Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: II

We study the rectilinear motion of a mass point with impacts against a stopper. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Grazing bifurcation and chaotic oscillations of vibro-impact systems with one degree of freedom

The bifurcations of dynamical systems, described by a second-order differential equation with periodic coefficients and an impact condition, are investigated. It is shown that a continuous change in the coefficients of the system, during which the number of impacts of the periodic solution increases, leads to the occurrence of a chaotic invariant set.     

A quantum model with one fermionic degree of freedom

A quantum model with one fermionic degree of freedom is discussed in detail. The operator action of the model has local operator gauge symmetry. A group of constrains on operator gauge potentialB ₀ and gauge transformation operatorU from some physical requirement are obtained. The Euler-Lagrange equation of motion of fermionic operator φ is just the usual equation of motion of fermion type. And the Euler-Lagrange equation of motion of operator gauge potentialB ₀ is just a constraint, which is just. the canonical quantization condition of fermion.     

Control of a system with one degree of freedom under complex restrictions

A very simple dynamical system with one degree of freedom, controlled by a force of bounded magnitude, is considered. It is assumed that the magnitude of the force may increase gradually at a finite rate and that the force is switched off instantaneously. Under these restrictions, which simulate real servo-systems, a control is constructed that steers the system to the origin and has the simplest possible structure.     

The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

In this paper we are concerned with the periodic Hamiltonian system with one degree of freedom, where the origin is a trivial solution. We assume that the corresponding linearized system at the origin is elliptic, and the characteristic exponents of the linearized system are $\pm i\omega$ with ω be a Diophantine number, moreover if the system is formally linearizable, then it is analytically linearizable. As a result, the origin is always stable in the sense of Liapunov in this case.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

Groups of symmetries for nonlinear thermodynamics with one degree of freedom

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 22, Dynamical Systems-3, 1995.     

Buffer phenomenon in systems with one and a half degrees of freedom

The buffer phenomenon is established for some classical mechanics problems that are described by pendulum-type equations with time-periodic small additive terms. This phenomenon is as follows: the systems under consideration can have an arbitrary fixed number of stable periodic modes if the system parameters are properly chosen.     

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.     

On constructing periodic solutions of quasi-linear non-self-contained systems with one degree of freedom, when amplitude equations possess multiple roots

Construction of periodic solutions of quasilinear non-self-contained systems with one degree of freedom, was investigated in [1 and 2]. In [1] the case of simple roots of amplitude equations was considered together with the case of a double root when the solution could be expanded into a series in integral powers of μ. In [2] the case of a double root is investigated in more detail Including expansions of solutions into series in μ^1/2. In the present paper, the case of arbitrary multiple roots for non-self-contained systems is reduced to the corresponding case for self-contained systems, which simplifies computations.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Forced vibration of plastic structures taking into account the elastic and damping properties of the material (systems with one degree of freedom)

A method, which takes into account the elastic properties and hysteresis losses of plastic materials, is proposed for calculating the forced vibrations of plastic structures with one degree of freedom. A solution of the equation of motion of the system is found in trigonometric form, and an algebraic equation for determining the vibration amplitude is obtained. In its general form the latter equation can be solved for special cases only; when it is required to determine the vibration amplitude in the general case, the successive approximations method is recommended, or rigorous solution of the algebraic equation for each specific case.Mekhanika Polimerov, Vol. 1, No. 3, pp. 101–106, 1965     

Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings. © 1996 John Wiley & Sons, Inc.