Homoclinic orbits in the forced pendulum system

1996 ◽  
pp. 113-126
Author(s):  
Daniel Offin ◽  
Hongfan Yu
Keyword(s):  
Homoclinic Orbits ◽  
Pendulum System ◽  
Forced Pendulum

scholarly journals Multiple Bifurcations of a Cylindrical Dynamical System

2016 ◽  
Vol 46 (1) ◽  
pp. 33-52
Author(s):  
Ning Han ◽  
Qingjie Cao
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Homoclinic Orbits ◽  
Complex Dynamic ◽  
Pendulum System ◽  
Sd Oscillator ◽  
Bistable State ◽  
Different Types ◽  
Coupling Dynamics ◽  
Dynamic Bifurcation

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

Periodic Motions to Chaos in Pendulum

2016 ◽  
Vol 26 (09) ◽  
pp. 1650159 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Complex Dynamics ◽  
Traditional Approach ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Pendulum System ◽  
Linear Dynamical Systems ◽  
Mapping Structures ◽  
Forced Pendulum ◽  
And Mathematics ◽  
Analytical Bifurcation Trees

It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions’ complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.

Predictor-network-based MRACS for inverted pendulum system.

1991 ◽  
Vol 111 (3) ◽  
pp. 221-229 ◽  
Author(s):  
Motomiki Uchida ◽  
Yukihiro Toyoda ◽  
Yoshikuni Akiyama ◽  
Kazushi Nakano ◽  
Hideo Nakamura
Keyword(s):  
Inverted Pendulum ◽  
Pendulum System ◽  
Inverted Pendulum System

Optimal PID Design Approaches for an Inverted Pendulum System

2016 ◽  
Vol 9 (3) ◽  
pp. 167 ◽  
Author(s):  
Muhammad Sani Gaya ◽  
Anas Abubakar Bisu ◽  
Syed Najib Syed Salim ◽  
I. S. Madugu ◽  
L. A. Yusuf ◽  
...  
Keyword(s):  
Inverted Pendulum ◽  
Pendulum System ◽  
Inverted Pendulum System

A quantitative measure of the degree of output controllability for output regulation control systems: Concept, approach, and applications

2021 ◽  
pp. 107754632110201
Author(s):  
Yaping Xia ◽  
Ruiyu Li ◽  
Minghui Yin ◽  
Yun Zou
Keyword(s):  
Structural Parameters ◽  
Quantitative Measure ◽  
Output Regulation ◽  
State Regulation ◽  
Control Effect ◽  
Pendulum System ◽  
Design And Optimization ◽  
Inverted Pendulum System ◽  
State Regulator ◽  
Output Controllability

Currently, many research studies reveal that for state regulator problems, the higher the degree of controllability is, the better the control effect likely is. Note that for the output regulator problems, the control performance is often evaluated by outputs. This article hence generalizes the concept and applications of degree of controllability to the case of output regulator. To this end, a kind of degree of output controllability is presented. Furthermore, simulations on wind turbines and the inverted pendulum system demonstrate that better control effect may be achieved by increasing the degree of output controllability measure. These results imply that similar to the case of degree of controllability for state regulation control, the degree of output controllability measure is likely a feasible candidate index for the design and optimization of the structural parameters of controlled plants in the case of output regulation control.

scholarly journals An Optimized Triggering Algorithm for Event-Triggered Control of Networked Control Systems

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1262
Author(s):  
Sunil Kumar Mishra ◽  
Amitkumar V. Jha ◽  
Vijay Kumar Verma ◽  
Bhargav Appasani ◽  
Almoataz Y. Abdelaziz ◽  
...  
Keyword(s):  
Control Systems ◽  
Networked Control Systems ◽  
Networked Control ◽  
Pendulum System ◽  
Uniform Sampling ◽  
Inverted Pendulum System ◽  
Event Triggering ◽  
Non Uniform Sampling ◽  
System States ◽  
Event Triggered

This paper presents an optimized algorithm for event-triggered control (ETC) of networked control systems (NCS). Initially, the traditional backstepping controller is designed for a generalized nonlinear plant in strict-feedback form that is subsequently extended to the ETC. In the NCS, the controller and the plant communicate with each other using a communication network. In order to minimize the bandwidth required, the number of samples to be sent over the communication channel should be reduced. This can be achieved using the non-uniform sampling of data. However, the implementation of non-uniform sampling without a proper event triggering rule might lead the closed-loop system towards instability. Therefore, an optimized event triggering algorithm has been designed such that the system states are always forced to remain in stable trajectory. Additionally, the effect of ETC on the stability of backstepping control has been analyzed using the Lyapunov stability theory. Two case studies on an inverted pendulum system and single-link robot system have been carried out to demonstrate the effectiveness of the proposed ETC in terms of system states, control effort and inter-event execution time.

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