A map-based algorithm for controlling low-dimensional chaos

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<ul><li><p>A mapbased algorithm for controlling lowdimensional chaosValery Petrov, Bo Peng, and Kenneth Showalter Citation: The Journal of Chemical Physics 96, 7506 (1992); doi: 10.1063/1.462402 View online: http://dx.doi.org/10.1063/1.462402 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Chaotic oscillations in a map-based model of neural activity Chaos 17, 043109 (2007); 10.1063/1.2795435 Mapbased neuron networks AIP Conf. Proc. 887, 69 (2007); 10.1063/1.2709587 Rules for controlling low-dimensional vocal fold models with muscle activation J. Acoust. Soc. Am. 112, 1064 (2002); 10.1121/1.1496080 Search for low-dimensional chaos in a reversed field pinch plasma Phys. Plasmas 6, 100 (1999); 10.1063/1.873264 A lowdimensional model for chaos in open fluid flows Phys. Fluids A 5, 1947 (1993); 10.1063/1.858821 </p><p> This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p>129.120.242.61 On: Tue, 25 Nov 2014 12:19:27</p><p>http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcovhttp://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/327320036/x01/AIP-PT/JCP_ArticleDL_101514/PT_SubscriptionAd_1640x440.jpg/47344656396c504a5a37344142416b75?xhttp://scitation.aip.org/search?value1=Valery+Petrov&amp;option1=authorhttp://scitation.aip.org/search?value1=Bo+Peng&amp;option1=authorhttp://scitation.aip.org/search?value1=Kenneth+Showalter&amp;option1=authorhttp://scitation.aip.org/content/aip/journal/jcp?ver=pdfcovhttp://dx.doi.org/10.1063/1.462402http://scitation.aip.org/content/aip/journal/jcp/96/10?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/chaos/17/4/10.1063/1.2795435?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.2709587?ver=pdfcovhttp://scitation.aip.org/content/asa/journal/jasa/112/3/10.1121/1.1496080?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/6/1/10.1063/1.873264?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pofa/5/8/10.1063/1.858821?ver=pdfcov</p></li><li><p>A map-based algorithm for controlling low-dimensional chaos Valery Petrov, Bo Peng, and Kenneth St1owaltera ) Department o/Chemistry, West Virginia University, Morgantown, West Virginia 26506-6045 </p><p>(Received 17 December 1991; accepted 5 February 1992) </p><p>A simple proportional-feedback algorithm for controlling chaos is presented. The scheme is a map-based variation ofa method recently proposed by Ott, Grebogi, and Yorke [Phys. Rev. Lett. 64, 1196 (1990) ] in which unstable periodic orbits embedded within a strange attractor are stabilized through deliberate perturbations of a system constraint. The simplified method offers advantages for control of systems in which more complicated algorithms might not be feasible due to short time scales or limited computational resources. Applications to chemical and biological models are presented to demonstrate the utility and limitations of the method. Low-dimensional chaos can usually be stabilized through proportional feedback of one parameter; in some cases, however, a linear combination of several parameters must be utilized. </p><p>I. INTRODUCTION </p><p>The ability to deliberately select and stabilize a particu-lar unstable periodic state from the aperiodic oscillations of a chaotic system offers intriguing possibilities for practical ap-plications. Ott, Grebogi, and Yorke (OG Y) have developed a general method for stabilizing unstable periodic orbits em-bedded within a strange attractor. 1 Their algorithm is based on targeting the stable manifold of a particular unstable orbit by introducing a controlled perturbation of a system con-straint. The general utility of the method has been demon-strated by stabilizing unstable orbits in the Henon map, 1 and controlling chaos in an experimental system consisting of a buckled magnetoelastic ribbon has been reported by Ditto, Rauseo, and Spano.2 </p><p>Controlling chaos in chemical systems is of special inter-est, both from a practical perspective, such as in tank reactor processes, and from the standpoint of implications for dy-namical behavior in biological systems, such as self-regula-tion. We recently proposed a simple geometric algorithm for controlling chaos in a chemical system based on targeting an unstable point in the return map corresponding to a particu-lar unstable orbit in the continuous system. 3 The map-based method offers significant advantages in applications where targeting procedures must be minimized, e.g., as in control-ling high frequency chaos. </p><p>Advantages and limitations of the map-based method for controlling chaos are further examined in this paper. The algorithm and its application to a simple three-variable chemical model45 are presented in Sec. II along with an analysis of the stability of the method to errors and perturba-tions. In Sec. III we generalize the algorithm to overcome limitations that may arise in applications of the simple one-parameter scheme. Application of the algorithm to a four-variable biological model recently studied by Lengyel and Epstein6 as an example of diffusion induced chaos is de-scribed in Sec. IV. A judicious choice of controlling param-eter, monitored variable, and Poincare section, or a linear </p><p>a) To whom correspondence should be addressed. </p><p>combination of two or more controlling parameters allows utilization of the map-based scheme for control. We con-clude in Sec. V with an assessment of the algorithm in poten-tial applications. </p><p>II. ALGORITHM AND APPLICATION TO A SIMPLE CHEMICAL SYSTEM </p><p>A. Proportional-feedback algorithm </p><p>Consider a ID map of the formxn + k (p) = F(xn,p) de-scribing the chaotic evolution of a system. The map can be linearized in the vicinity of any particular fixed point Xs; e.g., for the period-l unstable point, </p><p>xn+l(p)=f[xn-xs(P)]+xs(p), lfl&gt;l, (1) where p is a system constraint andfis the slope of the map at Xs' Suppose that varyingp by the small amount op moves the unstable point without significantly changing f, the linear-ized map becomes </p><p>xn+l (p+op) =f[xn -xs(P+op)] +xs(P+op), (2) </p><p>where </p><p>dxs (p) xs(P+op) =op +xs(p). </p><p>dp (3) </p><p>The transitivity of a chaotic system ensures that it will visit the vicinity of the fixed point at some point in time. Suppose Xn = Xs (p) + Ax, where Ax </p></li><li><p>PetroY, Peng, and Showalter: Controlling low-dimensional chaos 7507 </p><p>be stabilized and proportionally varies a control parameter according to the difference. We require this difference (or the parameter variation) to be smaller than some limit defin-ing the control range; once within the range, the proportion-al-feedback algorithm is switched on. The magnitude of the control range is arbitrary and can be made very small; the smaller the range, however, the longer the average transient before it is visited. Once the system is stabilized, the param-eter range defined by the subsequent perturbations necessary to maintain stabilization is independent of the predefined control range and is typically extremely small. 3 </p><p>The algorithm is applicable, in principle, to all the un-stable points corresponding to the unstable limit cycles of a strange attractor. The only requirement is that the attractor is described by effectively I D dynamics in the region around the unstable point of interest; whether or not the map is globally single valued is unimportant. The map need not be strictly single valued locally (as in a discrete system), but the extent of the local fractal structure should be small com-pared to the control range and the linear region of the fixed point. </p><p>The scheme can be further simplified by determining the proportionality factor g in Eq. (4) from the horizontal dis-tance of the shifted map to the original unstable point (see Fig. 3 of Ref. 3). In practice, the determination of g requires </p><p>1.4 </p><p>1.0 </p><p>0.6 . ' .. ": .. .... ". ". ". ". .. .. "" ..... " ............ .. </p><p>'. '. .. ' </p><p>measurements at two different parameter values to define two maps; the measured value of g is then used to calculate the necessary perturbations for control. </p><p>B. Application to a simple chemical system </p><p>The map-based algorithm has been applied to a three-variable model for isothermal chemical chaos. 3 The model is based on mass-action kinetics and can be studied as either an open4 or closed system.5 The rate equations for the open system are </p><p>da /3 2 -=fl(K+Y) -a-a , dr </p><p>(5) </p><p>a : = a + a/3 2 - /3, </p><p>dy O-=(3-y, </p><p>(6) </p><p>(7) dr </p><p>where a, (3, and yare dimensionless variables and fl, K, a, and 0 are dimensionless parameters. The general features of the attractor as well as the stabilized period-l limit cycle are shown in Fig. 1. The Poincare section defined by y = 15, r&gt; 0 gives rise to a map that closely obeys ID dynamics. Because fl is directly related to the concentration of the reac-tant, it is chosen to be the controlling parameter. The vari-</p><p>., . ... ~.:"I" </p><p>FIG. 1. Strange attractor of chemical system defined by Eqs. (5)-(7). Param-eters are (jl,K,a,o) = (0.1540,65,5 X 10 - \2 X 10 - 2); Poincare section defined by r = IS, r&gt; 0 shown as opaque sheet. Also shown is stabilized period I limit cycle (solid trajectory). .' .:;, JO </p><p>:. : . :r .. :, ' .. " ............ .. </p><p>0.2 </p><p>.... .. . ' . ...... .. .. .. . .. . . "~ " .. :, . </p><p>30 ,.4.0 10 ,.~O ,. </p><p>2.0 .... (J log </p><p>1.5 </p><p>J. Chern. Phys., Vol. 96, No.1 0, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:</p><p>129.120.242.61 On: Tue, 25 Nov 2014 12:19:27</p></li><li><p>7508 Petrev, Peng, and Showalter: Controlling low-dimensional chaos </p><p>ation of f.l is determined by monitoring the value of P on each return to the section, and period-I, period-2, and period-4 unstable limit cycles are readily stabilized (each with a dif-ferent value of g). The variable a may also be monitored for determining successive values of f.l for stabilization. </p><p>Intersections with the Poincare section before and dur-ing control yield values of Pn for construction of the map shown in Fig. 2(a). The nearly horizontal segment shows the mapping during control; it also defines the control range. Many different initial conditions were necessary to visualize the map in the control range, since once in the range the variation around Ps is extremely small. The enlargement in Fig. 2(b) shows that the map in the control range is not perfectly horizontal, corresponding to perfect control ac-cording to Eq. (10) (vide infra). The slope can be varied by </p><p>0 d 52 </p><p>0 d co </p><p>~o + . cO </p><p>c:o... lD </p><p>(a) </p><p>0 d "' </p><p>0 d N </p><p>'?i </p><p>0 </p><p>'4 </p><p>o t</p></li><li><p>Petrov, Peng, and Showalter: Controlling low-dimensional chaos 7509 </p><p>and the mapping becomes </p><p>x,,+~ = -/r(x" -x,) +xs The fixed point is stabilized when </p><p>Irl &lt; Ill/I </p><p>(16) </p><p>(17) </p><p>The nonzero slope of the map in Fig. 2 reflects a slight error in the value of g. We note that higher precision is required to stabilize fixed points with large slopes. </p><p>(d) Nonlinearity in mapping or response. If the original mapping over the control range involves a second-order term, </p><p>x" + k = x; + /(x" - x;) + I (x" - x; )2, (18) the new mapping becomes parabolic with the extremum lo-cated at the fixed point, </p><p>(19) </p><p>If the response of the system to the controlling perturbation is slightly nonlinear, </p><p>x; -x, = /~ 1 (x" -xs) +2(X" _X,)2, (20) the new mapping is also parabolic, </p><p>X,,+k =Xs +2(1-f)(X" _X,)2. (21) </p><p>In either case, the system is stabilized at the fixed point, only with a longer transient time. </p><p>We conclude that the proportional-feedback algorithm is robust in the presence of small errors, random perturba-tions, and nonlinearities. </p><p>III. UTILITY AND LIMITATIONS OF THE MAP-BASED ALGORITHM </p><p>A. Single-parameter scheme </p><p>The above discussion is based solely on maps. A map constructed from a Poincare section in the state space of a continuous system reflects the behavior of the system on the strange attractor. The behavior off the attractor, on the other hand, is affected by the stable manifolds, and is not described by the map. Because the map is not applicable for points off the attractor, it cannot be utilized when the system state is significantly displaced from the attractor. </p><p>The proportional-feedback algorithm, as presented above, requires any movement of the unstable point to be along the original section of the attractor as the control pa-rameter is varied; otherwise, there is no direct correspon-dence between the map and the original system once the parameter is varied. Furthermore, unless the unstable point moves along the attractor, monitoring different variables may suggest different op values when projecting the section to a particular axis. This can be understood by considering an extreme case where the unstable point on a Poincare sec-tion is shifted nearly perpendicular to the axis for a particu-lar variable. Monitoring that variable would suggest a large value for op, while monitoring another variable would result in a smaller value. </p><p>The requirement that the unstable point move along the attractor may seem to unduly restrict application of the algo-</p><p>rithm. However, provided that the shifted fixed point re-mains close to the attractor, the dynamical behavior is gov-erned primarily by the unstable manifold, and the associated unstable periodic orbit is readily stabilized by the algorithm. Figure 3 (a) shows 12 Poincare sections of the attractor in Fig. 1 for two values of /1, one corresponding to the center and the other to the maximum of the control range. The two sets of sections overlap at the resolution of this figure. With the higher resolution shown in Fig. 3(b), we see that the attractor is shifted only slightly away from its original posi-tion on varying /1; however, the unstable limit cycle shifts significantly along the unstable manifold. This behavior seems to be typical for many systems exhibiting low-dimen-sional chaos. For example, in the Lorenz7 and Rossler8 sys-tems one can easily find Poincare sections and control pa-rameters such that the unstable limit cycles are moved along the attractors. We have applied the single-parameter scheme to each of these systems and readily stabilized the corre-sponding period-l unstable orbits.'l </p><p>If all the accessible parameters significantly shift the at-tractor away from the unstable manifold of the fixed point, it is often possible to retain the simplicity of the scheme by monitoring another variable or by adjusting the value of g around its measured value. Consider, for example, a two-dimensional Poincare section parallel to the x-y plane in a three-dimensional x-y-z state space. If the stable eigenvec-tor is perpendicular to the x axis, then monitoring x results in targeting the stable eigenvector (as in the OGY scheme), and the orbit is stabilized. </p><p>B. Multiple-parameter scheme </p><p>We now describe a more general approach which relies on a...</p></li></ul>