Implicit Numerical Integration for Periodic Solutions of Autonomous Nonlinear Systems

1982 ◽  
Vol 49 (4) ◽  
pp. 861-866 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Nonlinear Systems ◽  
Critical Load ◽  
Periodic Solutions ◽  
Autonomous Systems ◽  
Conservative System ◽  
Constant Load ◽  
Second Order ◽  
Order System ◽  
Change Of Variables ◽  
System A

A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.

Periodic solutions for a quasilinear non-autonomous second-order system

2006 ◽  
Vol 22 (1-2) ◽  
pp. 263-271 ◽  
Author(s):  
Yu Tian ◽  
Guosheng Zhang ◽  
Weigao Ge
Keyword(s):  
Periodic Solutions ◽  
Second Order ◽  
Order System

scholarly journals Periodic Solutions for a System of Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Shugui Kang ◽  
Bao Shi
Keyword(s):  
Nonlinear Systems ◽  
Critical Point ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Existence Theorems ◽  
Point Theory ◽  
Second Order ◽  
System Of Difference Equations

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.

A NUMERICAL HOMOTOPY METHOD AND INVESTIGATIONS OF A SPRING-MASS SYSTEM

1993 ◽  
Vol 03 (03) ◽  
pp. 395-416
Author(s):  
B.D. DAVIDSON ◽  
D.E. STEWART
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Numerical Technique ◽  
A Priori ◽  
Autonomous Systems ◽  
Homotopy Method ◽  
Order System ◽  
Mass System ◽  
Highly Nonlinear ◽  
Local Methods

A numerical technique is developed to determine the behavior of periodic solutions to highly nonlinear non-autonomous systems of ordinary differential equations. The method is based on shooting in conjunction with a probability one homotopy method and an implementation of the topological index. It is shown that solutions may be characterized a priori in terms of an index and this is developed into a powerful numerical and investigative tool. This method is used to investigate the periodic solutions of a nonlinear fourth order system of differential equations. These equations describe the motion of a forced mechanical oscillator and are extremely difficult to evaluate numerically. Solutions are presented which could not be found using local methods. These include flip, saddle node and symmetry breaking pitchfork bifurcations.

Global Bifurcation of Periodic Solutions in Symmetric Reversible Second Order Systems with Delays

2021 ◽  
Vol 31 (12) ◽  
pp. 2150180
Author(s):  
Zalman Balanov ◽  
Joseph Burnett ◽  
Wiesław Krawcewicz ◽  
Huafeng Xiao
Keyword(s):  
Periodic Solutions ◽  
Autonomous Systems ◽  
Global Bifurcation ◽  
Degree Theory ◽  
Second Order ◽  
Equivariant Degree ◽  
Commensurate Delays ◽  
Spatio Temporal ◽  
Second Order Systems ◽  
The Right

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder [Formula: see text]-equivariant degree theory. Here, [Formula: see text] is related to the reversal symmetry combined with the autonomous form of the system, [Formula: see text] reflects extra spacial symmetries of the system, and [Formula: see text] is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with [Formula: see text] — the dihedral group of order 12.

scholarly journals Continuous Prescribed-Time Sliding Mode Control with A Prescribed-Time ESO for Second-Order Nonlinear Systems with Mismatched Disturbance

2021 ◽  
Author(s):  
Xiaozhe Ju ◽  
Feng Wang ◽  
Yingzi Guan ◽  
Shihao Xu
Keyword(s):  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Sliding Mode ◽  
Fixed Time ◽  
Second Order ◽  
Order System ◽  
Practical Applications ◽  
Mode Control ◽  
System States ◽  
Mismatched Disturbances

Abstract This paper aims to settle the continuous prescribed-time stabilization problem of second-order nonlinear systems with mismatched disturbances. A continuous prescribed-time sliding mode control (CPTSMC) method with a prescribed-time extended state observer (PTESO) is proposed. The PTESO can precisely estimate the unknown states and disturbances, with its upper bound for the settling time (UBST) prescribed by only one parameter more tightly than existing finite-time or fixed-time ESOs. Furthermore, as a common concern for ESOs, the peaking value problem is well addressed. Then, a novel prescribed-time convergent form with little conservatism and simple tuning procedures is designed, and the internal mechanism in acquiring higher transient performance is explicitly researched. By using the estimated states and disturbances, the CPTSMC makes system states converge in a chattering-alleviated manner following the novel prescribed-time form. In addition to proving that the UBST of the whole system is tightly prescribed by only one design parameter, we show the continuity of the CPTSMC and the boundedness of all system signals, which are vital for practical applications. Ultimately, numerical simulations on the second-order system and a DC motor servo verify the efficiency of the proposed control system.

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