JDCS, vol. 1, #3 (1995), pp. 295-446

1. L. Ortiz-Bobadilla
QUADRATIC VECTOR FIELDS IN CP-2 WITH TWO SADDLE-NODE TYPE. SINGULARITIES AT INFINITY, pp. 295 - 317

The work is devoted to the study of some group properties of the monodromy transformations of a family of vector fields in CP-2 at infinity. For additional information, click here

2. A.A. Agrachev
ON REGULARITY PROPERTIES OF EXTREMAL CONTROLS, pp. 319 - 324

We prove some regularity properties of the optimal controls for the smooth bracket generating systems with scalar control parameters, and show that the Cantor sets cannot be the sets of switching points.

3. A.V. Fursikov
EXACT BOUNDARY ZERO CONTROLLABILITY OF THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS, pp. 325 - 350

In a bounded three-dimensional domain U a solenoidal vector field v(x) is given. We construct a vector field z(t,x) defined on the lateral surface (0,T)xdU of the cylinder (0,T)xU which possesses the following property: he solution w(t,x) of the boundary value problem for the Navier-Stokes equation with the initial value v(x) and the boundary Dirichlet condition z(t,x) satisfies the relation w(T,x)=0 at the instant T.

4. M.B. Sevryuk
KAM-STABLE HAMILTONIANS, pp. 351 - 366

We present a simple proof of Ruessmanns theorem on invariant tori of analytic perturbations of analytic integrable Hamiltonian systems. Ruessmanns theorem asserts that if the image of a certain given mapping does not lie in any linear hyperplane passing through the origin, then any sufficiently small Hamiltonian perturbation of this integrable system possesses many invariant tori close to the unperturbed tori. The main idea of our proof is that we embed the perturbed Hamiltonian in a family of Hamiltonians depending on an external multidimensional parameter. We also show that the Ruessmann condition is necessary (i.e., not only sufficient) for the existence of perturbed tori and give analogs of Ruessmanns theorem for exact symplectic diffeomorphisms, reversible flows, and reversible deffeomorphisms.

5. W. Balser
AN INTEGRAL EQUATION FOR NORMAL SOLUTIONS TO MEROMORPHIC DIFFERENTIAL EQUATIONS, pp. 367 - 378

Normal solutions for meromorphic systems of linear ODE have been defined and studied by the author, in collaboration with W.B.Jurkat and D.A.Lutz. Here, we show that they satisfy a system of integral equations.

6. A.V. Pukhlikov
HAMILTONIAN STRUCTURES IN OPTIMAL CONTROL THEORY, pp. 379 - 401

The concept of a piecewise smooth Hamiltonian system, motivated by optimal control theory, is introduced and developed. It is proved that the Poisson bracket of two integrals of such a system is globally continuous. Integrable piecewise smooth systems are proved to be equivalent to smooth integrable systems.

7. M. Villarini
ALGEBRAIC NONSOLVABILITY OF THE PROBLEM OF EXISTENCE OF HOLOMORPHIC FIRST INTEGRALS, pp. 403 - 425

We prove that the existence of a holomorphic first integral of an analytic differential equation is an algebraically nonsolvable problem. Moreover, we provea perturbation lemma showing that a nilpotent singularity of a differential equation in the plane cannot have a holomorphic first integral independently of the prolongation of any n-th order jet. We give an application to the case of nilpotent centers. For additional information, click here

8. A.N. Starkov
FUCHSIAN GROUPS FROM THE DYNAMICAL VIEWPOINT, pp. 427 - 445

Here we survey the results on the structure of Fuchsian groups due to Hopf, Hedlund, Sullivan, Nicholls, Pommerenke, and others from the viewpoint of the dynamics of the geodesic and horocycle flows on the corresponding surfaces. Special attention is given to the structure of horocycle orbits; in particular, we construct Fuchsian groups with new types of horocycle orbits which are neither closed nor dense in the nonwandering set. We give a unique classification of Fuchsian groups from the dynamical viewpoint and indicate some open problems.