On Approximate Representation of Periodic Motions at a Saddle Point

The results of a qualitative analysis of a bistable oscillator are applied to the approximate integration of a system with a closed trajectory about a saddle point.     

On the large amplitude oscillations of a thin elastic beam

This paper is concerned with the finite amplitude, free, planar oscillations of a thin elastic beam. By assuming the motion to be inextensional but at the same time recognizing the existence of a resultant normal force acting on each cross-section of the beam a system of governing equations is derived which is manageable but still meaningful. For the case of the simply-supported beam a finite difference, Galerkin, and (regular) perturbation solutions are explicitly obtained. The results are compared and discussed. In the course of obtaining the various solutions it is found that an additional simplification in the form of the governing equations is possible. This simplification turns out to be quite important from a general point of view of obtaining approximate analytical solutions.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.     

Numerical analysis of the oscillations of human vocal cords

Nonlinear differential equations of the fifth order, which govern the oscillations of the human vocal cords, are analyzed. A combined numerical technique is used to observe the regular and irregular solutions of these equations. A scenario from periodic, via quasiperiodic to chaotic attractor is demonstrated. A perturbation of the strange chaotic attractor to principal, combination and primary resonances is also discussed and illustrated.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.     

Eigenoscillations of a fluid in a canonical domain and functional difference equations

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a fluid under gravity in such a domain. These solutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (by quadrature). The behavior of the solutions at large distances is studied by use of the saddle point technique. The corresponding eigenoscillations of a fluid are then interpreted as generalized eigenfunctions of the continuous spectrum.     

Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains

The exact solutions for the motion of a Maxwell fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of the Laplace transform. These solutions are presented as sum of the steady-state and transient solutions and describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion is described by the steady-state solution which is periodic in time and independent of the initial conditions. Finally, by means of graphical illustrations, the required times to reach the steady-state are determined for sine, cosine and combined oscillations of the boundary.     

Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

Large amplitude vibrations of a Timoshenko beam under an influence of temperature are analysed in this paper. In the considered model the temperature increases instantly and the heat is uniformly distributed along the beams length and cross-section. The mathematical model, represented by partial differential equations takes into account thermal and mechanical loadings. Next, the problem is reduced by means of the Galerkin method, considering the first three natural vibration modes of a simply supported beam in the both ends. The influence of the temperature on amplitudes and localisation of the resonance zones and stability of the solutions is studied numerically and analytically by the multiple time scale method. The bifurcation points, existence of unstable lobes and transition from regular to chaotic oscillations are shown.     

Analysis of a class of undesired multimode oscillations

The behavior of a negative-resistance circuit in which the parameters permit self-sustained oscillations to occur is discussed. In such a circuit, undesired oscillations of higher frequency caused by parasitic elements can exist in addition to normal oscillations. Parasitic oscillations are described by second-order simultaneous non-linear differential equations, taking into account of existence of the shunt capacitance, lead inductance and resistance of a negative-resistance element. Assuming that the frequency of the parasitic oscillations is sufficiently high compared with that of the desired normal oscillations, the approximate periodic solutions are obtained by using a method of averaging. In addition, the theoretical results are compared with the observed behavior of an experimental oscillator having similar parameters.     

Undercompressive shocks for nonstrictly hyperbolic conservation laws

We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.     

Regular oscillations or chaos in a fractional order system with any effective dimension

This paper introduces a fractional order system which can generate regular oscillations or create chaos. It shows that this system is capable to create regular or nonregular oscillations under suitable conditions. These necessary conditions are achieved by violation of the no-chaos criteria. The effective dimension of the proposed system can be chosen any order less than three. Therefore, this system is a good example for limit cycle or chaos generation via fractional-order systems with low orders. Numerical simulations illustrate behavior of the proposed system in different situations.     

Radiation from a ring source around a compressible plasma column

The electromagnetic waves and plasma oscillations excited by a magnetic current ring source around an infinitely long compressible plasma column are studied. The governing partial differential equations for the electromagnetic and plasma modes are first formulated. They are then converted to ordinary differential equations by means of the Fourier transformation. The fields outside the plasma column are composed of a surface wave part contributed by the poles on the path of integration and a radiated part arising from branch points. The far zone radiation field is evaluated by the method of steepest descent through a saddle point. The presence of the plasma column modifies the radiated electromagnetic field of the ring source, and the radiation pattern depends on the plasma parameters in a complicated way. This work was supported by the U.S. Army Research Office - Durham (N.C.) and by the National Science Foundation.     

Transversal oscillations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.     

Free oscillations of a fluid in a cylindrical container with arbitrary axisymmetric bottom and elastic elements on the free surface of the fluid

We consider the problem of free oscillations of an ideal fluid in a container that has the form of a right circular cylinder with arbitrary axisymmetric bottom in the case where the unperturbed free surface of the fluid is covered by an elastic membrane or plate. Using the expansion in eigenfunctions of an auxiliary spectral problem with a parameter in boundary conditions and the method of decomposition of the domain of meridional cross-section of a container, we obtain an analytic solution of the problem. Individual examples of mechanical systems are considered, for which we construct solutions by using the proposed algorithm, analyze these solutions, and compute the frequencies and forms of oscillations.     

Properties of Chaotic and Regular Boundary Crisis in Dissipative Driven Nonlinear Oscillators

The phenomenon of the chaotic boundary crisis and the related concept of the chaotic destroyer saddle has become recently a new problem in the studies of the destruction of chaotic attractors in nonlinear oscillators. As it is known, in the case of regular boundary crisis, the homoclinic bifurcation of the destroyer saddle defines the parameters of the annihilation of the chaotic attractor. In contrast, at the chaotic boundary crisis, the outset of the destroyer saddle which branches away from the chaotic attractor is tangled prior to the crisis. In our paper, the main point of interest is the problem of a relation, if any, between the homoclinic tangling of the destroyer saddle and the other properties of the system which may accompany the chaotic as well as the regular boundary crisis. In particular, the question if the phenomena of fractal basin boundary, indeterminate outcome, and a period of the destroyer saddle, are directly implied by the structure of the destroyer saddle invariant manifolds, is examined for some examples of the boundary crisis that occur in the mathematical models of the twin-well and the single-well potential nonlinear oscillators.     

Study of a transition in the qualitative behavior of a simple oscillator with Coulomb friction

A simple mass-spring system is submitted to a constant force in addition to a periodic perturbation of rectangular wave shape. It has been obtained in a previous study that the range of the period-amplitude plane of this perturbation, where the trajectories are sliding with no loss of contact, is divided into two parts, one in which there exist infinitely many equilibrium states and no periodic solutions, and another one where there exist periodic solutions and no equilibrium states. The present work focuses on the transition between these two parts. All along the transition line, there exists a single equilibrium state. Initial data out of equilibrium lead either to a periodic trajectory, or to a trajectory, which tends to the equilibrium or to a periodic solution, either in finite time or at infinity.     

On the non-linear vibration of a thin ring with steady pressurization

Large amplitude, flexural oscillations of an inextensible, linearly elastic, pressurized ring are analyzed. Non-linear governing equations describing the planar motion of a thin rod curved in its undeformed state and subject to a distributed load applied normal to the neutral axis are developed using Hamilton's extended principle. The equations are specialized to study the behavior of a circular ring, and approximate solutions are obtained for a single mode response by a perturbation technique. Free, undamped oscillations and forced response of the ring near resonance are discussed. The influence of the magnitude of pressurization on the non-linear character of the motion is investigated.     

参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时，会存在快慢耦合效应，与单项激励不同，参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化，也会产生一些特殊的非线性现象，为此，本文以两耦合Hodgkin-Huxley细胞模型为例，引入周期参外联合激励，探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统，得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为，结合转换相图，揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明，在激励幅值较小时，系统表现为概周期振荡，两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡，导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内，存在唯一稳定的平衡曲线，使得系统的轨迹逐渐趋向该平衡曲线，产生沉寂态，并随着慢变参数的变化，由分岔进入激发态．同时，快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同，导致不同形式的簇发振荡．另外，与单项激励下的情形不同，联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内，从而失去对簇发振荡的影响．     

Confinement of dynamical chaos expansion in rotifer populations inhabiting heterogeneous environments: an effect of attractor size invariance

We study the impact of inter-habitat interactions in rotifer populations with intrinsically different types of dynamics: chaotic and regular. For this purpose we use a modified version of the Consensus model. The Consensus model has been shown to be realistic enough to reproduce distinguishing features of the rotifer species dynamics. Being uncoupled, a habitat with chaotic dynamics and habitats with regular oscillations of the rotifer density do not affect each other. Migration of the rotifers between the habitats leads to the invasion of chaos into neighboring habitats, that is chaotic rotifer oscillations replace regular oscillations of the rotifer density in neighboring habitats. The invasion of chaos distorts spatial borders of the rotifer oscillations synchronization clusters. We show that the invasion of chaos is spatially confined. Invariance of the attractor size under transformation of regular oscillations into chaotic ones is demonstrated to be responsible for the chaos confinement.     

Regular and irregular motions in a duffing oscillator

This paper deals with the problems of identifying oscillations with regular and chaotic attractors in deterministic oscillating systems. It discusses a new approach based on the dynamical principle of symmetry, the construction of aperiodic solution domains, and the analysis of characteristic indices of quasistatic solutions for circular trajectories (in polar coordinates). The Duffing equation is examined within the framework of the dynamical principle of symmetry. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 139–144, January, 2000.