The Falkner-Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner-Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336-394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.     

Local maximality of hyperbolic sets

Two properties of a hyperbolic set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F, there exists a locally maximal set F′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the set F are related to the behavior of trajectories outside F, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on F.     

F-quadratic stochastic operators

In this paper, we introduce the notion of an F-quadratic stochastic operator. It is shown that each F-quadratic operator has a unique fixed point. Besides, it is proved that any trajectory of an F-quadratic stochastic operator exponentially rapidly converges to this fixed point.     

Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.     

Billiards in convex bodies with acute angles

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ? R^d has the property that the tangent cone of every non-smooth point q ? ?K is acute (in a certain sense), then there is a closed billiard trajectory in K.     

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory     

Globally and locally attractive solutions for quasi-periodically forced systems

We consider a class of differential equations, , with ω∈Rd, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x)=x2p+1, p∈N, we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g(x)=x² (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor.     

Semiclassical electron motion and Novikov's conjecture

The structure of nonclosed trajectories of semiclassical electron motion in a crystal in a weak constant and uniform magnetic field of irrationality degree 3 is considered. It is proved that two cases can exist. In the first case the set of energy levels which contain nonclosed trajectories is a closed interval and any regular nonclosed trajectory lies in a finite-wide stripe and comes through it in one direction. In the other case, there is only one energy level containing nonclosed trajectories. Bibliography:5 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 228–234     

Free and almost-free subsemigroups of a free semigroup

A subsemigroup S of a free semigroup F(Σ) is almost-free if there is a free subsemigroupT such that S?T?F(Σ) and T/S is finite. It is shown that it is decidable whether a subsemigroup generated by a regular subset of F(Σ) is almost-free. Sufficient- conditions are given such that if a family F of subsets of F(Σ) satisfies these conditions, then it is undecidable for L∈F whether the subsemigroup generated by L is free and also whether it is almost-free.     

Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems

It is shown that, if a parametrized fämily of extremals F can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity.     

Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation

The purpose of this paper is twofold. First, we present the existence theorem of an optimal trajectory in a nonconvex variational problem with recursive integral functionals by employing the norm-topology of a weighted Sobolev space. We show the continuity of the integral functional and the compactness of the set of admissible trajectories. Second, we show that a recursive integrand is represented by a normal integrand under the conditions guaranteeing the existence of optimal trajectories. We also demonstrate that if the recursive integrand satisfies the convexity conditions, then the normal integrand is a convex function. These results are achieved by the application of the representation theorem in Lp-spaces.     

Finite central height in linear groups

J.D. Dixon has characterized those pairs (n,F), where n is a positive integer and F a field, for which every locally nilpotent subgroup of GL(n,F) is nilpotent. He showed further that these pairs (n,F) have the stronger property that there is a bound on the nilpotency class of the nilpotent subgroups of GL(n,F). In this note we show that these pairs (n,F) have the still stronger property that every subgroup of GL(n,F) has finite bounded central height. Our main result generalizes to groups of automorphisms of Noetherian modules over commutative rings.     

An algorithm to recognize a middle graph

A graph F is called middle if there exists a graph G such that there is a one-to-one correspondence between the vertices of F and the vertices and edges of G such that two vertices of F are adjacent if and only if the corresponding elements (considered as subsets of the set of vertices) have a non-empty intersection.In this paper we present a linear time algorithm for the recognition of the middle graphs. The algorithm is based on a computer-oriented characterization of middle graphs. We show also how the algorithm can be generalized to recognize the middle graphs of hypergraphs.     

On the inverse Sturm-Liouville problem for spatially symmetric operators,III

In this note it is proved that x(·) a boundary trajectory of a Lipschitz-continuous differential inclusion ? ? F(t, x), x(0) = 0, the tangent cone to F(t, x(t)) at ?(t) that of attainable set E(t) at x(t) coincide for almost every t provided that ?F(t, x) is smooth (similar results with more stringent assumptions were obtained by H. Hermes (J. Differential Equations3 (1967), 256–270) and S. ?ojasiewicz, Jr. (Asterisque75–76 (1980), 187–197)). It is also proved that the outward normal to these cones along the trajectory is Lipschitz-continuous (in t). Moreover, using the lower, one-side, directional derivative instead of F. H. Clarke's generalised gradient, first-order necessary conditions are obtained, which can be stronger than those of Clarke (in “International Symposium on the Calculus of Variation and Optimal Control, University of Wisconsin, Madison, Wisconsin, September 1975”). The main ideas of this paper were presented in J. Hale's seminar at Brown University (March 1976).     

Extension Operators with Analytic Values

If F ? ? is a closed set such that the space of all Whitney jets on F admits an extension operator then there exists such an extension operator whose values are holomorphic in ?F if and only if ?F is compact. In the case F is a compact set, there is even an extension operatorfor which the extensions are holomorphic in (? ∪ {∞})F.     

Saturation problems with regularity constraints

For a graph F, we say that another graph G is F-saturated, if G is F-free and adding any edge to G would create a copy of F. We study for a given graph F and integer n whether there exists a regular n-vertex F-saturated graph, and if it does, what is the smallest number of edges of such a graph. We mainly focus on the case when F is a complete graph and prove for example that there exists a $K_3$-saturated regular graph on n vertices for every large enough n.We also study two relaxed versions of the problem: when we only require that no regular F-free supergraph of G should exist or when we drop the F-free condition and only require that any newly added edge should create a new copy of F.     

Allowable Interval Sequences and Line Transversals in the Plane

Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving two new results and generalizing three old ones. Two of the old results are Klee??s assertion that if F is totally separated then T(3) implies T, and the following variation of Hadwiger??s Transversal Theorem proved by Wenger and (independently) Tverberg: If F is ordered and each four sets of F have some transversal which respects the order on F, then there is a transversal for all of F which respects this order. The third old result (a consequence of an observation made by Kramer) and the first of the new results (which partially settles a 2008 conjecture of Eckhoff) deal with fractional transversals and share the following general form: If F has property T(k) and meets certain other conditions, then there exists a transversal of some m sets in F, with k<m<n. The second new result establishes a link between transversal properties and separation properties of certain families of convex sets.     

On Trajectories Entirely Situated Near a Hyperbolic Set

Let I ₁ be a set of points such that their trajectories under a diffeomorphism f ₁ are entirely close enough to a hyperbolic set F ₁ of this diffeomorphism. Then it is proved that the structure of I ₁ and restriction \( {f}_1\left|{}_{I_1}\right. \) (“motion in I ₁”) are essentially defined (up to an equivariant homeomorphism) by “internal dynamics” in F ₁ , i.e., by the restriction \( {f}_1\left|{}_{{}_{F_1}}\right. \) . (In more detail: the equivariant homeomorphism g ₁ of the set F ₁ on the hyperbolic set F ₂ of the second diffeomorphism f ₂ (probably, acting on another manifold M ₂) is extendable to an equivariant homeomorphic embedding I ₁ → M ₂ . The image of the imbedding contains all the trajectories f ₂ close enough to F ₂ .)     

Positive orthant scalar controllability of bilinear systems

For the bilinear control system $\dot x = \left( {A + uB} \right)x$ ,x ∈ ? ⁿ ,u ∈ ?, whereA is ann ×n essentially nonnegative matrix, andB is a diagonal matrix, the following controllability problem is investigated: can any two points with positive coordinates be joined by a trajectory of the system? Forn>2, the answer is negative in the generic case: hypersurfaces in ? ⁿ are constructed that are intersected by all the trajectories of the system in one direction.