PERIODIC SOLUTIONS OF THE FORCED PENDULUM EQUATION

1980 ◽  
pp. 149-160 ◽  
Author(s):  
Alfonso Castro
Keyword(s):  
Periodic Solutions ◽  
Pendulum Equation ◽  
Forced Pendulum

Periodic Solutions of Pendulum-Like Hamiltonian Systems in the Plane

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Alessandro Fonda ◽  
Rodica Toader
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Point Theorem ◽  
Elementary Level ◽  
Pendulum Equation ◽  
Forced Pendulum ◽  
The One

AbstractBy the use of a generalized version of the Poincaré-Birkhoff fixed point theorem, we prove the existence of at least two periodic solutions for a class of Hamiltonian systems in the plane, having in mind the forced pendulum equation as a particular case. Our approach is closely related to the one used by Franks in [15], but the proof remains at a more elementary level.

Periodic solutions of the forced pendulum equation with friction

2003 ◽  
Vol 14 (7) ◽  
pp. 311-320
Author(s):  
Pablo Amster ◽  
Maria Cristina Mariani
Keyword(s):  
Periodic Solutions ◽  
Pendulum Equation ◽  
Forced Pendulum

Prevalence of Non-degenerate Periodic Solutions in the Forced Pendulum Equation

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
Rafael Ortega
Keyword(s):  
Periodic Solutions ◽  
Main Tool ◽  
Main Question ◽  
Residual Set ◽  
Pendulum Equation ◽  
Transversality Theorem ◽  
Classical Duality ◽  
Forced Pendulum

AbstractMany known properties of the forced pendulum equation are generic. This means that they hold on a residual set of forcings. The classical duality category/measure inspires the main question in the paper: are these properties also valid for a prevalent set of forcings? This is the case for a key property: the non-degeneracy of periodic solutions. The main tool in the proof is a prevalent version of the parametric transversality theorem.

A “SHOOTING” APPROACH TO CHAOS

1994 ◽  
Vol 04 (01) ◽  
pp. 17-32
Author(s):  
S.P. HASTINGS
Keyword(s):  
Differential Equations ◽  
Chaotic Systems ◽  
Elementary Proof ◽  
Existence Of Solutions ◽  
Chaotic Behavior ◽  
Rigorous Proof ◽  
Lorenz Equations ◽  
One Dimensional ◽  
Pendulum Equation ◽  
Forced Pendulum

Toplogical shooting is a method for proving existence of solutions to certain types of problems in ordinary differential equations. We show how one-dimensional shooting provides an elementary proof that particular equations have solutions which exhibit properties associated with “chaos.” After an introductory illustration of how the method works for a forced pendulum equation, we outline a proof that the Lorenz equations have behavior typical of chaotic systems. Until 1993, there had been no rigorous proof that the Lorenz equations have any kind of chaotic behavior.

scholarly journals On the Stability of Periodic Solutions of the Damped Pendulum Equation

1997 ◽  
Vol 209 (2) ◽  
pp. 712-723 ◽  
Author(s):  
Jan Čepička ◽  
Pavel Drábek ◽  
Jolana Jenšı́ková
Keyword(s):  
Periodic Solutions ◽  
Pendulum Equation ◽  
The Stability
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