scholarly journals Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control

2018 ◽  
Vol 27 (2018) ◽  
pp. 73-78
Author(s):  
Dumitru Deleanu
Keyword(s):  
Periodic Orbits ◽  
Predictive Control ◽  
Nonnegative Integer ◽  
Strange Attractor ◽  
Discrete System ◽  
Control Method ◽  
Nonlinear Mapping ◽  
Unstable Periodic Orbits ◽  
Hénon Map ◽  
Henon Map

The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

CONTROL OF n-SCROLL CHUA'S CIRCUIT

2009 ◽  
Vol 19 (11) ◽  
pp. 3813-3822 ◽  
Author(s):  
ABDELKRIM BOUKABOU ◽  
BILEL SAYOUD ◽  
HAMZA BOUMAIZA ◽  
NOURA MANSOURI
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Predictive Control ◽  
Control Method ◽  
Sufficient Conditions ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Necessary And Sufficient

This paper addresses the control of unstable fixed points and unstable periodic orbits of the n-scroll Chua's circuit. In a first step, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed predictive control method. In addition, we show how a chaotic system with multiple unstable periodic orbits can be stabilized by taking the system dynamics from one UPO to another. Control performances of these approaches are demonstrated by numerical simulations.

EXPERIMENTAL VERIFICATION OF PYRAGAS’S CHAOS CONTROL METHOD APPLIED TO CHUA’S CIRCUIT

1994 ◽  
Vol 04 (06) ◽  
pp. 1703-1706 ◽  
Author(s):  
P. CELKA
Keyword(s):  
Periodic Orbits ◽  
Experimental Verification ◽  
Chaos Control ◽  
Control Method ◽  
Experimental Results ◽  
Experimental Setup ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit

We have built an experimental setup to apply Pyragas’s [1992, 1993] control method in order to stabilize unstable periodic orbits (UPO) in Chua’s circuit. We have been able to control low period UPO embedded in the double scroll attractor. However, experimental results show that the control method is useful under some restrictions we will discuss.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

The novel control method of three dimensional discrete hyperchaotic Hénon map

2014 ◽  
Vol 247 ◽  
pp. 487-493 ◽  
Author(s):  
Shao-Fu Wang ◽  
Xiao-Cong Li ◽  
Fei Xia ◽  
Zhan-Shan Xie
Keyword(s):  
Control Method ◽  
Three Dimensional ◽  
The Novel ◽  
Hénon Map ◽  
Henon Map

STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS

2000 ◽  
Vol 10 (03) ◽  
pp. 611-620 ◽  
Author(s):  
YU-PING TIAN ◽  
XINGHUO YU
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Control Method ◽  
Uncertain System ◽  
Parameter Estimates ◽  
System State ◽  
Unstable Periodic Orbits ◽  
Unknown Parameters ◽  
Model Following ◽  
Delayed System

A novel adaptive time-delayed control method is proposed for stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems with unknown parameters. We first explore the inherent properties of chaotic systems and use the system state and time-delayed system state to form an asymptotically stable invariant manifold so that when the system state enters the manifold and stays in it thereafter, the resulting motion enables the stabilization of the desired UPOs. We then use the model following concept to construct an identifier for the estimation of the uncertain system parameters. We shall prove that under the developed scheme, the system parameter estimates will converge to their true values. The effectiveness of the method is confirmed by computer simulations.

NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP

2013 ◽  
Vol 23 (07) ◽  
pp. 1330025 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
WARWICK TUCKER
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Interval Analysis ◽  
Numerical Study ◽  
Continuation Method ◽  
Basins Of Attraction ◽  
Hénon Map ◽  
Coexisting Attractors ◽  
Henon Map ◽  
Parameter Values

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically.

scholarly journals Control of Chaos in the Rössler System

1997 ◽  
Vol 50 (2) ◽  
pp. 263 ◽  
Author(s):  
Stuart Corney
Keyword(s):  
Periodic Orbits ◽  
Time Series Data ◽  
Control Method ◽  
Three Dimensional ◽  
Dynamical Variable ◽  
Series Data ◽  
External Parameter ◽  
Unstable Periodic Orbits ◽  
System A ◽  
Linear Differential

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

TIME DELAYED REPETITIVE LEARNING CONTROL FOR CHAOTIC SYSTEMS

2002 ◽  
Vol 12 (05) ◽  
pp. 1057-1065 ◽  
Author(s):  
YANXING SONG ◽  
XINGHUO YU ◽  
GUANRONG CHEN ◽  
JIAN-XIN XU ◽  
YU-PING TIAN
Keyword(s):  
Periodic Orbits ◽  
Adaptive Learning ◽  
Chaos Control ◽  
Chaotic Systems ◽  
Control Method ◽  
Learning Control ◽  
Control Technique ◽  
Unstable Periodic Orbits ◽  
Repetitive Learning ◽  
Control Principle

In this paper, a time-delayed chaos control method based on repetitive learning is proposed. A general repetitive learning control structure based on the invariant manifold of the chaotic system is given. The integration of the repetitive learning control principle and the time-delayed chaos control technique enables adaptive learning of appropriate control actions from learning cycles. In contrast to the conventional repetitive learning control, no exact knowledge (analytic representation) of the target unstable periodic orbits is needed, except for the time delay constant, which can be identified via either experiments or adaptive learning. The controller effectively stabilizes the states of the continuous-time chaos on desired unstable periodic orbits. Simulations on the Duffing and Lorenz chaotic systems are provided to verify the design and analysis.

A suspension of the Hénon map by periodic orbits

2012 ◽  
Vol 45 (12) ◽  
pp. 1486-1493 ◽  
Author(s):  
John Starrett ◽  
Craig Nicholas
Keyword(s):  
Periodic Orbits ◽  
Hénon Map ◽  
Henon Map
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