Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

An inverted pendulum on a fixed and a moving base

The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.     

Stepwise synthesis of constrained controls for single input nonlinear systems of special form

For control systems of the form \({dx/dt = a(x) + B(x)\beta(x, u)}\) with one-dimensional control, where a(x) is an n-dimensional vector function, B(x) is an \({(n \times m)}\)-matrix, and \({\beta(x, u)}\) is an m-dimensional vector function, the method of constructing of stepwise synthesis control is proposed. At first, under certain conditions we reduce such system to a system consisting of m subsystems; in each subsystem all equations are linear except of the last one. Further we propose the method for construction of controls which transfer an arbitrary initial point to the equilibrium point in a certain finite time. Each such control is constructed as a concatenation of a finite number of positional controls (we call it a stepwise synthesis control). On each step of our method we choose a new synthesis control. In this connection, nonlinearity of a system with respect to a control is essentially used. The obtained results are illustrated by examples. In particular, the problem of the complete stoppage of a two-link pendulum with the help of non-linear forces is solved. Finally, we introduce the class of nonlinear systems which is called the class of staircase systems that provides the applicability of our method.     

A construction scheme for linear and non-linear codes

A scheme for construcing linear and non-linear codes is presented. It constructs a code of block length 2n from two constituent codes of block length n. Codes so constructed can be either linear or non-linear even when the constituent codes are linear. The construction of many known linear and non-linear codes using this scheme will be shown.     

The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov's problem

Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov's problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version (n = 2) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version (2 < n ≤ 5 · 10⁵), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality 45000 ≤ n ≤ 62597 and the orbital eccentricities e < 0.25. Use of the Arnold–Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov's problem.     

The enumeration of certain sets of block maps

An n-block is a sequence b₁b₂…b_n, where b_i ϵ Z₂ for 1 ⩽ i ⩽ n, and an n-block map is a function from the set of n-blocks to the ring Z₂. The n-block maps form the ring GF(2) {x₁, x₂,…, x_n | x_i² = 1}. The set of all block maps with the operations of addition and polynomial substitution form a near-ring. The general problem of searching non-linear block maps for factors with respect to polynomial substitution seems to be extremely complex. However, effective search procedures have been described for factors within certain sets of block maps which are non-linear but are linear in the first variable. In this paper we show that the smallest of these sets contains just over 85% of the block maps which are linear in the first variable. The largest of the sets, for which there is an additional step in the search procedure, contains over 99% of the block maps which are linear in the first variable.     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

Complète intégrabilité singulière

We show that an holomorphic vector field in a neighbourhood of its singular point 0 0 ∈ ℂⁿ is analytically normalizable as soon as it has a sufficiently large number of commuting holomorphic vector fields, a sufficiently large number of formal first integrals, and that a Diophantian small divisors condition related to its linear part is satisfied.     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.     

The stability and stabilization of non-linear,non-stationary mechanical systems

Mechanical systems acted upon by extremely non-linear positional forces are considered. The decomposition method is used to determine the sufficient conditions for asymptotic stability of an equilibrium. Problems of stabilizing the equilibrium of non-linear, non-stationary systems with specified potential forces by adding forces of different structure are studied. For systems with a non-stationary, homogeneous, positive-definite potential, the possibility of stabilization by linear dissipative forces, uncharacteristic of linear systems, is established. For systems with an even number of coordinates n ≥ 4, in the presence of dissipative forces with complete dissipation, the possibility of vibrational stabilization by adding circular and gyroscopic forces with coefficients fluctuating about zero is demonstrated.     

Theory of annihilation games—I

Place tokens on distinct vertices of an arbitrary finite digraph with n vertices which may contain cycles or loops. Each of two players alternately selects a token and moves it from its present position u to a neighboring vertex v along a directed edge which may be a loop. If v is occupied, and u ≠ v, both tokens get annihilated and phase out of the game. The player first unable to move is the loser, the other the winner. If there is no last move, the outcome is declared a draw. An O(n⁶) algorithm for computing the previous-player-winning, next-player-winning and draw positions of the game is given. Furthermore, an algorithm is given for computing a best strategy in O(n⁶) steps and winning—starting from a next-player-winning position—in O(n⁵) moves.     

The structure of module derivations of Banach algebras of differentiable functions

This paper studies the structure and continuity of derivations of the Banach algebra Cⁿ(I) of n times continuously differentiable functions on an interval I into Banach Cⁿ(I)-modules. The structure of derivations into finite dimensional modules is completely determined. The question of when an arbitrary derivation splits into the sum of continuous and singular parts is discussed. An example is constructed of a derivation of C¹(I) which is discontinuous on every dense subalgebra.     

On a heterochromatic number for hypercubes

The neighbourhood heterochromatic numbernh_c(G) of a non-empty graph G is the smallest integer l such that for every colouring of G with exactly l colours, G contains a vertex all of whose neighbours have different colours. We prove that lim_n→∞(nh_c(Gⁿ)-1)/|V(Gⁿ)|=1 for any connected graph G with at least two vertices. We also give upper and lower bounds for the neighbourhood heterochromatic number of the 2ⁿ-dimensional hypercube.     

The non-linear oscillations of a pendulum of variable length on a vibrating base

A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone.     

On the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Representations of Boolean functions by exclusive-OR sums (modulo 2) of pseudoproducts is studied. An ExOR-sum of pseudoproducts (ESPP) is the sum modulo 2 of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this form, and the length of a Boolean function in the class of ESPPs is defined as the minimum length of an ESPP representing this function. The Shannon function L ^ESPP(n) of the length of Boolean functions in the class of ESPPs is considered, which equals the maximum length of a Boolean function of n variables in this class. Lower and upper bounds for the Shannon function L ^ESPP(n) are found. The upper bound is proved by using an algorithm which can be applied to construct representations by ExOR-sums of pseudoproducts for particular Boolean functions.     

The periodic motions of a non-autonomous Hamiltonian system with two degrees of freedom at parametric resonance of the fundamental type

The motion of an almost autonomous Hamiltonian system with two degrees of freedom, 2π-periodic in time, is considered. It is assumed that the origin is an equilibrium position of the system, the linearized unperturbed system is stable, and its characteristic exponents ±iωj (j = 1,2) are pure imaginary. In addition, it is assumed that the number 2ω₁ is approximately an integer, that is, the system exhibits parametric resonance of the fundamental type. Using Poincaré's theory of periodic motion and KAM-theory, it is shown that 4π-periodic motions of the system exist in a fairly small neighbourhood of the origin, and their bifurcation and stability are investigated. As applications, periodic motions are constructed in cases of parametric resonance of the fundamental type in the following problems: the plane elliptical restricted three-body problem near triangular libration points, and the problem of the motion of a dynamically symmetrical artificial satellite near its cylindrical precession in an elliptical orbit of small eccentricity.     

On a counterexample in analysis

An example of multivalued convex-valued Lipschitz mapping from ? ⁿ into ? ^m such that, at any point, the support function of this mapping has no mixed derivatives in the sense of Gâteaux with respect to the initial and conjugate variables is constructed.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.