Recent zbMATH articles in MSC 70Khttps://zbmath.org/atom/cc/70K2021-03-30T15:24:00+00:00WerkzeugEffect of size on the chaotic behavior of nano resonators.https://zbmath.org/1455.740452021-03-30T15:24:00+00:00"Alemansour, Hamed"https://zbmath.org/authors/?q=ai:alemansour.hamed"Miandoab, Ehsan Maani"https://zbmath.org/authors/?q=ai:miandoab.ehsan-maani"Pishkenari, Hossein Nejat"https://zbmath.org/authors/?q=ai:pishkenari.hossein-nejatSummary: Present study is devoted to investigate the size effect on chaotic behavior of a micro-electro-mechanical resonator under external electrostatic excitation. Using Galerkin's decomposition method, approximating the actuation force with a new effective lumped model, and neglecting higher order terms in the Taylor-series expansion, a simplified model of the main system is developed. By utilizing the Melnikov's method and based on the new form of the electrostatic force, an expression in terms of the system parameters is developed which can be used to rapidly estimate the chaotic region of the simplified system. Based on the analysis of the simple proposed model, it is shown that the effect of size on chaotic region varies significantly depending on bias voltage. By considering the size effect, it is demonstrated that chaotic vibration initiates at much higher constant voltages than predicted by classical theories; and, in high constant voltages, it is shown that strain gradient theory predicts occurrence of chaos at much lower amplitudes.Completely degenerate responsive tori in Hamiltonian systems.https://zbmath.org/1455.370482021-03-30T15:24:00+00:00"Si, Wen"https://zbmath.org/authors/?q=ai:si.wen"Yi, Yingfei"https://zbmath.org/authors/?q=ai:yi.yingfeiExistence of periodic solutions for the Newtonian equation of motion with \(p\)-Laplacian operator.https://zbmath.org/1455.340432021-03-30T15:24:00+00:00"Liu, Wenbin"https://zbmath.org/authors/?q=ai:liu.wenbin.1"Gao, Peng"https://zbmath.org/authors/?q=ai:gao.peng.1"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.19Summary: In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with \(p\)-Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fučik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems.Dynamics of the Chaplygin ball with variable parameters.https://zbmath.org/1455.370522021-03-30T15:24:00+00:00"Borisov, Alexei V."https://zbmath.org/authors/?q=ai:borisov.alexey-v"Mikishanina, Evgeniya A."https://zbmath.org/authors/?q=ai:mikishanina.evgeniya-arifzhanovnaSummary: This work is devoted to the study of the dynamics of the Chaplygin ball with variable moments of inertia, which occur due to the motion of pairs of internal material points, and internal rotors. The components of the inertia tensor and the gyrostatic momentum are periodic functions. In general, the problem is nonintegrable. In a special case, the relationship of the problem under consideration with the Liouville problem with changing parameters is shown. The case of the Chaplygin ball moving from rest is considered separately. Poincaré maps are constructed, strange attractors are found, and the stages of the origin of strange attractors are shown. Also, the trajectories of contact points are constructed to confirm the chaotic dynamics of the ball. A chart of dynamical regimes is constructed in a separate case for analyzing the nature of strange attractors.Analysis of special cases in the study of bifurcations of periodic solutions of the Ikeda equation.https://zbmath.org/1455.370462021-03-30T15:24:00+00:00"Kubyshkin, Evgenii P."https://zbmath.org/authors/?q=ai:kubyshkin.evgenii-p"Moriakova, Alena R."https://zbmath.org/authors/?q=ai:moriakova.alena-rSummary: This paper deals with bifurcations from the equilibrium states of periodic solutions of the Ikeda equation, which is well known in nonlinear optics as an equation with a delayed argument, in two special cases that have not been considered previously. Written in a characteristic time scale, the equation contains a small parameter with a derivative, which makes it singular. Both cases share a single mechanism of the loss of stability of equilibrium states under changes of the parameters of the equation associated with the passage of a countable number of roots of the characteristic equation through the imaginary axis of the complex plane, which are in this case in certain resonant relations. It is shown that the behavior of solutions of the equation with initial conditions from fixed neighborhoods of the studied equilibrium states in the phase space of the equation is described by countable systems of nonlinear ordinary differential equations that have a minimal structure and are called the normal form of the equation in the vicinity of the studied equilibrium state. An algorithm for constructing such systems of equations is developed. These systems of equations allow us to single out one ``fast'' variable and a countable number of ``slow'' variables, which makes it possible to apply the averaging method to the systems of equations obtained. Equilibrium states of the averaged system of equations of ``slow'' variables in the original equation correspond to periodic solutions of the same nature of sustainability. In the special cases under consideration, the possibility of simultaneous bifurcation from equilibrium states of a large number of stable periodic solutions (multistability bifurcation) and evolution of these periodic solutions to chaotic attractors with changing bifurcation parameters is shown. One of the special cases is associated with the formation of paired equilibrium states (a stable and an unstable one). An analysis of bifurcations in this case provides an explanation of the formation of the ``boiling points of trajectories'', when a periodic solution arises ``out of nothing'' at some point in the phase space under changes of the parameters of the equation and quickly becomes chaotic.Global dynamics of a non-smooth system with elastic and rigid impacts and dry friction.https://zbmath.org/1455.740702021-03-30T15:24:00+00:00"Li, Guofang"https://zbmath.org/authors/?q=ai:li.guofang"Wu, Shaopei"https://zbmath.org/authors/?q=ai:wu.shaopei"Wang, Hongbing"https://zbmath.org/authors/?q=ai:wang.hongbing"Ding, Wangcai"https://zbmath.org/authors/?q=ai:ding.wangcaiSummary: Global dynamics of a non-smooth dynamic model is discussed under the combined effects of three non-smooth factors of elastic impact, rigid impact and dry friction, which is a rarely discussed topic before. Six different basic phases, nine events and six segments of the system have been described. The stability of periodic motions in the system is analyzed theoretically. The motions distribution and the transition to the sticking-adhesion motions are studied. Influences of the sliding bifurcation and grazing bifurcation in the way to the transition to sticking-adhesion motions are demonstrated. The transition to sticking-adhesion motions composed of one or more of three basic routes is illustrated. The transition law of three kinds of fundamental periodic motions in the low-frequency and small-gap parameter domain is further analyzed so as to delineate the transition mechanism from fundamental periodic motions to sticking-adhesion motions. In high frequency region, the global motions distribution in the system is obtained through the global analysis method in parameter-state space. The transition and coexistence of the motions including stable and unstable periodic motions are further explored. In addition, the motions distribution and transition of large friction force versus forcing amplitude from the static state are discussed. The results will be beneficial for the motions control of the system.Chaotic dynamics of two coaxially-nested cylindrical shells reinforced by two beams.https://zbmath.org/1455.740462021-03-30T15:24:00+00:00"Awrejcewicz, J."https://zbmath.org/authors/?q=ai:awrejcewicz.jan"Krysko, A. V."https://zbmath.org/authors/?q=ai:krysko.anton-v"Saltykova, O. A."https://zbmath.org/authors/?q=ai:saltykova.olga-aleksandrovna"Vetsel, S. S."https://zbmath.org/authors/?q=ai:vetsel.s-s"Krysko, V. A."https://zbmath.org/authors/?q=ai:krysko.vadim-a-jun|krysko.vadim-anatolevich|krysko.vadim-aSummary: Non-linear dynamics and contact interactions of beam-shell structures composed of two closed cylindrical shells which are coaxially nested and reinforced by two beams located symmetrically on the shell external perimeter is studied. In the present work, clearances between the mentioned structural members are taken into account, two beams are subjected to harmonic loads, and the dissipation factors are neglected.
3D PDEs governing non-linear dynamics of beams and shells within the geometric theory of Novozhilov are employed, whereas the contact pressure is defined through Kantor's model. PDEs are reduced to ODEs by means of the FEM (finite element method), and the solution convergence is validated through different numbers of finite elements located along the structural members thickness and by employment of the Runge principle with respect to spatial coordinates. The Cauchy problem is solved by the explicit integration (Euler method), which allows one to carry out the computation without the need to define solutions in a few initial points.
Analysis of vibrations, including contact interactions, is realized with the use of methods of non-linear dynamics and the qualitative theory of differential equations, time histories/signals, phase portraits, Poincarè sections, Fourier spectra, wavelet-based analysis as well as the Lyapunov exponents.On the linearization of vector fields on a torus with prescribed frequency.https://zbmath.org/1455.370492021-03-30T15:24:00+00:00"Zhang, Dongfeng"https://zbmath.org/authors/?q=ai:zhang.dongfeng"Xu, Xindong"https://zbmath.org/authors/?q=ai:xu.xindongThe authors consider a vector field of the form \(X = N + P,\) where \(N = \omega(\xi)\) is a vector on the torus \(\mathbb{T}^n = \mathbb{R}^n/2\pi\mathbb{Z}^n\), describing uniform rotation motions with frequency \(\omega (\xi) = (\omega_1(\xi),\dots,\omega_n(\xi)),\) \(P(\theta, \xi)\) is a small perturbation, and the parameters \(\xi\) vary on some bounded closed connected domain \(\Pi.\) Let \(\Delta \colon [1,\infty) \to [1,\infty)\) be a continuous increasing unbounded function such that \(\Delta (1) = 1\) and \(\int_1^\infty \frac{\ln \Delta (t)}{t^2}dt < \infty\), \(D(s) = \{\theta \in \mathbb{C}^n/2\pi \mathbb{Z}^n \colon |Im \theta_j| \leq s, j - 1,\dots,n \},\) and \(\Pi_h = \{\xi \in \mathbb{C}^n \colon \mathrm{dist} (\xi, \Pi) \leq h \}.\)
The main result is as follows. Suppose that \(X\) is real analytic on \(D(s) \times \Pi_h\) and for \(\omega_0 = \omega (\xi_0)\) the following non-resonant condition
\[
|\langle k, \omega_0\rangle | \geq \frac{\alpha}{\Delta (|k|)} \quad \forall \,0 \neq k \in \mathbb{Z}^n
\]
is satisfied with \(\alpha >0\) and the Brouwer degree of \(\omega (\xi)\) at \(\xi_0\) on \(\Pi\) is non-zero. Then for a sufficiently small \(\|P\|_{D(s)\times\Pi_h} \neq 0\) there exists a real analytic diffeomorphism \(\Phi_{\omega_0}\) such that \(\Phi_{\omega_0}^\star(N+P) = N_\star\), and at least one \(\xi_\star \in \Pi\) such that the conjugated vector field \(N_\star\) at \(\xi = \xi_\star\) has a linear flow with \(\omega_0\) as its frequency.
Reviewer: Valerii V. Obukhovskij (Voronezh)The transgression effect in the problem of motion of an almost holonomic pendulum.https://zbmath.org/1455.700082021-03-30T15:24:00+00:00"Kuleshov, A. S."https://zbmath.org/authors/?q=ai:kuleshov.alexander-s"Ulyatovskaya, I. I."https://zbmath.org/authors/?q=ai:ulyatovskaya.i-iSummary: In 1986 Ya. V. Tatarinov presented the basis of the theory of weakly nonholonomic systems. Mechanical systems with nonholonomic constraints depending on a small parameter are considered. It is assumed that when the value of this parameter is zero, the constraints of such a system become integrable, i.e. in this case we have a family of holonomic systems depending on several arbitrary integration constants. We will assume that these holonomic systems are integrable hamiltonian systems. When the small parameter is not zero, the methods of perturbation theory can be used to represent, to a first approximation, the motion of the system with nonzero parameter values, as a combination of the motion of a slightly modified holonomic system with slowly varying previous integration constants (transgression effect). In this paper we describe the transgression effect in the problem of motion of an almost holonomic pendulum.