scholarly journals Control-Based Continuation of Unstable Periodic Orbits

2009 ◽  
Author(s):  
Jan Sieber ◽  
Bernd Krauskopf ◽  
David Wagg ◽  
Simon Neild ◽  
Alicia Gonzalez-Buelga
Keyword(s):  
Periodic Orbits ◽  
Control Parameter ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Experimental Procedure ◽  
Stable Period ◽  
Fold Bifurcation ◽  
Time Delayed Feedback ◽  
Unstable Branch ◽  
Pendulum Experiment

We present an experimental procedure to track periodic orbits through a fold (saddle-node) bifurcation, and demonstrate it with a parametrically excited pendulum experiment where the control parameter is the amplitude of the excitation. Specifically, we track the initially stable period-one rotation of the pendulum through its fold bifurcation and along the unstable branch. The fold bifurcation itself corresponds physically to the minimal amplitude that is able to support sustained rotation. Our scheme is based on a modification of time-delayed feedback in a continuation setting, and we show for an idealized model that it converges with the same efficiency as classical proportional-plus-derivative control.

scholarly journals Control-Based Continuation of Unstable Periodic Orbits

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Jan Sieber ◽  
Bernd Krauskopf ◽  
David Wagg ◽  
Simon Neild ◽  
Alicia Gonzalez-Buelga
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Experimental Procedure ◽  
Idealized Model ◽  
Stable Period ◽  
Fold Bifurcation ◽  
Time Delayed Feedback ◽  
Unstable Branch ◽  
Pendulum Experiment

We present an experimental procedure to track periodic orbits through a fold (saddle-node) bifurcation and demonstrate it with a parametrically excited pendulum experiment where the tracking parameter is the amplitude of the excitation. Specifically, we track the initially stable period-one rotation of the pendulum through its fold bifurcation and along the unstable branch. The fold bifurcation itself corresponds to the minimal amplitude that supports sustained rotation. Our scheme is based on a modification of time-delayed feedback in a continuation setting and we show for an idealized model that it converges with the same efficiency as classical proportional-plus-derivative control.

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.

AN ADAPTIVE CONTROL ALGORITHM FOR TRACKING UNSTABLE PERIODIC ORBITS

1994 ◽  
Vol 04 (05) ◽  
pp. 1311-1317 ◽  
Author(s):  
VALERY PETROV ◽  
MICHAEL F. CROWLEY ◽  
KENNETH SHOWALTER
Keyword(s):  
Stability Analysis ◽  
Periodic Orbits ◽  
Control Algorithm ◽  
Control Parameter ◽  
Unstable Periodic Orbits ◽  
Automatic Tracking ◽  
Bz Reaction ◽  
Control Scheme ◽  
Adaptive Control Algorithm ◽  
Predictor Corrector

A predictor-corrector control algorithm for stabilizing and tracking unstable periodic orbits is presented. Automatic tracking is made possible by incorporating a stability-analysis subroutine into a map-based control scheme. The method is used to track unstable orbits in the Belousov-Zhabotinsky (BZ) reaction as a laboratory control parameter is varied.

On Optimal Stabilization of Periodic Orbits via Time Delayed Feedback Control

1998 ◽  
Vol 08 (08) ◽  
pp. 1699-1706 ◽  
Author(s):  
M Basso ◽  
R. Genesio ◽  
L. Giovanardi ◽  
A. Tesi ◽  
G. Torrini
Keyword(s):  
Periodic Orbits ◽  
Feedback System ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Unstable Periodic Orbits ◽  
Stabilizing Controller ◽  
Circle Criterion ◽  
The One ◽  
Periodic Feedback ◽  
Time Delayed Feedback

The paper considers the problem of designing time delayed feedback controllers to stabilize unstable periodic orbits of a class of sinusoidally forced nonlinear systems. This problem is formulated as an absolute stability problem of a linear periodic feedback system, in order to employ the well-known circle criterion. In particular, once a single test is verified by an unstable periodic orbit of the chaotic system, our approach directly provides a procedure for designing the optimal stabilizing controller, i.e. the one ensuring the largest obtainable stability bounds. Even if the circle criterion provides a sufficient condition for stability and therefore the obtained stability bounds are conservative in nature, several examples, as the one presented in this paper, show that the performance of the designed controller is quite satisfactory in comparison with different approaches.

Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

2015 ◽  
Vol 25 (13) ◽  
pp. 1550185 ◽  
Author(s):  
Tetsushi Ueta ◽  
Daisuke Ito ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Transient Behavior ◽  
Control Input ◽  
Stable Limit ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Saddle Node

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

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