scholarly journals On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

2020 ◽  
Vol 16 (4) ◽  
pp. 607-623
Author(s):  
A.P. Markeev ◽  
◽  
T.N. Chekhovskaya ◽  
Keyword(s):  
Periodic Motion ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Natural Oscillations ◽  
Periodic Action ◽  
Straight Line Passing ◽  
Multiple Resonance ◽  
Horizontal Beam ◽  
Existence And Stability

The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).

The Existence and Stability of Ultraharmonics and Subharmonics in Forced Nonlinear Oscillations

1954 ◽  
Vol 21 (4) ◽  
pp. 327-335
Author(s):  
T. K. Caughey
Keyword(s):  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Forced Oscillations ◽  
Order System ◽  
Response Curves ◽  
Restoring Force ◽  
Amplitude Frequency Response ◽  
Existence And Stability ◽  
Linearized Theory ◽  
The Stability

Abstract A study is made of the forced oscillations of a second-order system having a small cubic nonlinearity in the restoring force. It is shown that under suitable conditions ultraharmonic or subharmonic motion exists in addition to the harmonic motion which a linearized theory would predict. By studying the stability of such motions it is shown that at points on the amplitude frequency-response curves having vertical tangents, instability and consequently “jumps” occur. A study of the dependence of the motion on the initial conditions reveals that while ultra-harmonic and harmonic motions are rather insensitive to initial conditions, the existence of subharmonic motion can be achieved only for a restricted set of initial conditions.

scholarly journals On the Dynamics of a Gravitational Dipole

2021 ◽  
Vol 17 (3) ◽  
pp. 247-261
Author(s):  
A. P. Markeev ◽  
Keyword(s):  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Center Of Mass ◽  
Kam Theory ◽  
Computer Algorithms ◽  
Periodic Motions ◽  
Small Eccentricity ◽  
Formal Stability ◽  
Nominal Mode

The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions.

STOCK-HARVEST DYNAMICS IN MULTIAGENT FISHERIES

2011 ◽  
Vol 21 (01) ◽  
pp. 363-372
Author(s):  
EN-GUO GU
Keyword(s):  
Initial Conditions ◽  
Profit Maximization ◽  
Fish Stock ◽  
Maximization Problem ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Fishery Resource ◽  
Global Dynamic ◽  
Existence And Stability ◽  
Fishery Model

In this work, we build a two-dimensional dynamical fishery model in which the total harvest is obtained by a multiagent game with best reply strategy and naive expectations, i.e. each agent decides the harvest quantity by solving a profit maximization problem. Special attention is paid to the global dynamic analysis in the light of feasible domains (initial conditions giving non-negative trajectories converging to an equilibrium), which is related to the crisis of extinction. We also study the existence and stability of non-negative equilibria for models through mathematical analysis and numerical simulations. We discover the increase in the margin price of fish stock may lead to instability of the fixed point and make the system sink into chaotic attractors. Thus the fishery resource may fluctuate in a stochastic form.

CHAOTIC BEHAVIOR AND DYNAMICS OF MAPS USED IN A METHOD OF SCRAMBLING SIGNALS

2010 ◽  
Vol 20 (12) ◽  
pp. 4097-4101
Author(s):  
REZA MAZROOEI-SEBDANI ◽  
MEHDI DEHGHAN
Keyword(s):  
Fixed Points ◽  
Limit Cycles ◽  
Secure Communication ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Chaotic Encryption ◽  
Input And Output ◽  
Sensitive Dependence ◽  
Close Relationship ◽  
Existence And Stability

The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. In this manuscript, we prove that a class of maps that have been proposed as suitable for scrambling signals possess the property of sensitive dependence on initial conditions (s.d.i.c.) necessary for chaos and cryptography. Our result can also be used for generating other maps with s.d.i.c., through a suitable semiconjugacy between their input and output parts. Using the condition of semiconjugacy we also establish for this class of maps rigorous criteria for the existence and stability of their fixed points and limit cycles.

scholarly journals Solitary Rossby Waves in the Lower Tropical Troposphere

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Andre Lenouo ◽  
Francois Kamga Nkankam
Keyword(s):  
West Africa ◽  
Large Scale ◽  
Rossby Waves ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Tropical Storm ◽  
Weakly Nonlinear ◽  
African Easterly Jet ◽  
Tropical Troposphere ◽  
Cascade Method

Weakly nonlinear approximation is used to study the theoretical comportment of large-scale disturbances around the intertropical midtropospheric jet. We show here that the Korteweg de Vries (KdV) theory is appropriated to describe the structure of the streamlines around the African easterly jet (AEJ) region. The introduction of the additional velocity of the soliton C1 permits to search the stage where the configuration of the wave structures is going to emerge out of specified initial conditions and this is the direct and inverse cascade method. It was also shown that the configurations of disturbances can be influenced by this parameter so that we can look if the disturbances are in the control or not of their dispersive effects. This permits to explain the evolution of initial conditions of the Tropical Storm (TS) Debby over West Africa from 20 to 24 August 2006.

Frequency analysis of nonlinear oscillations via the global error minimization

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
M Kalami Yazdi ◽  
P Hosseini Tehrani
Keyword(s):  
Variational Approach ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Global Error ◽  
Error Minimization ◽  
Free Oscillations ◽  
First Order ◽  
First Order Approximation

AbstractThe capacity and effectiveness of a modified variational approach, namely global error minimization (GEM) is illustrated in this study. For this purpose, the free oscillations of a rod rocking on a cylindrical surface and the Duffing-harmonic oscillator are treated. In order to validate and exhibit the merit of the method, the obtained result is compared with both of the exact frequency and the outcome of other well-known analytical methods. The corollary reveals that the first order approximation leads to an acceptable relative error, specially for large initial conditions. The procedure can be promisingly exerted to the conservative nonlinear problems.

Chaotic Motion of a Nonhomogeneous Torsional Pendulum

1997 ◽  
Vol 07 (03) ◽  
pp. 733-740 ◽  
Author(s):  
Jiin-Po Yeh
Keyword(s):  
Total Energy ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
Local Minimum ◽  
Phase Plane ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Chaotic Motions ◽  
Torsional Pendulum ◽  
Damped Systems

In this paper, the nonlinear oscillations of a nonhomogeneous torsional pendulum are investigated. Chaotic motions are shown to exist in both damped systems with two-well potential and undamped systems with one-well or two-well potential. Autocorrelations of the Poincaré mappings of the motion are presented and shown to be another useful tool to judge whether the system is chaotic. The total energy of the torsional pendulum is explored as well and it is conjectured that the irregularity of the total energy is probably one of the important factors which cause chaos. Lyapunov exponents are used as an indication of chaos in this paper. For systems with two-well potential, the phase-plane trajectories are found to stay in one well if the motion is regular, but jump from one well to another if the motion is chaotic. Making the initial conditions near the local minimum of the two-well potential is proved to be successful in preventing chaos from happening in the undamped systems.

scholarly journals Weak resonances of periodical nonlinear oscillations

2017 ◽  
Vol 58 ◽  
Author(s):  
Olga Lavcel-Budko ◽  
Aleksandras Krylovas
Keyword(s):  
Mathematical Model ◽  
Asymptotic Solution ◽  
Space Variable ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Perturbation Technique ◽  
The Mathematical Model

The mathematical model of nonlinear oscillations of weightless string is analyzed. Coefficients of the mathematical model and initial conditions are periodical functions of the space variable. A multiscale perturbation technique and integrating along characteristics are used to construct asymptotic solution without secular members.

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