1、 Larry Liebovitch,Ph.D.Florida Atlantic University 2019Data 1RANDOMrandomx(n)=RNDCHAOSDeterministicx(n+1)=3.95 x(n)1-x(n)Data 2etc.Data 1RANDOMrandomx(n)=RNDData 2CHAOSdeterministicx(n+1)=3.95 x(n)1-x(n)x(n+1)x(n)DefinitionCHAOSDeterministicpredict that valuethese valuesCHAOSSmall Number of Variable
2、sx(n+1)=f(x(n),x(n-1),x(n-2)DefinitionDefinitionCHAOSComplex OutputPropertiesCHAOSPhase Space is Low Dimensionalphase spaced ,randomd=1,chaosPropertiesCHAOSSensitivity to Initial Conditionsnearly identicalinitial valuesvery differentfinal valuesPropertiesCHAOSBifurcationssmall change in a parametero
3、ne patternanother patternTime SeriesX(t)Y(t)Z(t)embeddingPhase SpaceX(t)Z(t)phase space setY(t)Attractors in Phase SpaceLogistic EquationX(n+1)X(n)X(n+1)=3.95 X(n)1-X(n)Attractors in Phase SpaceLorenz EquationsX(t)Z(t)Y(t)X(n+1)X(n)Logistic Equationphase spacetime seriesdthe fractal dimension of the
4、 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)nData 1time seriesphase spaced Since ,the time series was producedby a randommechanism.d Data 2time seriesphase spaced=
5、1 Since d=1,the time series was produced by a deterministicmechanism.Constructed by direct measurement:Phase SpaceEach point in the phase space set has coordinatesX(t),Y(t),Z(t)Measure X(t),Y(t),Z(t)Z(t)X(t)Y(t)Constructed from one variablePhase SpaceTakens TheoremTakens 1981 In Dynamical Systems an
6、d 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(t+t),X(t+2 t)velocity(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 1
7、0-5displacement(cm)stimulus=171 Hzvelocity(cm/sec)Position and Velocity of the Surface of a Hair Cell in the Inner EarTeich et al.1989 Acta Otolaryngol(Stockh),Suppl.467;265-2795 x 10-6displacement(cm)stimulus=610 Hz-3 x 10-23 x 10-2-2 x 10-5Data 1RANDOMx(n)=RNDfractal demension of the phase space s
8、etfractal dimension of phase space setembedding dimension=number of values of the data taken at a time to produce the phase space setData 2CHAOSdeterministicx(n+1)=3.95 x(n)1-x(n)fractal dimension of phase space setfractal demension of the phase space set=1embedding dimension=number of values of the
9、 data taken at a time to produce the phase space setmicroelectrodechick heart cellcurrent sourcevoltmeterChick Heart CellsvGlass,Guevara,Blair&Shrier.1984 Phys.Rev.A29:1348-1357Spontaneous Beating,No External StlimulationChick Heart CellsvoltagetimePeriodically Stimulated2 stimulations-1 beatChick H
10、eart Cells2:1Chick Heart Cells1:1Periodically Stimulated1 stimulation-1 beatChick Heart Cells2:3Periodically Stimulated2 stimulations-3 beatsperiodic stimulation-chaotic responseThe Pattern of Beatingof Chick Heart CellsGlass,Guevara,Blair&Shrier.1984 Phys.Rev.A29:1348-1357=phase of the beat with re
11、spect to the stimulusThe Pattern of Beating of Chick Heart Cells continuedphase vs.previous phase0.500.51.01.000.51.0i+1experimentitheory(circle map)The Pattern of Beatingof Chick Heart CellsGlass,Guevara,Belair&Shrier.1984 Phys.Rev.A29:1348-1357Since the phase space set is 1-dimensional,the timing
12、between the beats of thesecells can be described by a deterministic relationship.XtimedXtimedx(t)x(t+t)x(t+2 t)Chanced(phase space set)Determinismd(phase space set)=lowDatax(t)t?C O L DModelHOT(Rayleigh,Saltzman)EquationsEquationsEquationsPhase SpaceZXYX 0cylinder of air rotating counter-clockwisecy
13、linder of air rotating clockwiseIXtop(t)-Xbottom(t)I e t =Liapunov ExponentX(t)X=1.00001Initial Condition:differentsameX(t)X=1.00X(n+1)=f X(n)Accuracy of values computed for X(n):1.736 2.345 3.2545.455 4.876 4.2343.212X(n+1)=f X(n)Accuracy of values computed for X(n):3.455 3.45?3.4?3.?Clockwork Univ
14、ersedetermimistic non-chaoticCancomputeall futureX(t),Y(t),Z(t).EquationsChaotic Universedetermimistic chaoticsensitivityto initial conditionsCan notcomputeall futureX(t),Y(t),Z(t).EquationsTrajectories from outside:pulled TOWARDS itwhy its called an attractorstarting away:Trajectories on the attrac
15、tor:pushed APART from each othersensitivity to initial conditionsstarting on:phase space setnot strangestrangetime seriesnot chaoticchaoticX(t)tX(t)tIf 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 ca
16、lculatedThere is an INFINITE number of trajectories on the attractor.When we go off the attractor,we are sucked back down exponentially fast.Were on an exact trajectory,just not on the one we thought we were on.4.We are on a“real”trajectory.3.Pulled backtowards the attractor.2.Error pushesus offthe
17、attractor.1.We start here.Trajectorythat we actuallycompute.Trajectory that we are trying to compute.TUESDAY10 lArT10 lWEDNESDAYArTA=3.22X(n)nX(n+1)=A X(n)1-X(n)A=3.42X(n)nX(n+1)=A X(n)1-X(n)A=3.62X(n)nBifurcationl Start with one value of A.l Start with x(1)=0.5.l Use the equation to compute x(2)fro
18、m x(1).l 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)l Ignore x(1)to x(50),these are the transient values off of the attractor.l Plot x(51)to x(300)on the Y-axis over the value of A on the X-axis.l Change the value of A,and repeat the procedure again.x(n+1)=A x
19、(n)1-x(n)Sudden changes of the pattern indicate bifurcations()x(n)x(n)The energy in glucose is transfered to ATP.ATP is used as an energy source to drive biochemical reactions.Glycolysis+-periodicTheoryMarkus and Hess 1985 Arch.Biol.Med.Exp.18:261-271Glycolysistimesugar inputATP outputchaotictimetim
20、etimeExperimentsHess and Markus 1987 Trends.Biomed.Sci.12:45-48cell-free extracts from bakers yeastGlycolysisATP measured by fluorescence glucose inputtimeExperimentsHess and Markus 1987 Trends.Biomed.Sci.12:45-48PeriodicfluorescenceGlycolysisVinGlycolysisExperimentsHess and Markus 1987 Trends.Biome
21、d.Sci.12:45-48Chaotic20 minGlycolysisMarkus et al.1985.Biophys.Chem 22:95-105Bifurcation DiagramchaostheoryexperimentGlycolysisMarkus 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 cycle#=pe
22、riod of the ATP concentrationfrequency of the input sugar flow cyclePhase 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 index finger out-of-phase with the tic
23、k of the metronome.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 motion.Position of Right Index FingerPosition of Left Index FingerA.TIME S
24、ERIESPhase TransitionsHaken 1983 Synergetics:An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT PressADDABDPosition of Right Index Finger360o0oB.POINT ESTIMATE OF RELATIVE PHASE180oSelf-Organized Phase TransitionsHaken 1983 Synergetics:An Introduction Springer-Verlag Kelso 2019 Dynamic
25、Patterns MIT Press2 secThis bifurcation can be explained as a change in a potential energy function similar to the change which occurs in a physical phase transition.system potentialscaling parameterPhase TransitionHaken 1983 Synergetics:An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MI
26、T PressSmall changes in parameters can produce large changes in behavior.+10cc ArT9cc ArTBifurcations can be used to test if a system is deterministic.Deterministic Mathematical ModelExperimentobserved bifurcationspredicted bifurcationsMatch?The fractal dimension of the phase space set tells us if t
27、he data was generated by a random or deterministic mechanism.ExperimentalDatax(t)tX(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.Mechanism that generated the experimental data.DeterministicRandomd=lowd
28、The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.EpidemicsSchaffer and Kot 1986 Chaos ed.Holden,Princeton Univ.Press40001500000measlesNew Yorktime series:phase space:chickenpoxEpidemicsOlsen and Schaffer 1990 Science 249:499-504
29、dimension of attractor in phase spacemeasles chickenpoxKobenhavn 3.1 3.4 Milwaukee 2.6 3.2St.Louis 2.2 2.7New York 2.7 3.3EpidemicsOlsen and Schaffer 1990 Science 249:499-504SEIR models -4 independent variables S susceptible E exposed,but not yet infectious I infectious R recoveredEpidemicsOlsen and
30、 Schaffer 1990 Science 249:499-504Conclusion:measles:chaotic chickenpox:noisy yearly cycletime series:voltageKaplan and Cohen 1990 Circ.Res.67:886-892normalfibrillation deathD=1chaosD=randomPhase spaceV(t),V(t+t)ElectrocardiogramECG:Electrical recording of the muscle activity of the heart.8time seri
31、es:voltageBabloyantz and Destexhe 1988 Biol.Cybern.58:203-211normalD=6chaosElectrocardiogramECG:Electrical recording of the muscle activity of the heart.ElectrocardiogramECG:Electrical recording of the muscle activity of the heart.time series:time between heartbeatsBabloyantz and Destexhe 1988 Biol.
32、Cybern.58:203-211normalD=6chaosfibrillation deathD=4chaosinduced arrhythmiasD=3chaosEvans,Khan,Garfinkel,Kass,Albano,and Diamond 1989 Circ.Suppl.80:II-134Zbilut,Mayer-Kress,Sobotka,OToole and Thomas 1989 Biol.Cybern,61:371-381ElectroencephalogramEEG:Electrical recording of the nerve activity of the
33、brain.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)Rapp,Bashore,Martinerie,Albano,Zimmerman,and Mees 1989 Brain Topography 2:99-118Babloyantz and Destexhe 1988 In:From Chemical to Biological Organization ed.Markus,Muller,and Nicolis,Springer-Ver
34、lagXu and Xu 1988 Bull.Math.Biol.5:559-565ElectroencephalogramEEG:Electrical recording of the nerve activity of the brain.Different groups find different dimensions under the same experimental conditions.ElectroencephalogramEEG:Electrical recording of the nerve activity of the brain.mental taskquiet
35、 awake,eyes closedquiet sleepbrain virus:Creutzfeld-JakobEpilepsy:petit malmeditation:Qi-kongElectroencephalogramEEG:Electrical recording of the nerve activity of the brain.perhaps:High DimensionLow DimensionRandom MarkovHow to compute the next x(n):Each t pick a random number 0 R 1 If open,and R pc
36、,then close.If closed,and R B1Control of Chaosmotion of a magnetoelastic ribbonDitto,Rauseo,and Spano 1990 Phys.Rev.Lett.65:3211-3214Control 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 Chaosmotion of a magne
37、toelastic ribbonDitto,Rauseo,and Spano 1990 Phys.Rev.Lett.65:3211-3214iterationnumber0-23592360-47994800-70997100-10000noneperiod 1period 2period 1controlControl of Chaosmotion of a magnetoelastic ribbonDitto,Rauseo,and Spano 1990 Phys.Rev.Lett.65:3211-32144.54.03.53.02.502000400060008000 10000Itera
38、tion NumberXnControl of Biological SystemsThe Old WayBrute Force Control.BIG machineBIG powerHeartAmpsControl of Biological SystemsThe New WayCleverly timed,delicate pulses.little machinelittle powermAHeartThe Old WayForces drive the system between stable states.How do we think of biological systems
39、?How do we think of biological systems?Force DForce EStable State BStable State AStable State CHow do we think of biological systems?The New WayHanging around for a while in one condition forces the system into another condition.Dynamics of ADynamics of BHow do we think of biological systems?Unstabl
40、e State BUnstable State AUnstable State CSummary of ChaosFEW INDEPENDENT VARIABLESBehavior is so complex that it mimics random behavior.Summary of ChaosThe value of the variables at the next instant in time can be calculated from their values at the previous instant in time.xi(t+t)=f(xi(t)DYNAMICAL
41、SYSTEMDETERMINISTICSummary of Chaosx1(t+t)-x2(t+t)=Ae tSENSITIVITY TO INITIAL CONDITIONSNOT PREDICTABLE IN THE LONG RUNSummary of ChaosSTRANGE ATTRACTOR Phase space is low dimensional(often fractal).Books About ChaosJ.GleickChaos:Making a New Science 1987 VikingintroductoryBooks About ChaosF.C.MoonC
42、haotic and Fractal Dynamics 1992 John Wiley&Sonsintermediate mathematicsBooks About ChaosJ.Guckenheimer&P.Holmes Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vector Fields 1983 Springer-VerlagE.Ott Chaos in Dynamical Systems 1993 Cambridge Univ.Press advanced mathematicsA.V.Holden Chaos 1986 Princeton Univ.PressE.&L.Moskilde Complexity,Chaos and Biological Evolution 1991 Plenumreviews of chaos in biologyBooks About ChaosBooks About ChaosJ.Bassingthwaighte,L.Liebovitch,&B.West Fractal Physiology 1994 Oxford Univ.Pressreviews of chaos in biology
