Wave localization in randomly disordered periodic layered piezoelectric structures

Considering the mechnoelectrical coupling, the localization of SH-waves in disordered periodic layered piezoelectric structures is studied. The waves propagating in directions normal and tangential to the layers are considered. The transfer matrices between two consecutive unit cells are obtained according to the continuity conditions. The expressions of localization factor and localization length in the disordered periodic structures are presented. For the disordered periodic piezoelectric structures, the numerical results of localization factor and localization length are presented and discussed. It can be seen from the results that the frequency passbands and stopbands appear for the ordered periodic structures and the wave localization phenomenon occurs in the disordered periodic ones, and the larger the coefficient of variation is, the greater the degree of wave localization is. The widths of stopbands in the ordered periodic structures are very narrow when the properties of the consecutive piezoelectric materials are similar and the intervals of stopbands become broader when a certain material parameter has large changes. For the wave propagating in the direction normal to the layers the localization length has less dependence on the frequency, but for the wave propagating in the direction tangential to the layers the localization length is strongly dependent on the frequency.The project supported by National Natural Science Foundation of China (10632020, 10672017 and 20451057).     

Dynamical properties of 2-torus parabolic maps

In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations. Mathematics Subject Classification: 37C15, 37D50     

一种值得注意的动力学行为

在讨论一类以ｐａｔｅｒｎ动力学为背景的二维平面映射时，发现了一种具有两个正Ｌｙａｐｕｎｏｖ特征指数的动力学行为．分析表明这种行为可能来自于ｓｎａｐｂａｃｋｒｅｐｅｌｅｒ．进一步的理论工作有待于深入     

C^r随机中心流形定理

随机中心流形定理在非线性随机分叉理论的研究中具有关键作用。本文对Ｃ＾ｒ非线性随机系统给出Ｃ＾ｒ中心流形的存在性定理，并讨论了中心流形的稳定性及例子。     

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped four-degree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reduced-order model which is signal-independent (doesn’t require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.     

Extremal exponents of random dynamical systems do not vanish

A family of random diffeomorphisms on a manifoldM is said to be a random dynamical system or RDS if it has the so-called cocycle property. The multiplicative ergodic theorem assignsd (=dimM) Lyapunov exponents to every invariant measure of the system. Take the maximum of the leading exponents associated with the various invariant measures. The resulting number is said to be the maximal exponent of the system. The minimal exponent is defined in a similar fashion. It is shown that the minimal exponent of an RDS on a compact manifold is negative, provided not all invariant measures are determined by the future of. A similar statement relates the maximal exponent with the past of. We proceed by introducing Markov systems and Markov measures. This notion covers flows of stochastic differential equations as well as products of random diffeomorphisms in Markovian dependence, in particular, products of iid diffeomorphisms. Markov measures are characterized by the fact that they are functionals of the past. Consequently, if there exists a non-Markovian invariant measure, then the maximal exponent does not vanish. Typically, Markov systems do have non-Markovian invariant measures. Finally, for linear systems we recover results of Ledrappier. In particular, these results provide another proof of Furstenberg's theorem on the positivity of the leading exponent of a product of iid unimodular matrices.     

On the stability of systems with stochastically variable parameters

The paper analyzes questions related to the construction of dynamic stability boundaries of elastic systems subjected to stochastic parametric excitation. It is supposed that the parametric action is a combination of a deterministic static component and a stochastic fluctuating component. The fluctuating component is taken to be a stationary ergodic process. The stability boundaries are built in the region of combination resonance using the stochastic averaging method and a probabilistic approach due to Khasminskii. In this connection, the stochastic averaging method based on the Stratonovich-Khasminskii theory is used. The probabilistic approach consists in using explicit asymptotic expressions for the largest Lyapunov exponent, from which the asymptotic stability boundaries are determined. As an application, the stability of a simply supported thin-walled bar subjected to a stochastically varying longitudinal load is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 128–138, December 2005.     

Characterization of Measures Satisfying the Pesin Entropy Formula for Random Dynamical Systems

In this paper we first give a formulation of SRB (Sinai–Ruelle–Bowen) property for invariant measures of stationary random dynamical systems and then prove that this property is sufficient and necessary for a formula of Pesin's type relating entropy and Lyapunov exponents of such dynamical systems. This result is a random version of the main result in Part I of Ledrappier and Young's celebrated paper [11].     

A generalized mathematical model to analyze the nonlinear behavior of a controlled gyroscope in gimbals

In this paper, the effect of the parameter variation on the stability and dynamic behavior of a gyroscope in gimbals with a feedback control system, formed by a Proportional + Integral $+$ Derivative (PID) controller and a DC motor with an ideal train gear is researched. The generalized mathematical model of the gyro is obtained from the Euler-Lagrange equations by using the nutation theory of the gyroscope. The use of approximated models of the control system are deduced from the mathematical model of the gyro, taking into account that the integral action of the PID controller is constrained and that the inductance of the DC motor may be negligible. The analysis and choice of appropriate state variables to simulate the dynamic behavior of different models of the gyro are also considered. The paper shows that from the analysis of the equilibrium points, a Bogdanov Takens and a Poincaré-Andronov-Hopf bifurcation can appear. These bifurcations are analyzed from the calculation of a parameter which is known as the first Lyapunov value, showing that it is possible to deduce a procedure to find out when a complicated model can be substituted by a simpler one. In particular, the possibility of self-oscillating and chaotic behavior for different models of the system by using the first Lyapunov value as a function of the parameters of the PID controller is researched. Numerical simulations have been performed to evaluate the analytical results and the mathematical discussions.     

Global dynamics of an autoparametric spring–mass–pendulum system

In this paper, a study of the global dynamics of an autoparametric four degree-of-freedom (DOF) spring–mass–pendulum system with a rigid body mode is presented. Following a modal decoupling procedure, typical approximate periodic solutions are obtained for the autoparametrically coupled modes in 1:2 internal resonance. A novel technique based on forward-time solutions for finite-time Lyapunov exponent is used to establish global convergence and domains of attraction of different solutions. The results are compared to numerically constructed domains of attraction in the plane of initial position and initial velocity for the pendulum. Simulations are also provided for a few interesting cases of interest near critical values of parameters. Results also shed some light on the role played by other modes present in a multi-DOF system in shaping the overall system response.