scholarly journals On subharmonic motions of a pendulum in a movable platform

2019 ◽  
Vol 486 (5) ◽  
pp. 547-553
Author(s):  
A. P. Markeev
Keyword(s):  
Small Amplitude ◽  
Linear Problem ◽  
Relative Equilibria ◽  
Unperturbed System ◽  
Periodic Motions ◽  
Rotating Platform ◽  
Stability Change ◽  
Existence And Stability ◽  
Movable Platform ◽  
Splitting Of Separatrices

The motion of a pendulum on a rotating platform is investigated in the presence of disturbances caused by its vertical harmonic oscillations of small amplitude. The parameters of the unperturbed system are considered close to the values, upon passing through which the number of relative equilibria of the pendulum and the nature of their stability change. The non-linear problem of the existence and stability of periodic motions of the pendulum relative to the platform with a period multiple to the period of its vertical oscillations is solved. The question of the splitting of separatrices of the unperturbed system is also considered.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

scholarly journals Periodically kicked feedforward chains of simple excitable FitzHugh-Nagumo neurons

2021 ◽  
Author(s):  
B. Ambrosio ◽  
S.M. Mintchev
Keyword(s):  
Numerical Investigation ◽  
Traveling Waves ◽  
Small Amplitude ◽  
Mixed Mode ◽  
Filtration Property ◽  
Periodic Traveling Waves ◽  
Mixed Mode Oscillations ◽  
Mode Oscillation ◽  
Existence And Stability ◽  
Mixed Mode Oscillation

Abstract This article communicates results on regular depolarization cascades in periodically-kicked feedforward chains of excitable two-dimensional FitzHugh-Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

Feedback Control of Grazing-Induced Chaos in the Single-Degree-of-Freedom Impact Oscillator

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Yongkang Shen ◽  
Shan Yin ◽  
Guilin Wen ◽  
Huidong Xu
Keyword(s):  
Feedback Control ◽  
Control Strategy ◽  
Dynamical Property ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
Impact Oscillator ◽  
Continuous Transition ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Existence And Stability

Based on the special dynamical property of continuous transition at certain degenerate grazing points in the single-degree-of-freedom impact oscillator, the control problem of the grazing-induced chaos is investigated in this paper. To design degenerate grazing bifurcations, we show how to obtain the degenerate grazing points of the 1/n impact periodic motions by the existence and stability analysis first. Then, a discrete-in-time feedback control strategy is used to suppress the grazing-induced chaos into the 1/n impact periodic motions precisely by the desired degenerate grazing bifurcation. The feasibility of the control strategy is verified by numerical simulations.

Periodic motions of vortices on surfaces with symmetry

2002 ◽  
Vol 460 ◽  
pp. 83-92 ◽  
Author(s):  
ANIK SOULIÈRE ◽  
TADASHI TOKIEDA
Keyword(s):  
Ideal Fluid ◽  
Relative Equilibria ◽  
Point Vortices ◽  
Two Dimensional ◽  
Periodic Motions

The theory of point vortices in a two-dimensional ideal fluid has a long history, but on surfaces other than the plane no method of finding periodic motions (except relative equilibria) of N vortices is known. We present one such method and find infinite families of periodic motions on surfaces possessing certain symmetries, including spheres, ellipsoids of revolution and cylinders. Our families exhibit bifurcations. N can be made arbitrarily large. Numerical plots of bifurcations are given.

Vibration Suppression of Rotating Machinery Utilizing an Automatic Ball Balancer and Discontinuous Spring Characteristics

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Jun Liu ◽  
Yukio Ishida
Keyword(s):  
Rotational Speed ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Critical Speed ◽  
Vibration Suppression ◽  
Rotating Machinery ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Speed Range ◽  
Suppression Method

Automatic ball balancer is a balancing device where two balls inside a hollow rotor move to optimal rest positions automatically to eliminate unbalance. As a result, vibrations are suppressed to the zero amplitude in the rotational speed range higher than the major critical speed. However, it has the following defects. The amplitude of vibration increases in the rotational speed range lower than the major critical speed. In addition, almost periodic motions with large amplitude occur in the vicinity of the major critical speed due to the rolling of balls inside the rotor. Because of these defects, an automatic ball balancer has not been used widely. This paper proposes the vibration suppression method utilizing the discontinuous spring characteristics together with an automatic ball balancer to overcome these defects and to suppress vibration. The validity of the proposed method is confirmed theoretically, numerically, and experimentally. The results show that amplitude of vibration can be suppressed to a small amplitude in the vicinity of the major critical speed and the zero amplitude in the range higher than the major critical speed.

scholarly journals Libration Points Inside a Spherical Cavity of a Uniformly Rotating Gravitating Ball

2021 ◽  
Vol 17 (4) ◽  
pp. 413-427
Author(s):  
A. A. Burov ◽  
◽  
V. I. Nikonov ◽  
Keyword(s):  
Spherical Cavity ◽  
Center Of Mass ◽  
Relative Equilibria ◽  
The Body ◽  
Libration Points ◽  
Homogeneous Ball ◽  
Existence And Stability ◽  
Axis Of Symmetry ◽  
Stability And Bifurcations

The problem of the existence and stability of relative equilibria (libration points) of a uniformly rotating gravitating body, which is a homogeneous ball with a spherical cavity, is considered. It is assumed that the rotation is carried out around an axis perpendicular to the axis of symmetry of the body and passing through its center of mass. The libration points located inside the cavity are investigated. Families of both isolated and nonisolated relative equilibria are found. Their stability and bifurcations are investigated. Realms of possible motion are constructed.

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