Contact Transformations and Local Reducibility of ODE to the Form y‴=0

Orbits of the ODE y=0 in corresponding jet bundles are investigated. Explicit relations for the right-hand side of an arbitrary 3-order ODE necessary and sufficient for the existence of a contact transformation reducing this equation locally to the form y=0 are obtained.     

The nonautonomous function-theoretic center problem

We study the linearizability and stability of a nonautonomous dynamical system in the neighborhood of a neutral fixed point. Our results generalize the classical results of Schröder and Siegel in the case when the linear part of the mapping is an irrational rotation, well known results in the rational case and the fundamental result on the representation of the system as a translation in the neighborhood of a fixed point at infinity.     

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.     

4-Webs in the Plane and Their Linearizability

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply the linearizability conditions, recently found by Akivis, Goldberg and Lychagin, to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is always linearizable (this result was first obtained by Mayrhofer in 1928; it also follows from the papers of Nakai). Using the same conditions, we further prove that such a 4-web with a constant basic invariant (Nakai's web) is linearizable if and only if it is parallelizable. Next we study four classes of the so-called almost parallelizable 4-webs APW _a ,a=1,2,3,4 (for them the curvature K=0 and the basic invariant is constant on the leaves of the web foliation X _a ), and prove that a 4-web APW _a is linearizable if and only if it coincides with a 4-web MW _a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper.     

Some new results in differential algebraic control theory

In the differential algebra framework, static state and dynamicstate feedback linearization are considered. The relationshipbetween dynamic feedback and flatness defined by Fliess is discussed.The existence of an equivalent proper differential I–Osystem of a given differential I–O system is discussed,which is closely related to the choice of a proper fictitiousoutput in control design. The concept flatness and its relationwith dynamic feedback linearizability, controllability, observability,invertibility and minimal realization are discussed. Finally,it is demonstrated that many fundamental control concepts andtheir interrelationships can be incorporated into an extendedcontrol diagram.     

Controlling multiparticle system on the line. I

We study classical multiparticle system (e.g. Toda lattice) on the line whose dynamics will be controlled by forces applied to few particles of the system. Various problem settings, typical for control theory are posed for this model; among those: studying accessibility and controllability properties, structure properties and feedback linearization of respective control system, time-optimal relocation of particles. We obtain complete or partial answers to the posed questions; criteria and methods of geometric control theory are employed. In the present part I we consider nonperiodic multiparticle system. In the forthcoming part II we address controllability issue for multiparticle system subject to periodic boundary conditions.     

Linear extensions of partial orders and reverse mathematics

We introduce the notion of τ‐like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω‐like means that every element has finitely many predecessors, while being ζ‐like means that every interval is finite. We consider statements of the form “any τ‐like partial order has a τ‐like linear extension” and “any τ‐like partial order is embeddable into τ” (when τ is ζ this result appears to be new). Working in the framework of reverse mathematics, we show that these statements are equivalent either to $\mathsf {B}{\Sigma }^{0}_{2}$ or to $\mathsf {ACA}_0$ over the usual base system $\mathsf {RCA}_0$.