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.     

一类非自治系统概周期解的存在唯一性

研究高压输电网中出现的概周期振荡现象,结合运用 Ｌｉａｐｕｎｏｖ 函数,获得了系统产生概周期振荡的先兆性条件,为避免系统产生概周期振荡提供了参考数据·     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Multiple periodic oscillations in a nonlinear suspension bridge system

In this paper, we study periodic oscillations in a suspension bridge system governed by the coupled nonlinear wave and beam equations describing oscillations in the supporting cable and roadbed under periodic external forces. By applying a variational reduction method, it is proved that the suspension bridge system has at least three periodic oscillations.     

Bifurcation of slow motions in a stiff spring pendulum

We consider free oscillations of a double pendulum, where one of the pendula is modelled as a very stiff spring. Contrary to a single spring pendulum numerical simulations show an unexpected large influence of the fast longitudinal oscillations on the slow pendulum oscillations even for extremely large values of the stiffness. The transition from the regular motion, which is governed by the dynamics of a rigid double pendulum close to a periodic orbit, to the irregular motion with large contributions from the longitudinal oscillations occurs due to a subcritical symmetry breaking bifurcation of the periodic solution. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE

Normal form method is first employed to study the Hopf-pitchfork bifurcation in neutral functional differential equation (NFDE), and is proved to be an efficient approach to show the rich dynamics (periodic and quasi-periodic oscillations) around the bifurcation point. We give an algorithm for calculating the third-order normal form in NFDE models, which naturally arise in the method of extended time delay autosynchronization (ETDAS). The existence of Hopf-pitchfork bifurcation in a van der Pol’s equation with extended delay feedback is given and the unfoldings near this critical point is obtained by applying our algorithm. Some interesting phenomena, such as the coexistence of several stable periodic oscillations (or quasi-periodic oscillations) and the existence of saddle connection bifurcation on a torus, are found by analyzing the bifurcation diagram and are illustrated by numerical method.     

Locally periodic thin domains with varying period

We analyze the behavior of the solutions of the Laplace equation with Neumann boundary conditions in a thin domain with a highly oscillatory behavior. The oscillations are locally periodic in the sense that both the amplitude and the period of the oscillations may not be constant and actually they vary in space. We obtain the asymptotic homogenized limit and provide some correctors. To accomplish this goal, we extend the unfolding operator method to the locally periodic case. The main ideas of this extension may be applied to other cases like perforated domains or reticulated structures, which are locally periodic with not necessarily a constant period.     

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.     

Using chaos to reduce oscillations: Experimental results

Chaotic dynamic systems are usually controlled in a way, which allows the replacement of chaotic behavior by the desired periodic motion. We give the example in which an originally regular (periodic) system is controlled in such a way as to make it chaotic. This approach based on the idea of dynamical absorber allows the significant reduction of the amplitude of the oscillations in the neighborhood of the resonance. We present experimental results, which confirm our previous numerical studies [D?browski A, Kapitaniak T. Using chaos to reduce oscillations. Nonlinear Phenomen Complex Syst 2001;4(2):206–11].     

Bifurcation analysis of the mathematical model of the NO+CO/Pt(100) reaction

The article presents a detailed bifurcation analysis of steady and periodic states for a new mathematical model of the NO+CO/Pt(100) reaction. Various bifurcation diagrams are constructed in the planes of partial reagent pressures and surface temperature. Regions with oscillations and multiple steady solutions are investigated. Isolated branches of steady and periodic states are identified. An “explosive” bifurcation of the periodic solution leading to chaotic alternation of small-and large-amplitude oscillations is detected and analyzed for the first time. A good quantitative fit is demonstrated between modeling results and experimental data. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 52–78.     

Structure and properties of four-kink multisolitons of the sine-Gordon equation

The dynamics of nonlinear waves of the sine-Gordon equation with a spatially modulated periodic potential are studied using analytical and numerical methods. The structure and properties of four-kink multisolitons excited on two identical attracting impurities are determined. For small-amplitude oscillations, an analytical spectrum of the oscillations is obtained, which is in qualitatively agreement with the numerical results.     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

Spatially periodic fundamental solutions of the theory of oscillations

Spatially periodic fundamental solutions of the theory of oscillations are constructed, applicable to anisotropic elastic media with a general form of anisotropy. The results are compared with the isotropic case.     

Epidemic dynamics in a heterogeneous incompletely isolated population with allowance for seasonal variations in the infection rate

Mathematical parasite-host models are generalized to the case when the population members differ in susceptibility and contagiousness, there is an external source of infection, and the model parameters depend periodically (seasonally) on time. The model is proved to have a periodic solution that is unique and exponentially stable for sufficiently small periodic oscillations of the coefficients.     

具脉冲扰动的时滞Ivlev型捕食系统

构建了具脉冲扰动的时滞Ivlev型捕食系统,获得了捕食者灭绝周期解全局渐近吸引和系统持续生存的充分条件.数值例子验证了理论结果,揭示了系统诸如吸引子突变,高倍周期振动,分支等复杂的动力学行为,最后进行了总结与讨论.     

The oscillations of a satellite about a direction fixed in absolute space

The stability of the plane oscillations of a satellite about the centre of mass in a central Newtonian gravitational field is investigated. The orbit of the centre of mass is circular and the principal central moments of inertia of the satellite are different. In unperturbed motion, one of the axes of inertia is perpendicular to the plane of the orbit, while the satellite performs periodic oscillations about a direction fixed in absolute space. The problem of the stability of these oscillations with respect to plane and spatial perturbations is investigated.     

Modeling chemical reactions by forced limit-cycle oscillator: synchronization phenomena and transition to chaos

The lattice limit-cycle (LLC) model is introduced as a minimal mean-field scheme which can model reactive dynamics on lattices (low dimensional supports) producing non-linear limit cycle oscillations. Under the influence of an external periodic force the dynamics of the LLC may be drastically modified. Synchronization phenomena, bifurcations and transitions to chaos are observed as a function of the strength of the force. Taking advantage of the drastic change on the dynamics due to the periodic forcing, it is possible to modify the output/product or the production rate of a chemical reaction at will, simply by applying a periodic force to it, without the need to change the support properties or the experimental conditions.     

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.