scholarly journals PULSATING FEEDBACK CONTROL FOR STABILIZING UNSTABLE PERIODIC ORBITS IN A NONLINEAR OSCILLATOR WITH A NONSYMMETRIC POTENTIAL

2007 ◽  
Vol 17 (08) ◽  
pp. 2797-2803 ◽  
Author(s):  
G. LITAK ◽  
M. ALI ◽  
L. M. SAHA
Keyword(s):  
Feedback Control ◽  
Periodic Orbits ◽  
Efficient Method ◽  
Chaotic Attractor ◽  
Nonlinear Oscillator ◽  
Chaos Control ◽  
Degree Of Freedom ◽  
Unstable Periodic Orbits ◽  
Periodic State

We examine a strange chaotic attractor and its unstable periodic orbits in case of one-degree of freedom nonlinear oscillator with nonsymmetric potential. We propose an efficient method of chaos control stabilizing these orbits by a pulsating feedback technique. Discrete set of pulses enable us to transfer the system from one periodic state to another.

Global Bifurcation Approximation in Controlling Chaotic Systems

1998 ◽  
Vol 08 (05) ◽  
pp. 1013-1023
Author(s):  
Byoung-Cheon Lee ◽  
Bong-Gyun Kim ◽  
Bo-Hyeun Wang
Keyword(s):  
Feedback Control ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Global Bifurcation ◽  
Good Control ◽  
Hysteresis Phenomenon ◽  
Unstable Periodic Orbits ◽  
Tracking Method ◽  
Adaptive Tracking

In our previous research [Lee et al., 1995], we demonstrated that return map control and adaptive tracking method can be used together to locate, stabilize and track unstable periodic orbits (UPO) automatically. Our adaptive tracking method is based on the control bifurcation (CB) phenomenon which is another route to chaos generated by feedback control. Along the CB route, there are numerous driven periodic orbits (DPOs), and they can be good control targets if small system modification is allowed. In this paper, we introduce a new control concept of global bifurcation approximation (GBA) which is quite different from the traditional local linear approximation (LLA). Based on this approach, we also demonstrate that chaotic attractor can be induced from a periodic orbit. If feedback control is applied along the direction to chaos, small erratic fluctuations of a periodic orbit is magnified and the chaotic attractor is induced. One of the special features of CB is the existence of irreversible orbit (IO) which is generated at the strong extreme of feedback control and has irreversible property. We show that IO induces a hysteresis phenomenon in CB, and we discuss how to keep away from IO.

Control Bifurcation Structure of Return Map Control in Chua's Circuit

1997 ◽  
Vol 07 (04) ◽  
pp. 903-909 ◽  
Author(s):  
Byoung-Cheon Lee ◽  
Ki-Hak Lee ◽  
Bo-Hyeun Wang
Keyword(s):  
Feedback Control ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Control Parameters ◽  
Unstable Periodic Orbits ◽  
Bifurcation Structure ◽  
Return Map ◽  
Adaptive Tracking ◽  
Route To Chaos ◽  
Prior Analysis

We demonstrate that return map control and adaptive tracking can be used together to locate, stabilize, and track unstable periodic orbits (UPOs). Through bifurcation studies as a function of some control parameters of return map control, we observe the control bifurcation (CB) phenomenon which exhibits another route to chaos. Nearby an UPO there are a lot of driven periodic orbits (DPOs) along the CB route. DPOs are not embedded in the original chaotic attractor, but they are generated artificially by driving the system slightly in a direction with feedback control. Based on the CB phenomenon, our adaptive tracking algorithm searches for the location and the exact control condition of the UPO by minimizing feedback perturbations. We discuss the universality of the CB phenomenon and the possibility of immediate control which does not require much prior analysis of the system.

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.

DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

2012 ◽  
Vol 22 (01) ◽  
pp. 1230001
Author(s):  
BENJAMIN COY
Keyword(s):  
Learning Community ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Topological Analysis ◽  
Three Dimensions ◽  
Locally Linear Embedding ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Linear Embedding ◽  
Locally Linear

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

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