Riemann’s Zeta function $\zeta \left(x\right)={\sum }_{k=0}^{\infty }\frac{1}{k^x}$ do not satisfy any polynomial DAE, this argument has often been used to demonstrate in the past that the GPAC is less powerful than digital computation.It was proved in Bournez et al. (2007), that if a more modern notion of computation is considered, i.e. in particular if computability is not restricted to real-time generation of functions, the GPAC is actually equivalent to Turing machines.Our purpose is first to discuss the robustness of the notion of computation involved in Bournez et al. (2007), by establishing that many natural variants of the notion of computation from this paper lead to the same computability result.Second, to go from these computability results towards considerations about (time) complexity: we explore several natural variants for measuring time/space complexity of a computation.Quite surprisingly, whereas defining a robust time complexity for general continuous time systems is a well known open problem, we prove that all variants are actually equivalent even at the complexity level. As a consequence, it seems that a robust and well defined notion of time complexity exists for the GPAC, or equivalently for computations by polynomial ordinary differential equations.Another side effect of our proof is also that we show in some way that polynomial ordinary differential equations can actually be used as a kind of programming model, and that there is a rather nice and robust notion of ordinary differential equation (ODE) programming.     

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.