Control methods on unstable periodic orbits of a chaotic dynamical system — control chaos in buck converter

2008 ◽  
Author(s):  
Cosmin Ivan ◽  
Alexandru Serbanescu
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Buck Converter ◽  
Control Methods ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical System

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

A New Property of Noninvasive Control Methods Applied to Stabilize Unstable Periodic Orbits

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Saeed Montazeri ◽  
Ali Rahmani Hanzaki
Keyword(s):  
Energy Consumption ◽  
Periodic Orbits ◽  
Parameter Optimization ◽  
Control Methods ◽  
System State ◽  
Unstable Periodic Orbits ◽  
Optimal Energy

This paper is devoted to present a theorem on a new property of noninvasive control methods applied to stabilize unstable periodic orbits (UPOs). This property is related to the optimal energy consumption of the controller in the presence of noise. The approach of parameter optimization is applied to study optimal energy consumption of the controller. Throughout this paper, the problem of energy consumption of the controller is studied when the system state is close to the UPO, and fluctuates around it because of the presence of noise.

FEEDBACK CONTROL OF THE LIU CHAOTIC DYNAMICAL SYSTEM

2010 ◽  
Vol 24 (03) ◽  
pp. 397-404 ◽  
Author(s):  
XINGYUAN WANG ◽  
XINGUANG LI
Keyword(s):  
Dynamical System ◽  
Feedback Control ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Unstable Periodic Orbits ◽  
Chaotic Dynamical System ◽  
Feedback Method ◽  
Liu System

Classical feedback method is used to control chaos in the Liu dynamical system. Based on the Routh–Hurwitz criteria, the conditions of the asymptotic stability of the steady states of the controlled Liu system are discussed, and they are also proved theoretically. Numerical simulations show that the method can suppress chaos to both unstable equilibrium points and unstable periodic orbits (limit cycles) successfully.

scholarly journals Feedback control modulation for controlling chaotic maps

2021 ◽  
Vol 26 (3) ◽  
pp. 419-439
Author(s):  
Roberta Hansen ◽  
Graciela A. González
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Periodic Orbits ◽  
Waiting Time ◽  
Control Parameter ◽  
Chaotic Maps ◽  
Noise Sensitivity ◽  
Control Methods ◽  
Unstable Periodic Orbits ◽  
Discrete Maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

THE PENDULUM WEAVES ALL KNOTS AND LINKS

2005 ◽  
Vol 14 (07) ◽  
pp. 869-881
Author(s):  
JOHN STARRETT
Keyword(s):  
Dynamical System ◽  
Strange Attractor ◽  
Complex Behavior ◽  
Unstable Periodic Orbits ◽  
Physical Pendulum ◽  
Chaotic Dynamical System ◽  
Manifold With Boundary ◽  
Infinite Set ◽  
Local Stable Manifolds ◽  
Knots And Links

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible.

scholarly journals Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map

2021 ◽  
Vol 31 (04) ◽  
pp. 2150110
Author(s):  
Yuu Miino ◽  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Hiroshi Kawakami
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Ad Hoc ◽  
Computation Method ◽  
Control Input ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Controlling Chaos ◽  
Dimensional Dynamical System ◽  
The Given

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

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