scholarly journals Localization of periodic orbits of autonomous systems based on high-order extremum conditions

2004 ◽  
Vol 2004 (3) ◽  
pp. 277-290 ◽  
Author(s):  
Konstantin E. Starkov
Keyword(s):  
Periodic Orbits ◽  
Algebraic Surface ◽  
Polynomial System ◽  
Autonomous Systems ◽  
High Order ◽  
Mass Action ◽  
Nonsingular Solution ◽  
Localization Analysis ◽  
The Mathematical Model ◽  
Extremum Conditions

This paper gives localization and nonexistence conditions of periodic orbits in some subsets of the state space. Mainly, our approach is based on high-order extremum conditions, on high-order tangency conditions of a nonsingular solution of a polynomial system with an algebraic surface, and on some ideas related to algebraically-dependent polynomials. Examples of the localization analysis of periodic orbits are presented including the Blasius equations, the generalized mass action (GMA) system, and the mathematical model of the chemical reaction with autocatalytic step.

Delayed Feedback Control of Periodic Orbits in Autonomous Systems

1998 ◽  
Vol 81 (3) ◽  
pp. 562-565 ◽  
Author(s):  
Wolfram Just ◽  
Dirk Reckwerth ◽  
Johannes Möckel ◽  
Ekkehard Reibold ◽  
Hartmut Benner
Keyword(s):  
Feedback Control ◽  
Periodic Orbits ◽  
Autonomous Systems ◽  
Delayed Feedback ◽  
Delayed Feedback Control

KNOTTED PERIODIC SOLUTIONS OF A LINEAR NONAUTONOMOUS SYSTEM AND SOME RELATED THREE-DIMENSIONAL FLOWS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250280
Author(s):  
JIBIN LI ◽  
XIAOHUA ZHAO
Keyword(s):  
Phase Space ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Distinct Type ◽  
Frequency Parameter ◽  
Bounded Region ◽  
Nonautonomous Systems ◽  
Dimensional Phase Space

This paper considers a three-dimensional linear nonautonomous systems. It shows that for every integer frequency parameter value, this system has a distinct type of knotted periodic solutions, which lie in a bounded region of R3. Exact explicit parametric representations of the knotted periodic solutions are given. By using these parametric representations, two series of three-dimensional flows are constructed, such that these three-dimensional autonomous systems have knotted periodic orbits in the three-dimensional phase space.

scholarly journals A SIMPLE APPROACH TO CALCULATION AND CONTROL OF UNSTABLE PERIODIC ORBITS IN CHAOTIC PIECEWISE-LINEAR SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 215-224 ◽  
Author(s):  
TETSUSHI UETA ◽  
GUANRONG CHEN ◽  
TOHRU KAWABE
Keyword(s):  
Periodic Orbits ◽  
State Feedback ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Simple Approach ◽  
State Feedback Control ◽  
Unstable Periodic Orbits ◽  
Simple Method ◽  
Controlled System ◽  
And Control

This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor, and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given.

scholarly journals Analysis of chosen numerical methods for the application in high order interior ballistics simulations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012112
Author(s):  
M Krol
Keyword(s):  
High Order ◽  
Test Problems ◽  
Numerical Accuracy ◽  
Order Of Accuracy ◽  
Computing Power ◽  
Interior Ballistics ◽  
Depth Analysis ◽  
Catastrophic Failures ◽  
Volume Method ◽  
The Mathematical Model

Abstract Considering constant development of the interior ballistics, along with new gun and ammunition designs, the necessity of in-depth analysis of the shot event is continuously increasing. Numerical simulations of interior ballistics problems are useful for optimising new designs or explaining complex issues, regarding performance instabilities and catastrophic failures. With the rise of the computing power, there is a significant urge to drive the numerical errors towards machine zero. This goal demands using methods of high order of accuracy in both space and time. Current methods allow to achieve an arbitrary order of numerical accuracy, thus allowing to shift the focus towards sophistication of the mathematical model of the studied phenomenon. Therefore, in this work, some numerical schemes, in context of finite volume method, are reviewed and studied using well established test problems. The results of the presented analysis are meant to become the basis for future development of a high order numerical scheme for simulation of interior ballistics problems.

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