Fractals-and-Chaos-Simplified-for-the-Life-S分形與混沌簡化生活的課件

1、Introduction to CHAOSLarry Liebovitch, Ph.D.Florida Atlantic University2019Introduction to CHAOSLarry LieThese two sets of data have the samemeanvariancepower spectrumThese two sets of data have thFractals-and-Chaos-Simplified-for-the-Life-S分形與混沌簡化生活的課件Data 1RANDOMrandomx(n) = RNDData 1RANDOMCHAOSDe

2、terministicx(n+1) = 3.95 x(n) 1-x(n)Data 2CHAOSData 2etc.etc.Fractals-and-Chaos-Simplified-for-the-Life-S分形與混沌簡化生活的課件Data 1RANDOMrandomx(n) = RNDData 1RANDOMData 2CHAOSdeterministicx(n+1) = 3.95 x(n) 1-x(n)x(n+1)x(n)Data 2CHAOSx(n+1)x(n)DefinitionCHAOSDeterministicpredict that valuethese valuesDefin

3、itionCHAOSDeterministicprCHAOSSmall Number of Variablesx(n+1) = f(x(n), x(n-1), x(n-2)DefinitionCHAOSSmall Number of VariablesDefinitionCHAOSComplex OutputDefinitionCHAOSComplex OutputPropertiesCHAOSPhase Space is Low Dimensionalphase spaced , randomd = 1, chaosPropertiesCHAOSPhase Space is Properti

4、esCHAOSSensitivity to Initial Conditionsnearly identicalinitial valuesvery differentfinal valuesPropertiesCHAOSSensitivity to PropertiesCHAOSBifurcationssmall change in a parameterone patternanother patternPropertiesCHAOSBifurcationssmaTime SeriesX(t)Y(t)Z(t)embeddingTime SeriesX(t)Y(t)Z(t)embeddiPh

5、ase SpaceX(t)Z(t)phase space setY(t)Phase SpaceX(t)Z(t)phase Y(t)Attractors in Phase SpaceLogistic EquationX(n+1)X(n)X(n+1) = 3.95 X(n) 1-X(n)Attractors in Phase SpaceLogisAttractors in Phase SpaceLorenz EquationsX(t)Z(t)Y(t)Attractors in Phase SpaceLorenX(n+1)X(n)Logistic Equationphase spacetime se

6、riesdthe fractal dimension of the attractord the fractal dimension of the attractord = 2.03, therefore, the equation of the time series that produced this attractor depends on 3 independent variables. X(t)Z(t)Y(t)X(n+1)nLorenz Equationsphase spacetimData 1time seriesphase spaced Since ,the time seri

7、es was producedby a randommechanism.d Data 1time seriesphase spaced Data 2time seriesphase spaced = 1 Since d = 1,the time series was produced by a deterministicmechanism.Data 2time seriesphase spaced Constructed by direct measurement:Phase SpaceEach point in the phase space set has coordinatesX(t),

8、 Y(t), Z(t)Measure X(t), Y(t), Z(t)Z(t)X(t)Y(t)Constructed by direct measuremConstructed from one variablePhase SpaceTakens TheoremTakens 1981 In Dynamical Systems and Turbulence Ed. Rand & Young, Springer-Verlag, pp. 366 - 381X(t+ t)X(t+2 t)X(t)Each point in thephase space sethas coordinatesX(t), X

9、(t + t), X(t+2 t) Constructed from one variablePvelocity (cm/sec)Position and Velocity of the Surface of a Hair Cell in the Inner EarTeich et al. 1989 Acta Otolaryngol (Stockh), Suppl. 467 ;265 - 27910-1-10-1-10-43 x 10-5displacement (cm)stimulus = 171 Hzvelocity (cm/sec)Position and velocity (cm/se

10、c)Position and Velocity of the Surface of a Hair Cell in the Inner EarTeich et al. 1989 Acta Otolaryngol (Stockh), Suppl. 467 ;265 - 2795 x 10-6displacement (cm)stimulus = 610 Hz-3 x 10-23 x 10-2-2 x 10-5velocity (cm/sec)Position and Data 1RANDOMx(n) = RNDfractal demension of the phase space setfrac

11、tal dimension of phase space setembedding dimension = number of values of the data taken at a time to produce the phase space setData 1RANDOMfractal demension Data 2CHAOSdeterministicx(n+1) = 3.95 x(n) 1 - x(n)fractal dimension of phase space setfractal demension of the phase space set = 1embedding

12、dimension = number of values of the data taken at a time to produce the phase space setData 2CHAOSfractal dimension fmicroelectrodechick heart cellcurrent sourcevoltmeterChick Heart CellsvGlass, Guevara, Blair & Shrier.1984 Phys. Rev. A29:1348 - 1357microelectrodechick heart cellSpontaneous Beating,

13、 No External StlimulationChick Heart CellsvoltagetimeSpontaneous Beating, Chick HeaPeriodically Stimulated2 stimulations - 1 beatChick Heart Cells2:1Periodically StimulatedChick HChick Heart Cells1:1Periodically Stimulated1 stimulation - 1 beatChick Heart Cells1:1PeriodicalChick Heart Cells2:3Period

14、ically Stimulated2 stimulations - 3 beatsChick Heart Cells2:3Periodicalperiodic stimulation - chaotic responseThe Pattern of Beatingof Chick Heart CellsGlass, Guevara, Blair & Shrier.1984 Phys. Rev. A29:1348 - 1357periodic stimulation - chaotic= phase of the beat with respect to the stimulusThe Patt

15、ern of Beating of Chick Heart Cells continuedphase vs. previous phase0.500.51.01.000.51.0i + 1experimentitheory (circle map)= phase of the beat with respeThe Pattern of Beatingof Chick Heart CellsGlass, Guevara, Belair & Shrier.1984 Phys. Rev. A29:1348 - 1357Since the phase space set is 1-dimensiona

16、l, the timing between the beats of thesecells can be described by a deterministic relationship.The Pattern of BeatingGlass, GProcedureTime seriese.g. voltage as a function of timeTurn the Time Series into a Geometric ObjectThis is called embedding.ProcedureTime seriesProcedureDetermine the Topologic

17、al Properties of this ObjectEspecially, the fractal dimension. High Fractal Dimension = Random = chance Low Fractal Dimension = Chaos = deterministicProcedureDetermine the TopologThe Fractal Dimension is NOT equal to The Fractal DimensionThe Fractal Dimension is NOT Fractal Dimension:How many new pi

18、eces of the Time Series are found when viewed at finer time resolution.XtimedFractal Dimension:How many neFractal Dimension:The Dimension of the Attractor in Phase Space is related to theNumber of Independent Variables. Xtimedx(t)x(t+ t)x(t+2 t)Fractal Dimension:The DimensiMechanism that Generated t

19、he DataChanced(phase space set)Determinismd(phase space set) = lowDatax(t)t?Mechanism that Generated the DC O L DLorenz1963 J. Atmos. Sci. 20:13-141ModelHOT(Rayleigh, Saltzman)C O L DLorenz1963 J. AtmosLorenz1963 J. Atmos. Sci. 20:13-141EquationsLorenz1963 J. Atmos. Sci. 20:X = speed of the convecti

20、ve circulation X 0 clockwise, X 0 counterclockwiseY = temperature difference between rising and falling fluidEquationsLorenz1963 J. Atmos. Sci. 20:13-141X = speed of the convective ciZ = bottom to top temperature minus the linear gradientEquationsLorenz1963 J. Atmos. Sci. 20:13-141Z = bottom to top

21、temperature Phase SpaceLorenz1963 J. Atmos. Sci. 20:13-141ZXYPhase SpaceLorenz1963 J. AtmoLorenz AttractorX 0cylinder of air rotating counter-clockwisecylinder of air rotating clockwiseLorenz AttractorX 0cyliIXtop(t) - Xbottom(t)I e t = Liapunov ExponentSensitivity to Initial ConditionsLorenz Equati

22、onsX(t)X= 1.00001Initial Condition:differentsameX(t)X= 1.00IXtop(t) - Xbottom(t)I e Deterministic, Non-ChaoticX(n+1) = f X(n)Accuracy of values computed for X(n):1.736 2.345 3.2545.455 4.876 4.2343.212Deterministic, Non-ChaoticX(n+Deterministic, ChaoticX(n+1) = f X(n)Accuracy of values computed for

23、X(n):3.455 3.45? 3.4? 3.? ? ? ?Deterministic, ChaoticX(n+1) =Initial Conditions X(t0), Y(t0), Z(t0).Clockwork Universedetermimistic non-chaoticCancomputeall futureX(t), Y(t), Z(t).EquationsInitial Conditions X(t0), Y(tInitial Conditions X(t0), Y(t0), Z(t0).Chaotic Universedetermimistic chaoticsensit

24、ivityto initial conditionsCan notcomputeall futureX(t), Y(t), Z(t).EquationsInitial Conditions X(t0), Y(tLorenz Strange AttractorTrajectories from outside:pulled TOWARDS itwhy its called an attractorstarting away:Lorenz Strange AttractorTrajecLorenz Strange AttractorTrajectories on the attractor:pus

25、hed APART from each othersensitivity to initial conditionsstarting on:Lorenz Strange AttractorTrajec“Strange”attractor is fractalphase space setnot strangestrange“Strange”attractor is fractal“Chaotic”sensitivity to initial conditionstime seriesnot chaoticchaoticX(t)tX(t)t“Chaotic”sensitivity to init

26、iShadowing TheoremIf the errors at each integration step are small, there is an EXACT trajectory which lies within a small distance of the errorfull trajectory that we calculatedShadowing TheoremIf the errorsThere is an INFINITE number of trajectories on the attractor. When we go off the attractor,

27、we are sucked back down exponentially fast. Were on an exact trajectory, just not on the one we thought we were on.Shadowing TheoremThere is an INFINITE number of4. We are on a “real” trajectory.3. Pulled backtowards the attractor.2. Error pushesus offthe attractor.1. We start here.Trajectorythat we

28、 actuallycompute.Trajectory that we are trying to compute.4. We are on a “real” trajectoSensitivity to initial conditions means that the conditions of an experiment can be quite similar, but that the results can be quite different.Sensitivity to initial conditiTUESDAY+10 lArTTUESDAY+10 lArT10 lWEDNE

29、SDAYArT+10 lWEDNESDAYArT+A = 3.22X(n)nX(n + 1) = A X(n) 1 -X (n)A = 3.22X(n)nX(n + 1) = A X(n)A = 3.42X(n)nX(n + 1) = A X(n) 1 -X (n)A = 3.42X(n)nX(n + 1) = A X(n)A = 3.62X(n)nBifurcationA = 3.62X(n)nBifurcation Start with one value of A. Start with x(1) = 0.5. Use the equation to compute x(2) from

30、x(1). Use the equation to compute x(3) from x(2) and so on. up to x(300).x(n + 1) = A x(n) 1 -x(n) Start with one value of A.x(n Ignore x(1) to x(50), these are the transient values off of the attractor. Plot x(51) to x(300) on the Y-axis over the value of A on the X-axis. Change the value of A, and

31、 repeat the procedure again.x(n + 1) = A x(n) 1 -x(n) Ignore x(1) to x(50), thesex(Sudden changes of the pattern indicate bifurcations ( ) x(n)x(n)Sudden changes of the pattern The energy in glucose is transfered to ATP. ATP is used as an energy source to drive biochemical reactions.Glycolysis+-The

32、energy in glucose is transperiodicTheoryMarkus and Hess 1985 Arch. Biol. Med. Exp. 18:261-271Glycolysistimesugar inputATP outputchaotictimetimetimeperiodicTheoryGlycolysistimesuExperimentsHess and Markus 1987 Trends. Biomed. Sci. 12:45-48cell-free extracts from bakers yeastGlycolysisATP measured by

33、fluorescence glucose inputtimeExperimentscell-free extracts ExperimentsHess and Markus 1987 Trends. Biomed. Sci. 12:45-48PeriodicfluorescenceGlycolysisVinExperimentsPeriodicfluorescencGlycolysisExperimentsHess and Markus 1987 Trends. Biomed. Sci. 12:45-48Chaotic20 minGlycolysisExperimentsChaotic20Gl

34、ycolysisMarkus et al. 1985. Biophys. Chem 22:95-105Bifurcation DiagramchaostheoryexperimentGlycolysisBifurcation DiagramcGlycolysisMarkus et al. 1985. Biophys. Chem 22:95-105ADP measured at the same phase each time of the input sugar flow cycle(ATP is related to ADP)period of the input sugar flow cy

35、cle# =period of the ATP concentrationfrequency of the input sugar flow cycleGlycolysisADP measured at periPhase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT PressTap the left index fingerin-phase with the tickof the metronome.Try to tap the right

36、 index finger out-of-phase with the tick of the metronome.Phase TransitionsTap the left Phase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT PressAs the frequency of the metronome increases, the right finger shifts from out-of-phase to in-phase mot

37、ion.Phase TransitionsAs the frequePosition of Right Index FingerPosition of Left Index FingerA. TIME SERIESPhase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT PressADDABDPosition of Right Index FingerPosition of Right Index Finger360o0oB. POINT ES

38、TIMATE OF RELATIVE PHASE180oSelf-Organized Phase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT Press2 secPosition of Right Index FingerThis bifurcation can be explained as a change in a potential energy function similar to the change which occurs

39、in a physical phase transition.system potentialscaling parameterPhase TransitionHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT PressThis bifurcation can be explaiSmall changes in parameters can produce large changes in behavior.+10cc ArT+9cc ArTSmall changes

40、in parameters caBifurcations can be used to test if a system is deterministic.Deterministic Mathematical ModelExperimentobserved bifurcationspredicted bifurcationsMatch ?Bifurcations can be used to teThe fractal dimension of the phase space set tells us if the data was generated by a random or deter

41、ministic mechanism.ExperimentalDatax(t)tThe fractal dimension of the pX(t+ t)Phase SpaceSetX(t)The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.X(t+ t)Phase SpaceX(t)The frMechanism that generated the experimental data.Determini

42、sticRandomd = lowd The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.Mechanism that generated the eEpidemicsSchaffer and Kot 1986 Chaos ed. Holden, Princeton Univ. Press40001500000measlesNew Yorktime series:phase space:chickenpox

43、Epidemics40001500000measlesNewEpidemicsOlsen and Schaffer 1990 Science 249:499-504dimension of attractor in phase spacemeasleschickenpoxKobenhavn 3.1 3.4 Milwaukee 2.6 3.2St. Louis 2.2 2.7New York 2.7 3.3EpidemicsmeasleschickenpoxKobeEpidemicsOlsen and Schaffer 1990 Science 249:499-504SEIR models -

44、4 independent variables S susceptible E exposed, but not yet infectious I infectious R recoveredEpidemicsSEIR models - 4 indEpidemicsOlsen and Schaffer 1990 Science 249:499-504Conclusion: measles: chaotic chickenpox: noisy yearly cycleEpidemicsConclusion:time series: voltageKaplan and Cohen 1990 Cir

45、c. Res. 67:886-892normalfibrillation deathD = 1chaosD = randomPhase spaceV(t), V(t+ t)ElectrocardiogramECG: Electrical recording of the muscle activity of the heart.8time series: voltagenormalfibrtime series: voltageBabloyantz and Destexhe 1988 Biol. Cybern. 58:203-211normalD = 6chaosElectrocardiogr

46、amECG: Electrical recording of the muscle activity of the heart.time series: voltagenormalD = ElectrocardiogramECG: Electrical recording of the muscle activity of the heart.time series: time between heartbeatsBabloyantz and Destexhe 1988 Biol. Cybern. 58:203-211normalD = 6chaosfibrillation deathD =

47、4chaosinduced arrhythmiasD = 3chaosEvans, Khan, Garfinkel, Kass, Albano, and Diamond 1989 Circ. Suppl. 80:II-134Zbilut, Mayer-Kress, Sobotka, OToole and Thomas 1989 Biol. Cybern, 61:371-381Electrocardiogramtime series: ElectroencephalogramEEG: Electrical recording of the nerve activity of the brain.

48、Mayer-Kress and Layne 1987 Ann. N.Y. Acad. Sci. 504:62-78time series: V(t)phase space: D=8 chaosV(t)V(t+ t)Electroencephalogramtime serieRapp, Bashore, Martinerie, Albano, Zimmerman, and Mees 1989 Brain Topography 2:99-118Babloyantz and Destexhe 1988 In: From Chemical to Biological Organization ed.

49、Markus, Muller, and Nicolis, Springer-VerlagXu and Xu 1988 Bull. Math. Biol. 5:559-565ElectroencephalogramEEG: Electrical recording of the nerve activity of the brain.Rapp, Bashore, Martinerie, AlbDifferent groups find different dimensions under the same experimental conditions.ElectroencephalogramE

50、EG: Electrical recording of the nerve activity of the brain.Different groups find differenmental taskquiet awake, eyes closedquiet sleepbrain virus: Creutzfeld- JakobEpilepsy: petit malmeditation: Qi-kongElectroencephalogramEEG: Electrical recording of the nerve activity of the brain.perhaps:High Di

51、mensionLow Dimensionmental taskElectroencephalograRandom MarkovHow to compute the next x(n):Each t pick a random number 0 R 1 If open, and R pc, then close. If closed, and R B1Control of Chaosmotion of a magnetoelastic ribbonDitto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214B B1Control of C

52、haosControl of Chaosmotion of a magnetoelastic ribbonDitto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214sensorXXn = X (t = nT)2 TB = Bo sin ( t)Control of ChaossensorXXn = X Control of Chaosmotion of a magnetoelastic ribbonDitto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-3214iterationn

53、umber0 - 23592360 - 47994800 - 70997100 - 10000noneperiod 1period 2period 1controlControl of Chaositeration0 - 2Control of Chaosmotion of a magnetoelastic ribbonDitto, Rauseo, and Spano 1990 Phys. Rev. Lett. 65:3211-32144.54.03.53.02.50200040006000800010000Iteration NumberXnControl of Chaos4.54.03.5

54、3.02.Control of Biological SystemsThe Old WayBrute Force Control.BIG machineBIG powerHeartAmpsControl of Biological SystemsTControl of Biological SystemsThe New WayCleverly timed, delicate pulses.little machinelittle powermAHeartControl of Biological SystemsTThe Old WayForces drive the system between stable states.How do we think of biological systems?The Old WayHow do we think of How do we think of biological systems?Force DForc