A. Hubler, G. Foster, K. Phelps, Managing chaos: Thinking out of the box, Complexity 12, (2007). Managing Chaos

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1 A. Hubler, G. Foster, K. Phelps, Maagig chaos: Thikig out of the box, Complexity 12, (2007). Maagig Chaos Alfred W. Hübler, Gle C. Foster, ad Kirsti C. Phelps Alfred Hubler ad Gle Foster are at Ceter for Complex Systems Research ad with the Departmet of Physics at the Uiversity of Illiois at Urbaa-Champaig, Urbaa, IL, 61801, U.S.A. Kirsti Phelps is at the Illiois Leadership Ceter at the Uiversity of Illiois at Urbaa-Champaig, Urbaa, IL, 61801, U.S.A. Chaos is ievitable. I the sese that perturbatio is evolutioary, it's also desirable. But maagig it is essetial. It's o use for ay of us to hope that someoe else will do it. Do you have your ow persoal strategies i place? C. P. Brikworth, 2006 [1] Chaos meas that strategies go wildly astray. It is ofte associated with missed deadlies, uderstaffig, ruaway costs, ad similar situatios geerally cosidered egative. Uder these circumstaces Chaos describes a situatio where the goals of a strategy are uachievable ad therefore the outcomes become radom, upredictable ad ofte udesirable. This is exemplified i a recet message by Bill Ford to all of Ford Motor s employees sayig: "The busiess model that sustaied us for decades is o loger sufficiet to sustai profitability." Geoffrey Colvi, seior editor at Fortue magazie, aalyzes Ford s problems i his article Maagig i Chaos [2]. But what if the goals of a strategy are achievable, but a small deviatio from the pla or regulatio leads to very differet outcomes? [3] This behavior is called determiistic

2 chaos [4]. Without maagemet, determiistic chaos ca produce arbitrary outcomes, some may be very positive some may be very egative. For istace, the evolutio of a orgaizatio is determiistic chaos, if it ecourages thikig out of the box ad implemets these ew ideas rapidly, such as Google s Chaos by desig strategy [5]. If the maagemet does a good job i prioritizig ideas for implemetatio the overall outcome is positive. This appears to be a particularly good recipe for success at research facilities ad educatioal istitutios [6]. I the followig we discuss the maagemet of some very simple determiistic chaotic agets which are subject to a small amout of oise. The agets ca be thought of as busiess uits or other oliear dyamical systems. The chaotic agets are cotrolled by a cotrol uit, which could be a maager or a computer algorithm. This is by o meas a simulatio of maagig a real world social orgaizatio or busiess etity. Models where simple cotrol uits maage a set of simple determiistic chaotic agets may provide ituitio or illustrate a paradigm for the maagemet of etities which have realistic strategies, but where a small deviatio from the pla or regulatio leads to a very differet outcome. Fially we discuss maagig etworks of determiistic chaotic agets. Predictig chaos is hard, cotrollig chaos is easy. More precisely, log term predictios of determiistic chaos are hard, sice eve very small amouts of oise ca chage the motio sigificatly. Short term predictios ad eve medium term predictios of chaos are ot that difficult, sice the motio is govered by a determiistic equatio, plus some small oise [8]. I cotrast, cotrollig the chaotic motio of a

3 aget is ofte easy, both short term ad log term. Just apply a cotrol force which is equal to the differece betwee the ext state of the aget ad the target, ad it will go to the target [8]. This requires predictig the ext state, which is a short term predictio, ad therefore possible for chaotic agets. This cotrol algorithm would ot work for a radom motio, sice radom motio ca ot be predicted, ot eve for oe time step. This chaos cotrol algorithm was itroduced by Hubler i 1989 [9] ad sice has bee further developed ad widely used [10, 11]. Fig. 1 shows ope loop cotrol of chaotic logistic map dyamics for three differet targets [8]. Ope loop cotrol is ot always stable, oly if the target is i the coverget regio of the state space [12]. I coverget regio two eighborig states get eve closer at the ext time step [13]. The state space of a chaotic aget ca be divided ito two regios, the coverget regio ad the rest, the diverget regio. The dashed area i Fig. 1 is the coverget regio. If the target is i the coverget regio, the chaos cotrol is stable. Eve if the target dyamics is chaotic or radom, the cotrol is stable if the target dyamics is i the coverget regio. Fig. 1c shows the coversio of a ucotrolled chaotic logistic map dyamics ito a cotrolled chaotic logistic map dyamics. The cotrol uit, ad everyoe who has access to the target dyamics, ca make log term predictios of cotrolled chaos, whereas ayoe else ca oly make short term predictios of cotrolled chaos. For the cotrol uit, cotrolled chaos is predictable, ad still has most of the beefits of chaos. Chaotic agets costatly explore the state- space

4 ad have high potetial for improvig their performace, i particular i evolvig eviromets [14]. What would happe if a cotrol uit tries to cotrol a set of slightly differet chaotic agets, with a sigle cotrol force? If a cotrol force is desiged for a aget, ad the applied to agets with differet parameter values the cotrol may or may ot work. For simple systems, such as logistic map chaos with parameter value a=3.8, cotrollig chaos works as log as the differece betwee the agets is less tha 25%. This meas a cotrol uit ca cotrol a set of chaotic agets with a sigle cotrol force if the differece betwee the agets is less tha 25%. However, if the diversity of the agets greater, the cotrol fails. I this umerical example we use a chaotic target, which is a rather sophisticated cotrol. If the target is simpler the cotrol works for groups of agets with a larger diversity. Aother iterestig questio is the cotrol of simple aget etworks, for istace the cotrol of a chaotic leader-follower system. We cosider the situatio where the dyamics of a chaotic aget (leader) is imitated by a secod aget (follower), ad we assume there is some feedback from the follower to the leader. The Heo map is a simple model for such a system. We study the dyamics a chaotic leader-follower etwork which is cotrolled by a cotrol uit [15]. I this case the coverget regio depeds oly o the state of the leader. Therefore a stable cotrol of the leader-follower etwork ca be easily achieved by cotrollig the leader. Fig. 2 shows the ucotrolled ad cotrolled chaos i a leader-follower etwork. The state of the leader

5 ad the follower are plotted versus time. For the first 20 time steps there is o cotrol, ad the dyamics is Heo map chaos. Afterward a cotrol force is applied, where the target dyamics is a chaotic logistic map dyamics iside the coverget regio of the Heo map. The plot illustrates that the leader-follower system quickly approaches the target, ad thus behaves like a chaotic logistic map. This cotrolled chaotic dyamics is predictable for the cotrol uit, eve for a log period of time. Log term predictio of ucotrolled chaos is virtually impossible i large etworks of chaotic agets. However it appears to be possible to switch such etworks to cotrolled chaos, which makes them predictable, without losig the beefits of chaotic systems. Eve though the dyamics of social orgaizatios are much more complicated tha these simple chaotic models, it is coceivable that a similar approach ca be used to predict ad cotrol them. ACKNOWLEDGEMENT This work was supported by the Natioal Sciece Foudatio through grat No. DMS ITR ad grat No. DGE REFERENCES 1. Brikworth, C. P. Maagig chaos. URL as of 10/2006: 2. Colvi, G. Maagig i chaos. Fortue 2006, 154. URL as of 10/2006: 3. Wheeler, D. J. Uderstadig variatio: The key to maagig chaos, 2d Rev editio; SPC Press, Koxville, TN, 1999.

6 4. Schuster, H.G. Determiistic chaos, 2Rev Ed editio. Wiley-VCH: Weiheim Lashisky, A. Chaos by desig. Fortue 2006, 154. URL as of 10/2006: 6. Mossberg, B. Chaos o campus: A prescriptio for global leadership. Educatioal Record 1993, 74, Strelioff, C.; Hübler, A. Medium term predictio of chaos. Phys. Rev. Lett. 2006, 96, The dyamics of a aget or some other dyamical system is modeled with a map with additive forcig, x 1 f ( x, a) F, =0,1,2,,N, where x is the state at time step. a is the aget parameter. For the logistic map the mappig fuctio is f ( x, a) a x (1 x ), where 0 x 1, 0 a 4. The chaotic logistic map is used to describe growth with limited resources. For a=3.8 the dyamics is chaotic ad this value is used i the figures. F X 1 f ( X, b) is a cotrol force, where X is the target dyamics, ad b is a estimate of the aget parameter a. f ( X, b) is a short term predictio of the chaotic system. I Fig. 1a the target is X 0. 6 ad i Fig. 1b the target is X I Fig. 1c the target X X X 1 20 is chaotic, where the distace betwee eighborig trajectories icreases by a factor of two each time step. The distace betwee the trajectories is d 2 2 ( xn X N ) ( y N YN ). 9. Hübler, A. Adaptive cotrol of chaotic systems. Helv. Phys. Acta 1989, 62, Breede, J. L. ; Dikelacker F. ; Hübler A. Noise i the modelig ad cotrol of

7 dyamical systems. Phys. Rev. A 1990, 42, Ott, E. ; Grebogi, C.; Yorke, J. A. Cotrollig chaos. Phys. Rev. Lett. 1990, 64, Jackso, E.A. Cotrols of dyamic flows with attractors. Phys. Rev. A 1991, 44, The coverget regio is that part of the state space where eighborig trajectories get closer withi oe time step, i.e. f / x 1. The coverget regio for the logistic map is (2a) x (2a). 14. Hübler, A.; Pies, D. Predictio ad adaptatio i a evolvig chaotic eviromet. I: Cowa, G.; Pies, D.; Meltzer D. (Eds), Complexity: metaphors, models, ad reality. Addiso-Wesley: Readig, MA, 1994, pp The Heo map has two variables, x the state of the leader at time step =0,1,2,, N, ad y, the state of the follower. The follower imitates the leader, i.e. y 1 b x Fy,. The dyamics of the leader is modeled by a logistic map, plus some feedback from the follower, 2 x 1 1 a x y Fx,. Typical values are a=1.1, ad b=0.3. The coverget regio is ( b 1) /(2a) x (1 b) /(2a). The cotrol forces are F 2 x, X 1 (1 a X Y ) ad Fy, Y 1 b X, where X the target state of the leader ady, the target state of the follower. Cotrollig the leader meas, that X ca be ay dyamics, statioary, periodic, chaotic, or radom withi the coverget regio, ad there is o cotrol of the follower, i.e. Y b 1 X.

9 Figure 1. The state of a chaotic aget versus time [8]. The cotrol starts at time step 20. The cotiuous lie is the target. I plot (a) the target is iside the coverget regio (gray area) ad the cotrol is stable. I plot (b), the target is outside the coverget regio, ad the cotrol is ustable. I this case the dyamics does ot get closer ad closer to the target. I plot (c) the target is chaotic. Sice the target is iside the coverget regio, the cotrol is stable, eve if the target is chaotic.

10 Figure 2. Cotrollig a chaotic leader-follower etwork [14]. This plot shows the state of the leader x ad the state of the follower y versus time step. Before time step =20 the chaotic etwork is ucotrolled ad hard to predict, after time step =20 the chaotic etwork is cotrolled ad predictable for the cotrol uit. The red lie is the chaotic target dyamics. I this computer simulatio the chaotic etwork dyamics is very close to the target.

AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS

http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC