1. Guiding an adaptive system through chaos Alfred Hubler, Ctr for Complex Systems Research Kirstin Phelps, Illinois Leadership Center University of Illinois at Urbana-Champaign

2. Summary System: Self-adjusting map dynamics with open loop parametric controls Problem: Target of the control is time dependent and passes through chaos. Is the control stable? Answer: No, not in the chaotic regime. But the control is still effective, since the systems stays at the nearest edge chaos: Adaptation to the edge of chaos- self-adjusting systems avoid chaos

3. Chaos: plans go wildly astray opposing opinions, strong emotions, and high stakes small events can have large unexpected consequences - missed deadlines, cost overruns + thinking out of the box, rapid implementation of new ideas, such as Google’s “Chaos by design” strategy => Chaos needs to be managed “Chaos is inevitable. In the sense that perturbation is evolutionary, it's also desirable. But managing it is essential. It's no use for any of us to hope that someone else will do it. Do you have your own personal strategies in place?” C. P. Brinkworth, 2006

4. System Chaotic map F with time dependent parameter x n+1 = F(x n, a n ) where x n = state at time step n=0,1,…,N-1 Parameter dynamics with control and self-adjustment: a n+1 = a n +c(A n -a n )+f w(x n, x n-1, …) where a n = parameter at time step n=0,1,…,N-1, c = magnitude of control force, A n =target f = magnitude of self-adjustment, w = low pass filter Target dynamics A n+1 = A n +r where r = rate of change of target, x n =chaos for a range of targets

5. Results: soft control Figure 1. Dynamics of the self-adjusting system which is guided through a chaotic region by a soft control: the dynamics of the state (a), the dynamics of the parameter (b), and a histogram of the parameter values (c). The peaks in the histogram at the boundary of the chaotic region indicate that the system avoids chaos (F=3.8(1-a n 2 )x n (1-x n ), w=x n -x n-32 ).

6. Results: strong control Figure 2. Dynamics of the self-adjusting system which is guided through a chaotic region by a strong control: the dynamics of the state (a), the dynamics of the parameter (b), and a histogram of the parameter values (c). The peaks in the histogram at the boundary of the chaotic region indicate are much less pronounced than in Figure 1.

7. Boundary between adaptation-to-the- edge-of-chaos and stable control Figure 3. The boundary between the adaptation to the edge of chaos and stable control. In the area labeled “control stable” the parameter stays close to the target, even in the chaotic regime. In the area labeled “adaptation to the edge of chaos” the parameter stays outside the chaotic regime, even if the target is inside the chaotic regime.

8. Soft controls=>good predictability Figure 4. The strength of the chaos versus the strength of the control (a), and the deviation of the parameter from the target versus the control strength (b). If the control strength is small the deviation from the target is large but the system is more predictable.

9. Summary Chaos & management: [1]Brinkworth, C. P. Managing chaos. URL as of 10/2006: http://catherinepalinbrinkworth.com/managing-chaos.html, [2] Patterson, K. Crucial Conversations: Tools for talking when stakes are high, McGraw- Hill: Heights Town, N.J., 2002 [3] Wheeler, D. J. Understanding variation: The key to managing chaos, 2nd Rev edition; SPC Press, Knoxville, TN, 1999. [4] Schuster, H.G. Deterministic chaos, 2Rev Ed edition. Wiley-VCH: Weinheim 1987. [5] Lashinsky, A. Chaos by design. Fortune 2006, 154. URL as of 10/2006: http://money.cnn.com/magazines/fortune/fortune_archive/2006/10/02/8387489/index.htmLashinsky http://money.cnn.com/magazines/fortune/fortune_archive/2006/10/02/8387489/index.htm Prediction of chaos: [6] Strelioff, C.; Hübler, A. Medium term prediction of chaos. Phys. Rev. Lett., 2006, 96, 044101-044104. Control of Chaos: [7] Hübler, A. Adaptive control of chaotic systems. Helv. Phys. Acta 1989, 62, 343-346. [8] Breeden, J. L. ; Dinkelacker F. ; Hübler A. Noise in the modeling and control of dynamical systems. Phys. Rev. A 1990, 42, 5827-5836. [9] Ott, E. ; Grebogi, C.; Yorke, J. A. Controlling chaos. Phys. Rev. Lett. 1990, 64, 1196– 1199.

10. Adaptation to the edge of chaos: [10] Kauffman, S. A. The origins of order: Self-organization and selection in evolution; Oxford University Press: New York, 1993. [11] Packard, N. H. in Dynamic patterns in complex systems, edited by J. A. S. Kelso, A. J. Mandell, and M. F. Schlesinger (World Scientific, Singapur, 1988), pp. 293–301. [12] Mitchell, M. ; Hraber, P. T.; Crutchfield J. P. Revisiting the edge of chaos: Evolving cellular automata to perform computations. Complex Systems 1993, 7, 89-130. [13] Bak, P. ; Tang, C. ; Wiesenfeld K. Self-organized criticality. Phys. Rev. A 1988, 38, 364-374. [14] Zhigulin, V. P. ; Rabinovich, M. I. ; Huerta, R. ; Abarbanel, H. D. I. Robustness and enhancement of neural synchronization by activity-dependent coupling. Phys. Rev. E, 2003, 67, 021901-021904. [15] Pierre, D. ;Hubler, A. A theory for adaptation and competition applied to logistic Map Dynamics. Physica 75D, 1994, 343-360. [16] Langton, C. A. Computation at the edge of chaos. Physica 42D, 1990, 12-37. [17] Adamatzky, A.; Holland, O. Chaos, phenomenology of excitation in 2-D cellular automata and swarm systems. Solitons and Fractals, 1993, 9, 1233-1265.phenomenology of excitation in 2-D cellular automata and swarm systems Quantitative Model forAdaptation to the edge of chaos: [18] Melby, P. ; Kaidel, J.; Weber, N.; Hubler, A. Adaptation to the edge of chaos in the self-adjusting logistic map. Phys. Rev. Lett., 2000, 84, 5991-5993.Adaptation to the edge of chaos in the self-adjusting logistic map [19] Melby, P. ; Weber, N. ; Hubler A. Robustness of adaptation in controlled self- adjusting chaotic systems. Fluct. Noise Lett., 2002, 2, L285-L292. [20] Baym, M.; Hübler, A. W. Conserved quantities and adaptation to the edge of chaos, Phys. Rev. E, 2006, 73, 056210-056217.BaymHübler

11. This work: System: Self-adjusting map dynamics with open loop parametric controls Problem: Target of the control is time dependent and passes through chaos. Is the control stable? Answer: No, not in the chaotic regime. But the control is still effective, since the systems stays at the nearest edge chaos: Soft controls=>high predictability, but large deviation from target