On the Motion of the Pendulum in an Alternating, Sawtooth Force Field

2020 ◽  
Vol 30 (09) ◽  
pp. 2050135
Author(s):  
Alexander A. Burov ◽  
Vasily I. Nikonov
Keyword(s):  
Nonlinear Dynamics ◽  
Iterative Method ◽  
Force Field ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Melnikov Method ◽  
Stability Conditions ◽  
Poincaré Maps ◽  
Separatrix Splitting

The motion of the pendulum in a variable sawtooth force field is considered. For the “lower” equilibrium, the necessary stability conditions are investigated numerically, the results are presented in the form of an Ince–Strutt diagram. Using the Poincaré–Melnikov method separatrix splitting is studied analytically. Numerically, for some values of parameters, the nonlinear dynamics is studied using Poincaré maps, the regions of regular and chaotic behavior are revealed. The iterative method earlier proposed is used for the localization of periodic solutions, located inside the numerically identified “invariant tori”.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (12) ◽  
pp. 4381-4386 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Neural Networks ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Cellular Neural Networks ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Poincaré Maps ◽  
New Class ◽  
Numerical Studies ◽  
Connection Matrices

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.

CHAOS IN THE REORIENTATION PROCESS OF A DUAL-SPIN SPACECRAFT WITH TIME-DEPENDENT MOMENTS OF INERTIA

2000 ◽  
Vol 10 (05) ◽  
pp. 997-1018 ◽  
Author(s):  
M. IÑARREA ◽  
V. LANCHARES
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Center Of Mass ◽  
Melnikov Method ◽  
Moments Of Inertia ◽  
Suitable Parameter ◽  
Nutation Angle ◽  
Principal Axes ◽  
Reorientation Process ◽  
Subharmonic Resonances

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

scholarly journals Reflection of Solar Activity Dynamics in Radionuclide Data

Radiocarbon ◽  
1992 ◽  
Vol 34 (2) ◽  
pp. 207-212 ◽  
Author(s):  
A. V. Blinov ◽  
M. N. Kremliovskij
Keyword(s):  
Nonlinear Dynamics ◽  
Solar Activity ◽  
Numerical Methods ◽  
Chaotic Behavior ◽  
Magnetic Activity ◽  
Data Sets ◽  
Cosmogenic Nuclide ◽  
Proxy Records ◽  
Solar Magnetic Activity ◽  
Activity Dynamics

Variability of solar magnetic activity manifested within sunspot cycles demonstrates features of chaotic behavior. We have analyzed cosmogenic nuclide proxy records for the presence of the solar activity signals. We have applied numerical methods of nonlinear dynamics to the data showing the contribution of the chaotic component. We have also formulated what kind of cosmogenic nuclide data sets are needed for investigations on solar activity.

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

1995 ◽  
Vol 05 (02) ◽  
pp. 321-348 ◽  
Author(s):  
JAMES A. WALSH
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Mode Locking ◽  
Rotation Vector ◽  
Higher Genus ◽  
Rotation Vectors ◽  
Dissipative Dynamical Systems ◽  
Rotation Set

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincaré. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 791-795
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Chaotic Behavior ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Pipe Conveying Fluid ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Transition To Chaos ◽  
Subharmonic Bifurcations ◽  
Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

TRANSITION TO CHAOS IN A CURVED CARBON NANOTUBE UNDER HARMONIC EXCITATION

2012 ◽  
Vol 26 (32) ◽  
pp. 1250210 ◽  
Author(s):  
YIXIANG GENG ◽  
LIXIANG ZHANG
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Periodic Attractor ◽  
Runge Kutta Method ◽  
Transition To Chaos ◽  
Melnikov Criterion

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge–Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out.

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