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 orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

Bifurcation of the self-excited oscillations of a plate with slight damping in a supersonic gas flow

A non-linear boundary-value problem is considered which simulates the oscillations of a plate in a supersonic gas flow. The classical version of the formulation of the problem, proposed by Bolotin, as well as several of its modifications considered by Holmes and Marsden, are taken as a basis. The oscillations of the plate are studied assuming that the damping coefficient is small. This version of the formulation of the problem leads to the need to investigate the bifurcations of the self-excited oscillations in a non-linear boundary-value problem in a case which is close to the critical case of a double pair of pure imaginary values of the stability spectrum. The bifurcation problem is reduced to the investigation of a complex second order non-linear differential equation by the method of normal forms. All the stages in the investigation are carried out without using the Bubnov method.     

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.     

Self-excited oscillations of a third-order system

A third-order self-excited oscillatory system in the neighbourhood of a stable local integral manifold is investigated. A periodic manifold and a corresponding system in amplitude-phase variables are constructed with approximately the required accuracy. Using the procedure of separation of variables (averaging) the conditions for the existence, uniqueness and stability of self-excited oscillation modes are established, and critical cases of the degeneracy of these conditions are considered. A thermomechanical model of the self-excitation of oscillations inherent in gas-dynamic systems with a heat source is taken as an example. The bifurcation pattern of self-excited oscillations in the space of the governing parameters of the system is investigated in the second approximation.     

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.     

The critical case of a pair of zero roots in a two-degree-of-freedom hamiltonian system

The problem of the motion of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its equilibrium position is considered. It is assumed that the characteristic equation of the linearized system has a pair of pure imaginary roots. The roots of the other pair are assumed to be close to or equal to zero, and in the latter case non-simple elementary dividers correspond to these roots. The problem of the existence, bifurcations and orbital stability of families of periodic motions, generated from the equilibrium position, is solved. Conditionally periodic motions are analysed. The problem of the boundedness of the trajectories of the system in the neighbourhood of the equilibrium position in the case when it is Lyapunov unstable, is considered. Non-linear oscillations of an artificial satellite in the region of its steady rotation around the normal to the orbit plane are investigated as an application.     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

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.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

Perturbations and Hopf bifurcation of the planar discontinuous dynamical system

The objective of the paper is to obtain results on the behavior of a specific plane discontinuous dynamical system in the neighbourhood of the singular point. A new technique of investigation is presented. Conditions for existence of the foci and centres are proposed. The focus-centre problem and Hopf bifurcation are considered. Appropriate examples are given to ilustrate the bifurcation theorem.     

A mechanical system containing weakly coupled subsystems

The concept of a mechanical system (model), containing coupled subsystems (MSCCS) is introduced. Examples of such a system are the Sun–planets–satellites system, a system of interacting moving objects, a system of translationally and rotationally moving celestial bodies, chains of coupled oscillators, Sommerfeld pendulums, spring systems, etc. The MSCCS subsystems and the entire system are analysed, and problems related to the investigation of the oscillations, bifurcation, stability, stabilization and resonance are stated. A solution of the oscillations problem is given for a class of MSCCS, described by reversible mechanical systems. It is proved that the autonomous MSCCS retains its family of symmetrical periodic motions (SPMs) under parametric perturbations, while in the periodic MSCCS a family of SPMs bifurcates by producing two families of SPMs. The two-body problem and the N-planet problem are investigated as applications. The generating properties of the two-body problem are established. For the N-planet problem it is proved that an (N + 1)-parametric family of orbits exists, close to elliptic orbits of arbitrary eccentricity, the family being parametrized by energy integral constant, and a syzygy of planets occurs.     

The increasingly complex structure of the bifurcation tree of a piecewise-smooth system

A new approach to the study of the dynamics of a piecewise-smooth system is proposed, which uses the a priori known possible bifurcation structures of the parameter space. In Section 1 the synthesis of the structures of the bifurcation tree of the system is considered, namely, the local structures, bifurcation bands, sources and nodes. It is shown that a node corresponding to a doubling bifurcation with reorientation of the domain of existence can generate a sequence of increasingly complex structures. Then the increasing number of unstable orbits serves as one of the mechanisms giving rise to the chaotic behaviour of the dynamical system. In Section 2 the procedure for synthesizing the structures of the bifurcation tree of a piecewise-smooth system proposed in the first part of the paper is applied to the problem of the forced vibrations of a linear oscillator with impacts against a stopping device. Period-doubling cascades are discovered, which are accompanied by the reorientation of the domain of existence of a solution relative to some bifurcation surface, namely, the trunk of the tree. A set of frequency intervals is distinguished on the bifurcation trunk, each containing an infinite sequence of increasingly complex local structures appearing and disappearing at the nodes. This specific mechanism, giving rise to the chaotic motion of the oscillator, is realized in neighbourhoods of the limiting nodal bifurcation points.     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

Effect of bifurcation on the semi-active optimal control problem

For nonlinear dynamic systems near bifurcation, the basins of attraction of fixed points as well as the steady-state responses can change considerably with a small variation of the bifurcating parameter. This paper studies the effect of bifurcation on the semi-active optimal control problem with fixed final state by using the cell mapping method. A system parameter is taken as the control. The admissible control values considered encompass a bifurcation point of the system. Global changes in the optimal control solution for different targets are studied. Saddle node, supercritical pitchfork and subcritical Hopf bifurcations are considered in the examples. It has been found that the global topology of the optimal control solution is strongly dependent on the state of the target.     

Forced oscillations of a non-linear system with a repulsive positional force

Non-linear systems with one degree of freedom, in which the positional force is directed away from the equilibrium position of the system, are considered. The existence of forced periodic oscillations, their Lyapunov stability, and the behaviour of amplitude-frequency characteristics are investigated. It is shown that stable periodic oscillations are possible in the case when the positional force has non-monotonic properties. Forced oscillations of a pendulum with respect to the upper equilibrium position are considered as an example.     

Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay

The parabolic functional differential equation $\frac{{\partial u}} {{\partial t}} = D\frac{{\partial ^2 u}} {{\partial x^2 }} - u + K(1 + \gamma \cos u(x + \theta ,t - T)) $ is considered on the circle [0, 2π]. Here, D > 0, T > 0, K > 0, and γ ∈ (0, 1). Such equations arise in the modeling of nonlinear optical systems with a time delay T > 0 and a spatial argument rotated by an angle θ ∈ [0, 2π) in the nonlocal feedback loop in the approximation of a thin circular layer. The goal of this study is to describe spatially inhomogeneous rotating-wave solutions bifurcating from a homogeneous stationary solution in the case of a Andronov-Hopf bifurcation. The existence of such waves is proved by passing to a moving coordinate system, which makes it possible to reduce the problem to the construction of a nontrivial solution to a periodic boundary value problem for a stationary delay differential equation. The existence of rotating waves in an annulus resulting from a Andronov-Hopf bifurcation is proved, and the leading coefficients in the expansion of the solution in powers of a small parameter are obtained. The conditions for the stability of waves are derived by constructing a normal form for the Andronov-Hopf bifurcation for the functional differential equation under study.     

Efficient methods in the search for periodic oscillations in dynamical systems

Efficient methods in the search for the periodic oscillations of dynamical systems are described. Their application to the sixteenth Hilbert problem for quadratic systems and the Aizerman problem is considered. A synthesis of the method of harmonic linearization with the applied bifurcation theory and numerical methods for calculting periodic oscillations is described.