scholarly journals A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems

2022 ◽  
Author(s):  
Xu Zhang ◽  
Guanrong Chen
Keyword(s):  
Periodic Orbits ◽  
Continuous Time ◽  
Autonomous Systems ◽  
Geometric Criterion ◽  
Time Autonomous

scholarly journals CHAOTIFYING CONTINUOUS-TIME NONLINEAR AUTONOMOUS SYSTEMS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250232 ◽  
Author(s):  
SIMIN YU ◽  
GUANRONG CHEN
Keyword(s):  
Linear Systems ◽  
Lyapunov Exponents ◽  
Continuous Time ◽  
Autonomous Systems ◽  
Nonlinear Function ◽  
Positive Zero ◽  
State Variables ◽  
Nonlinear Feedback ◽  
Special Cases ◽  
Time Autonomous

Based on the principle of chaotification for continuous-time autonomous systems, which relies on two basic properties of chaos, i.e. being globally bounded with necessary positive-zero-negative Lyapunov exponents, this paper derives a feasible and unified chaotification method for designing a general chaotic continuous-time autonomous nonlinear system. For a system consisting of a linear and a nonlinear subsystems, chaotification is achieved using separation of state variables, which decomposes the system into two open-loop subsystems interacting through mutual feedback resulting in an overall closed-loop nonlinear feedback system. Under the condition that the nonlinear feedback control output is uniformly bounded where the nonlinear function is of bounded-input/bounded-output, it is proved that the resulting system is chaotic in the sense of being globally bounded with a required placement of Lyapunov exponents. Several numerical examples are given to verify the effectiveness of the theoretical design. Since linear systems are special cases of nonlinear systems, the new method is also applicable to linear systems in general.

scholarly journals A geometric characterisation of persistently exciting signals generated by continuous-time autonomous systems

2016 ◽  
Vol 49 (18) ◽  
pp. 826-831 ◽  
Author(s):  
Alberto Padoan ◽  
Giordano Scarciotti ◽  
Alessandro Astolfi
Keyword(s):  
Continuous Time ◽  
Autonomous Systems ◽  
Time Autonomous

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

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.

Optimal periodic orbits of continuous time chaotic systems

2000 ◽  
Vol 62 (2) ◽  
pp. 1950-1959 ◽  
Author(s):  
Tsung-Hsun Yang ◽  
Brian R. Hunt ◽  
Edward Ott
Keyword(s):  
Periodic Orbits ◽  
Continuous Time ◽  
Chaotic Systems

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
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