Bifurcations of periodic orbits in autonomous systems

1979 ◽  
pp. 371-377
Author(s):  
Yieh-Hei Wan
Keyword(s):  
Periodic Orbits ◽  
Autonomous Systems

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.

The Poincaré–Bendixson theorem for certain differential equations of higher order

1979 ◽  
Vol 83 (1-2) ◽  
pp. 63-79 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equations ◽  
Vector Function ◽  
Periodic Orbits ◽  
Jacobian Matrix ◽  
Autonomous Systems ◽  
Higher Order

SynopsisBy adapting its well-known proof, the Poincaré–Bendixson theorem, on the existence of periodic orbits of plane autonomous systems, is extended to vector differential equations of the form f(D)x + bφ(g(D)x) = 0. The only restrictions placed on the vector function φ(y) are that its Jacobian matrix should be continuous and lie within a suitably chosen ellintic ball.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1281-1293 ◽  
Author(s):  
FENGSHAN BAI ◽  
GABRIEL J. LORD ◽  
ALASTAIR SPENCE
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete System ◽  
Numerical Technique ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Connecting Orbits

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

CONTROLLING CHAOS IN CHUA'S CIRCUIT

1993 ◽  
Vol 03 (01) ◽  
pp. 109-117 ◽  
Author(s):  
G. A. JOHNSON ◽  
T. E. TIGNER ◽  
E. R. HUNT
Keyword(s):  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Autonomous Systems ◽  
Control Technique ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Periodically Driven ◽  
Single Orbit ◽  
Occasional Proportional Feedback ◽  
Control Circuits

The occasional proportional feedback (OPF) control technique has been successful in stabilizing periodic orbits in both periodically driven and autonomous systems undergoing chaotic behavior. By applying this technique to the well-known Chua's circuit, we are able to control a variety of periodic orbits including single-correction, low-period orbits and multiple-correction, high-period orbits. Also, by employing two control circuits, we are able to stabilize orbits that visit both regions of Chua's circuit's double-scroll attractor, applying corrections in each of these regions during a single orbit.

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