Suppression of chaotic vibrations in nonlinear systems by the example of a mechanical tachometer

A method for controlling dynamic chaos is proposed by introducing state feedback and changing the spectrum of Lyapunov characteristic parameters of a closed system to achieve the desired result - the transition from chaotic mode to regular motion. The solution of this problem is considered on the example of stabilization of a mechanical tachometer. The parameters of the controller in the feed-back circuit are determined by the method of modal con-trol synthesis.

Rhythm: Types, Hierarchy and Language Diversity

This article deals with the factors which cause the emergence of rhythm, its types, hierarchy and the relation of rhythm to life and the human language diversity. According to Aristotle “all types of rhythm are measured by certain movements”. So all events and processes connected with rhythm are rhythmical in nature. Rhythm is a regular reiteration of identical cases, processes and events within the boundaries of time and space. Rhythm is the form of regular motion. However, rhythm is not the result of motion. It is just the movement itself. All types of rhythm or movement, to our mind, are based on energy the absence of which excludes movements, rhythms, accordingly then life, human language, as well as language diversity. Thus, studying rhythm, its types and systemic hierarchy, to our mind, enables us to reveal the mechanism of transition from inanimate nature to animate one, on the one hand, and creation of the styduing diversity principle of nature, as well as language diversity, on the other hand. The main task of a linguistic scholar, as defined by David Crystal, is great interest. To this linguist “the main task of the linguistic scholar is not to improve the language teaching situation … etc., his task is basically to study and understand the general principles upon which all languages are built. What are the “design features” of human language? What are the differences between languages? How can we describe and classify this? How far are they fundamental? What concepts do we have before we can begin to talk about language at all” (D.Crystal, 1997). Our aim, accordingly, is to make an attempt to study the types of rhythm, its systemic hierarchy and the relation of rhythm to emergence of life and language diversity.

Mapping exomoon trajectories around Earth-like exoplanets

ABSTRACT We consider a system in which both the parent star and the Earth-like exoplanet move on circular orbits. Using numerical methods, such as the orbit classification technique, we study all types of trajectories of possible exomoons around the exoplanet. In particular, we scan the phase space around the exoplanet and we distinguish between bounded, collisional, and escaping trajectories, considering both retrograde and prograde types of motion. In the case of bounded regular motion, we also use the grid method and a standard predictor-corrector procedure for revealing the corresponding network of symmetric periodic solutions, while we also compute their linear stability.

Effect of a Constant Bias on the Nonlinear Dynamics of a Biharmonically Driven Sinusoidal Potential System

The nonlinear dynamics of an underdamped sinusoidal potential system is experimentally and numerically studied. The system shows regular (nonchaotic) periodic motion when driven by a small amplitude ([Formula: see text]) sinusoidal force (frequency [Formula: see text]). However, when the system is driven by a similarly small amplitude biharmonic force (frequencies [Formula: see text] and [Formula: see text] with amplitudes [Formula: see text] and [Formula: see text], respectively) chaotic motion appear as a function of amplitude ([Formula: see text]) of the [Formula: see text]-frequency component for a fixed [Formula: see text]. We investigate the effect of an additional constant force [Formula: see text] on the dynamics of the system in the ([Formula: see text]) space. We find that [Formula: see text] can cause chaotic motion to move to regular motion and regular motion can also become chaotic in certain ([Formula: see text]) domains.

On the Behavior of the Generalized Alignment Index (GALI) Method for Regular Motion in Multidimensional Hamiltonian Systems

We investigate the behavior of the Generalized Alignment Index of order k (GALIk ) for regular orbits of multidimensional Hamiltonian systems. The GALIk is an efficient chaos indicator, which asymptotically attains positive values for regular motion when 2≤k ≤N, with N being the dimension (D) of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model for various values of the system's degrees of freedom, we show that the asymptotic GALIk values decrease when the order k of the index increases and when the orbit's energy approaches the periodic orbit's destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, by performing extensive numerical simulations we show that the behavior of the index does not depend on the choice of the initial deviation vectors needed for its evaluation.

Nonlinear analysis of electrostatic micro-electro-mechanical systems resonators subject to delayed proportional–derivative controller

The present study deals with the nonlinear analysis of electrostatic micro-electro-mechanical systems resonators with two symmetric electrodes and subjected to delayed proportional–derivative controller. After a brief description of the model, the stability analysis of the linearized system is presented to depict the stability charts in the parameter space of proportional gain and time delay. The bifurcation diagram is used to confirm the existence of the delay-dependent and delay-independent regions and to analyze the effect of proportional–derivative gains and time delay on the dynamics of the system. Using Melnikov’s theorem, the criterion for the appearance of horseshoe chaos from homoclinic and heteroclinic bifurcations is presented. Melnikov’s predictions are confirmed by using the numerical simulations based on the basin of attraction of initial conditions. It is found that the increase in proportional gain contributes to increase the region of regular motion in both bifurcations. However, the increase in derivative gain contributes rather to reduce the region of regular motion for homoclinic bifurcation, although it increases rather this region in the case of heteroclinic bifurcation. Moreover, it is also observed, depending on proportional–derivative gains, the existence of a critical value of the delay where before it, the region of regular motion increases and after it, decreases rather.

Nonstatistical Reaction Dynamics

Nonstatistical dynamics is important for many chemical reactions. The Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolecular kinetics assumes a reactant molecule maintains a statistical microcanonical ensemble of vibrational states during its dissociation so that its unimolecular dynamics are time independent. Such dynamics results when the reactant's atomic motion is chaotic or irregular. Intrinsic non-RRKM dynamics occurs when part of the reactant's phase space consists of quasiperiodic/regular motion and a bottleneck exists, so that the unimolecular rate constant is time dependent. Nonrandom excitation of a molecule may result in short-time apparent non-RRKM dynamics. For rotational activation, the 2J + 1 K levels for a particular J may be highly mixed, making K an active degree of freedom, or K may be a good quantum number and an adiabatic degree of freedom. Nonstatistical dynamics is often important for bimolecular reactions and their intermediates and for product-energy partitioning of bimolecular and unimolecular reactions. Post–transition state dynamics is often highly complex and nonstatistical.