Homoclinic twisting bifurcations and cusp horseshoe maps

Chaotic dynamics arises when the unstable manifold of a hyperbolic equilibrium point changes its twist type along a homoclinic orbit as some generic parameter is varied. Such bifurcation points occur naturally in singularly perturbed systems. Some quotient symbolic systems induced from the Bernoulli symbolic system on two symbols are proved to be characteristic for this new mechanism of chaos generation. Combination of geometrical and analytical methods is proved to be more fruitful.     

The discussion about horseshoes and finite subharmonic bifurcations

In this paper, finite subharmonic bifurcations have been discussed by means of some examples. It is found that for centrally symmetric system, under small disturbance, if it has two independant sequences of subharmonic bifurcations, the system passes to chaos (horeseshoe) through finite subharmonic bifurcations, and that for noncentrally symmetric system, the relation between subharmonic bifurcations and horseshoe is complicated.     

Nonlinear analysis of unbalanced mass of vertical conveyor: primary, subharmonic, and superharmonic response

In the area of mechanics and electronics, the behaviors of mechanical systems under periodic loadings have been examined by many researchers. Vertical conveyors are effective examples observing various kinds of parameters of this problem. In this study, primary, subharmonic, and superharmonic responses have been investigated with multiple scales along with numerical methods for vertical conveyors. The change in the parameters of motion, stability condition, and jump phenomena has been shown graphically by Mathematica software for comparing the results. Both analytical and numerical results obtained had good agreement.     

Symmetry breaking bifurcations of a parametrically excited pendulum

This paper examines the bifurcation behavior of a planar pendulum subjected to high-frequency parametric excitation along a tilted angle. Parametric nonlinear identification is performed on the experimental system via an optimization approach that utilizes a developed approximate analytical solution. Experimental and theoretical efforts then consider the influence of a subtle tilt angle in the applied parametric excitation by contrasting the predicted and observed mean angle bifurcations with the bifurcations due to excitation applied in either the vertical or horizontal direction. Results show that small deviations from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations as opposed to pitchfork bifurcations.     

Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum

Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability.     

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov'smethod is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system for various resonant cases.Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.     

Generalized thermoelastic functionally graded solid with a periodically varying heat source

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity without energy dissipation (TEWOED). The governing equations of generalized thermoelasticity without energy dissipation (GN model type II) for a functionally graded materials (FGM) (i.e. material with spatially varying material properties)are established. The governing equations are expressed in Laplace–Fourier double transform domain and solved in that domain. Now, the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand is obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking nonhomogeneity parameter suitably. Finally the results obtained are presented graphically to show the effect of nonhomogeneity on displacement, temperature, stress and strain.     

Acoustic waves in a cylindrical duct with periodically varying cross section

The null field approach is used to study the propagation of acoustic waves in a rotationally symmetric hard-walled duct with a periodically varying cross section. For a radius that varies sinusoidally along the axial distance, numerical computations give the axial wave number and the passbands and stopbands of the modes of the duct. In particular, small passbands are seen to exist even for very large variations in radius, and probably all the way to the point where the duct is cut off. We also present some plots which show the pressure pattern inside the duct.     

Hamiltonian long wave expansions for internal waves over a periodically varying bottom

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.     

The behaviour of a simple pendulum with uniformly shortening string length

Whilst watching the loading of a ship by crane it is often observed that the cargo shifts abruptly during the lifting stage. In order to try to understand this kind of behaviour the motion of a simple undamped pendulum, which is being shortened at a constant speed, is determined. It is found that the tension in the string always increases initially from its static value and that this increase in tension is very large even when the initial angular speed is small. Finally, it is proved that the linearised approximation is most unsatisfactory even when the initial amplitude of the swing and the shortening rate are small.     

Sound emission in a nonequilibrium medium from a plane surface with periodically varying temperature

Any change in temperature T of a continuous medium is accompanied by a change in its density , which gives rise to motion and, in particular, to sound emission. This question, which has been examined for the usual equilibrium medium, in [1], for instance, is extended in the present paper to the case of a nonequilibrium medium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 195–199, May–June, 1975.     

On the motion of incompressible-medium particles in a domain with a periodically varying boundary

The motion of incompressible-medium particles in a cavity bounded by a plane bottom, vertical lateral walls, and a top boundary deformed in accordance with an arbitrary periodic law is investigated. The problem is reduced to solving the Hamilton equations with a time-periodic Hamiltonian. To study the system, the Hamilton system averaging method and the KAM (Kolmogorov, Arnold, Mozer) theory are used. A novel modification of the averaging procedure using the Poincaré point map is proposed. Correct to an exponentially small cavity boundary deformation amplitude, the Poincaré map points lie on the closed integral curves of an averaged autonomous Hamiltonian system. An asymptotic expansion of the averaged Hamiltonian in the amplitude is written down. The method is applied to the solution of the following problems: (i) the Stokes problem of mass transfer by a progressive wave on the surface of a heavy finite-depth fluid and (ii) the problems of particle motion in a thin layer of viscous or viscoplastic medium with a deformable boundary. For the case of finite amplitudes, qualitative agreement between the results of the asymptotic theory and the numerical calculations is obtained. The reasons for the appearance of a stochastic regime are discussed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–19, July–August, 2000. The work received financial support from the Russian Foundation for Basic Research (project 99-01-00250).     

Oscillations of an ice sheet under a periodically varying load

Translated from Prikiadnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 141–146, September–October, 1992.     

Harmonic,subharmonic, superharmonic,simultaneous sub/super harmonic and combination resonances of self-excited two coupled second order systems to multi-frequency excitation

Harmonic, subharmonic, superharmonic, simultaneous sub/super harmonic, and combination resonances of the additive type of self-excited two coupled-second order systems to multi-frequency excitation are investigated. The theoretical results are obtained by the multiple-scales method. The steady state amplitudes for each resonance are plotted, showing the influence of the different parameters. Analysis for each figure is given. Approximate solution corresponding to each type of resonance is determined. Stability analyses are carried out for each case.     

Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.     

Generalized thermoviscoelastic interaction due to periodically varying heat source with three-phase-lag effect

The present paper aims at studying the thermo-visco-elastic interaction in a homogeneous, infinite Kelvin-Voigt-type viscoelastic, thermally conducting medium due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, GN model II (TEWOED) and GN model III (TEWED) are employed to study the thermomechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalized theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace-Fourier double transform domain and are solved in that domain. The inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature and stress are obtained for a hypothetical material. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of viscosity is also shown. In absence of viscous effect the results corresponding to GN model II and GN model III agree with the results of the existing literature.