Resonant forcing of multidimensional chaotic dynamics多维混沌动力学强迫共振

1、guiding an adaptive system through chaosalfred hubler, ctr for complex systems researchkirstin phelps, illinois leadership centeruniversity of illinois at urbana-champaignsummary system: self-adjusting map dynamics with open loop parametric controls problem: target of the control is time dependent a

2、nd 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 chaoscfnnchaos: plans go wildly astray opposing opinions, strong e

3、motions, 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 googles “chaos by design” strategy = chaos needs to be managed “chaos is inevitable. in the sense that perturbation is

4、evolutionary, its also desirable. but managing it is essential. its 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 system chaotic map f with time dependent parameter xn+1 = f(xn, an) where xn = state at time step

5、n=0,1,n-1 parameter dynamics with control and self-adjustment: an+1= an +c(an-an)+f w(xn, xn-1, )where an = parameter at time step n=0,1,n-1,c = magnitude of control force, an=targetf = magnitude of self-adjustment, w = low pass filter target dynamicsan+1= an+rwhere r = rate of change of target, xn=

6、chaos for a range of targetsresults: soft controlfigure 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 th

7、e boundary of the chaotic region indicate that the system avoids chaos (f=3.8(1-an2)xn(1-xn), w=xn-xn-32).results: strong controlfigure 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

8、(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. )1 ()1 (8 . 3),(2nnnnnxxaaxfboundary between adaptation-to-the-edge-of-chaos and stable controlfigure 3. the boundary between the ada

9、ptation 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 re

10、gime. soft controls=good predictabilityfigure 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 predic

11、table.summarychaos & management:1brinkworth, c. p. managing chaos. url as of 10/2006: http:/ 2 patterson, k. crucial conversations: tools for talking when stakes are high, mcgraw-hill: heights town, n.j., 20023 wheeler, d. j. understanding variation: the key to managing chaos, 2nd rev edition; s

12、pc 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:/ of chaos:6 strelioff, c.; hbler, a. medium term prediction of chaos. phys. rev. lett., 2006, 96, 044101-044104.c

13、ontrol of chaos:7 hbler, a. adaptive control of chaotic systems. helv. phys. acta 1989, 62, 343-346.8 breeden, j. l. ; dinkelacker f. ; hbler 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

14、. lett. 1990, 64, 11961199.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. schle

15、singer (world scientific, singapur, 1988), pp. 293301.12 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 198

16、8, 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 ma

17、p 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.quantitative model foradaptation

18、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, 5991-5993.19 melby, p. ; weber, n. ; hubler a. robustness of adaptation in controlled self-adjusting chaotic systems. fluct. noise let

19、t., 2002, 2, l285-l292.20 baym, m.; hbler, a. w. conserved quantities and adaptation to the edge of chaos, phys. rev. e, 2006, 73, 056210-056217.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 deviati