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1、Introduction to CHAOS,Larry Liebovitch, Ph.D.Florida Atlantic University2019,Introduction to CHAOSLarry Lie,These two sets of data have the same,meanvariancepower spectrum,These two sets of data have th,Fractals-and-Chaos-Simplified-for-the-Life-S分形与混沌简化生活的课件,Data 1,RANDOMrandomx(n) = RND,Data 1RAND
2、OM,CHAOSDeterministicx(n+1) = 3.95 x(n) 1-x(n),Data 2,CHAOSData 2,etc.,etc.,Fractals-and-Chaos-Simplified-for-the-Life-S分形与混沌简化生活的课件,Data 1,RANDOMrandomx(n) = RND,Data 1RANDOM,Data 2,CHAOSdeterministicx(n+1) = 3.95 x(n) 1-x(n),x(n+1),x(n),Data 2CHAOSx(n+1)x(n),Definition,CHAOS,Deterministic,predict
3、that value,these values,DefinitionCHAOSDeterministicpr,CHAOS,Small Number of Variables,x(n+1) = f(x(n), x(n-1), x(n-2),Definition,CHAOSSmall Number of Variables,Definition,CHAOS,Complex Output,DefinitionCHAOSComplex Output,Properties,CHAOS,Phase Space is Low Dimensional,phase space,d , random,d = 1,
4、 chaos,PropertiesCHAOSPhase Space is,Properties,CHAOS,Sensitivity to Initial Conditions,nearly identicalinitial values,very differentfinal values,PropertiesCHAOSSensitivity to,Properties,CHAOS,Bifurcations,small change in a parameter,one pattern,another pattern,PropertiesCHAOSBifurcationssma,Time Se
5、ries,X(t),Y(t),Z(t),embedding,Time SeriesX(t)Y(t)Z(t)embeddi,Phase Space,X(t),Z(t),phase space set,Y(t),Phase SpaceX(t)Z(t)phase Y(t),Attractors in Phase Space,Logistic Equation,X(n+1),X(n),X(n+1) = 3.95 X(n) 1-X(n),Attractors in Phase SpaceLogis,Attractors in Phase Space,Lorenz Equations,X(t),Z(t),
6、Y(t),Attractors in Phase SpaceLoren,X(n+1),X(n),Logistic Equation,phase space,time series,d1,The number of independent variables = smallest integer the fractal dimension of the attractor,d 1, therefore, the equation of the time series that produced this attractor depends on 1 independent variable.,X
7、(n+1)X(n)Logistic Equationpha,Lorenz Equations,phase space,time series,d =2.03,The number of independent variables = smallest integer the fractal dimension of the attractor,d = 2.03, therefore, the equation of the time series that produced this attractor depends on 3 independent variables.,X(t),Z(t)
8、,Y(t),X(n+1),n,Lorenz Equationsphase spacetim,Data 1,time series,phase space,d,Since ,the time series was producedby a randommechanism.,d,Data 1time seriesphase spaced,Data 2,time series,phase space,d = 1,Since d = 1,the time series was produced by a deterministicmechanism.,Data 2time seriesphase sp
9、aced,Constructed by direct measurement:,Phase Space,Each 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 by direct measurem,Constructed from one variable,Phase Space,Takens TheoremTakens 1981 In Dynamical Systems and Turbulence Ed. Ran
10、d & Young, Springer-Verlag, pp. 366 - 381,X(t+ t),X(t+2 t),X(t),Each point in thephase space sethas coordinatesX(t), X(t + t), X(t+2 t),Constructed from one variableP,velocity (cm/sec),Position and Velocity of the Surface of a Hair Cell in the Inner Ear,Teich et al. 1989 Acta Otolaryngol (Stockh), S
11、uppl. 467 ;265 - 279,10-1,-10-1,-10-4,3 x 10-5,displacement (cm),stimulus = 171 Hz,velocity (cm/sec)Position and,velocity (cm/sec),Position and Velocity of the Surface of a Hair Cell in the Inner Ear,Teich et al. 1989 Acta Otolaryngol (Stockh), Suppl. 467 ;265 - 279,5 x 10-6,displacement (cm),stimul
12、us = 610 Hz,-3 x 10-2,3 x 10-2,-2 x 10-5,velocity (cm/sec)Position and,Data 1,RANDOMx(n) = RND,fractal demension of the phase space set,fractal dimension of phase space set,embedding dimension = number of values of the data taken at a time to produce the phase space set,Data 1RANDOMfractal demension
13、,Data 2,CHAOSdeterministicx(n+1) = 3.95 x(n) 1 - x(n),fractal dimension of phase space set,fractal demension of the phase space set = 1,embedding dimension = number of values of the data taken at a time to produce the phase space set,Data 2CHAOSfractal dimension f,microelectrode,chick heart cell,cur
14、rent source,voltmeter,Chick Heart Cells,v,Glass, Guevara, Blair & Shrier.1984 Phys. Rev. A29:1348 - 1357,microelectrodechick heart cell,Spontaneous Beating, No External Stlimulation,Chick Heart Cells,voltage,time,Spontaneous Beating, Chick Hea,Periodically Stimulated2 stimulations - 1 beat,Chick Hea
15、rt Cells,2:1,Periodically StimulatedChick H,Chick Heart Cells,1:1,Periodically Stimulated1 stimulation - 1 beat,Chick Heart Cells1:1Periodical,Chick Heart Cells,2:3,Periodically Stimulated2 stimulations - 3 beats,Chick Heart Cells2:3Periodical,periodic stimulation - chaotic response,The Pattern of B
16、eatingof Chick Heart Cells,Glass, Guevara, Blair & Shrier.1984 Phys. Rev. A29:1348 - 1357,periodic stimulation - chaotic,= phase of the beat with respect to the stimulus,The Pattern of Beating of Chick Heart Cells continued,phase vs. previous phase,0.5,0,0.5,1.0,1.0,0,0.5,1.0,i + 1,experiment,i,theo
17、ry (circle map),= phase of the beat with respe,The Pattern of Beatingof Chick Heart Cells,Glass, Guevara, Belair & Shrier.1984 Phys. Rev. A29:1348 - 1357,Since the phase space set is 1-dimensional, the timing between the beats of thesecells can be described by a deterministic relationship.,The Patte
18、rn of BeatingGlass, G,Procedure,Time seriese.g. voltage as a function of timeTurn the Time Series into a Geometric ObjectThis is called embedding.,ProcedureTime series,Procedure,Determine the Topological Properties of this ObjectEspecially, the fractal dimension. High Fractal Dimension = Random = ch
19、ance Low Fractal Dimension = Chaos = deterministic,ProcedureDetermine the Topolog,The Fractal Dimension is NOT equal to The Fractal Dimension,The Fractal Dimension is NOT,Fractal Dimension:How many new pieces of the Time Series are found when viewed at finer time resolution.,X,time,d,Fractal Dimensi
20、on:How many ne,Fractal Dimension:The Dimension of the Attractor in Phase Space is related to theNumber of Independent Variables.,X,time,d,x(t),x(t+ t),x(t+2 t),Fractal Dimension:The Dimensi,Mechanism that Generated the Data,Chanced(phase space set),Determinismd(phase space set) = low,Data,x(t),t,?,M
21、echanism that Generated the D,C O L D,Lorenz1963 J. Atmos. Sci. 20:13-141,Model,HOT,(Rayleigh, Saltzman),C O L DLorenz1963 J. Atmos,Lorenz1963 J. Atmos. Sci. 20:13-141,Equations,Lorenz1963 J. Atmos. Sci. 20:,X = speed of the convective circulation X 0 clockwise, X 0 counterclockwiseY = temperature d
22、ifference between rising and falling fluid,Equations,Lorenz1963 J. Atmos. Sci. 20:13-141,X = speed of the convective ci,Z = bottom to top temperature minus the linear gradient,Equations,Lorenz1963 J. Atmos. Sci. 20:13-141,Z = bottom to top temperature,Phase Space,Lorenz1963 J. Atmos. Sci. 20:13-141,
23、Z,X,Y,Phase SpaceLorenz1963 J. Atmo,Lorenz Attractor,X 0,X 0,cylinder of air rotating counter-clockwise,cylinder of air rotating clockwise,Lorenz AttractorX 0cyli,IXtop(t) - Xbottom(t)I e t = Liapunov Exponent,Sensitivity to Initial ConditionsLorenz Equations,X(t),X= 1.00001,Initial Condition:,diffe
24、rent,same,X(t),X= 1.,0,0,IXtop(t) - Xbottom(t)I e,Deterministic, Non-Chaotic,X(n+1) = f X(n),Accuracy of values computed for X(n):,1.736 2.345 3.2545.455 4.876 4.2343.212,Deterministic, Non-ChaoticX(n+,Deterministic, Chaotic,X(n+1) = f X(n),Accuracy of values computed for X(n):,3.455 3.45? 3.4? 3.?
25、? ? ?,Deterministic, ChaoticX(n+1) =,Initial Conditions X(t0), Y(t0), Z(t0).,Clockwork Universedetermimistic non-chaotic,Cancomputeall futureX(t), Y(t), Z(t).,Equations,Initial Conditions X(t0), Y(t,Initial Conditions X(t0), Y(t0), Z(t0).,Chaotic Universedetermimistic chaotic,sensitivityto initial c
26、onditions,Can notcomputeall futureX(t), Y(t), Z(t).,Equations,Initial Conditions X(t0), Y(t,Lorenz Strange Attractor,Trajectories from outside:pulled TOWARDS itwhy its called an attractor,starting away:,Lorenz Strange AttractorTrajec,Lorenz Strange Attractor,Trajectories on the attractor:pushed APAR
27、T from each othersensitivity to initial conditions,starting on:,Lorenz Strange AttractorTrajec,“Strange”attractor is fractal,phase space set,not strange,strange,“Strange”attractor is fractal,“Chaotic”sensitivity to initial conditions,time series,not chaotic,chaotic,X(t),t,X(t),t,“Chaotic”sensitivity
28、 to initi,Shadowing Theorem,If 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 calculated,Shadowing TheoremIf the errors,There is an INFINITE number of trajectories on the attractor. When we go off the
29、 attractor, we are sucked back down exponentially fast. Were on an exact trajectory, just not on the one we thought we were on.,Shadowing Theorem,There is an INFINITE number of,4. We are on a “real” trajectory.,3. Pulled backtowards the attractor.,2. Error pushesus offthe attractor.,1. We start here
30、.,Trajectorythat we actuallycompute.,Trajectory that we are trying to compute.,4. We are on a “real” trajecto,Sensitivity 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 conditi,TUESDAY,+,10 l,
31、ArT,TUESDAY+10 lArT,10 l,WEDNESDAY,ArT,+,10 lWEDNESDAYArT+,A = 3.22,X(n),n,X(n + 1) = A X(n) 1 -X (n),A = 3.22X(n)nX(n + 1) = A X(n),A = 3.42,X(n),n,X(n + 1) = A X(n) 1 -X (n),A = 3.42X(n)nX(n + 1) = A X(n),A = 3.62,X(n),n,Bifurcation,A = 3.62X(n)nBifurcation,Start with one value of A. Start with x(
32、1) = 0.5. Use the equation to compute x(2) from 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 val
33、ue of A on the X-axis. Change the value of A, and 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 s
34、ource to drive biochemical reactions.,Glycolysis,+,-,-,The energy in glucose is trans,periodic,TheoryMarkus and Hess 1985 Arch. Biol. Med. Exp. 18:261-271,Glycolysis,time,sugar input,ATP output,chaotic,time,time,time,periodicTheoryGlycolysistimesu,ExperimentsHess and Markus 1987 Trends. Biomed. Sci.
35、 12:45-48,cell-free extracts from bakers yeast,Glycolysis,ATP measured by fluorescence glucose input,time,Experimentscell-free extracts,ExperimentsHess and Markus 1987 Trends. Biomed. Sci. 12:45-48,Periodic,fluorescence,Glycolysis,Vin,ExperimentsPeriodicfluorescenc,Glycolysis,ExperimentsHess and Mar
36、kus 1987 Trends. Biomed. Sci. 12:45-48,Chaotic,20 min,GlycolysisExperimentsChaotic20,GlycolysisMarkus et al. 1985. Biophys. Chem 22:95-105,Bifurcation Diagram,chaos,theory,experiment,GlycolysisBifurcation Diagramc,GlycolysisMarkus et al. 1985. Biophys. Chem 22:95-105,ADP measured at the same phase e
37、ach time of the input sugar flow cycle(ATP is related to ADP),period of the input sugar flow cycle,# =,period of the ATP concentration,frequency of the input sugar flow cycle,GlycolysisADP measured at peri,Phase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Pa
38、tterns MIT Press,Tap the left index fingerin-phase with the tickof the metronome.,Try to tap the right 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 Pres
39、s,As the frequency of the metronome increases, the right finger shifts from out-of-phase to in-phase motion.,Phase TransitionsAs the freque,Position of Right Index FingerPosition of Left Index Finger,A. TIME SERIES,Phase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 D
40、ynamic Patterns MIT Press,ADD,ABD,Position of Right Index Finger,Position of Right Index Finger,360o,0o,B. POINT ESTIMATE OF RELATIVE PHASE,180o,Self-Organized Phase TransitionsHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT Press,2 sec,Position of Right Index
41、 Finger,This bifurcation can be explained as a change in a potential energy function similar to the change which occurs in a physical phase transition.,system potential,scaling parameter,Phase TransitionHaken 1983 Synergetics: An Introduction Springer-Verlag Kelso 2019 Dynamic Patterns MIT Press,Thi
42、s bifurcation can be explai,Small changes in parameters can produce large changes in behavior.,+,10cc ArT,+,9cc ArT,Small changes in parameters ca,Bifurcations can be used to test if a system is deterministic.,Deterministic Mathematical Model,Experiment,observed bifurcations,predicted bifurcations,M
43、atch ?,Bifurcations can be used to te,The fractal dimension of the phase space set tells us if the data was generated by a random or deterministic mechanism.,ExperimentalData,x(t),t,The fractal dimension of the p,X(t+ t),Phase SpaceSet,X(t),The fractal dimension of the phase space set tells us if th
44、e data was generated by a random or a deterministic mechanism.,X(t+ t)Phase SpaceX(t)The fr,Mechanism that generated the experimental data.,Deterministic,Random,d = low,d,The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.,Mechani
45、sm that generated the e,EpidemicsSchaffer and Kot 1986 Chaos ed. Holden, Princeton Univ. Press,4000,15000,0,0,measles,New York,time series:,phase space:,chickenpox,Epidemics40001500000measlesNew,EpidemicsOlsen and Schaffer 1990 Science 249:499-504dimension of attractor in phase space,measles,chicken
46、pox,Kobenhavn 3.1 3.4 Milwaukee 2.6 3.2St. Louis 2.2 2.7New York 2.7 3.3,EpidemicsmeasleschickenpoxKobe,EpidemicsOlsen and Schaffer 1990 Science 249:499-504,SEIR models - 4 independent variables S susceptible E exposed, but not yet infectious I infectious R recovered,EpidemicsSEIR models - 4 ind,Epi
47、demicsOlsen and Schaffer 1990 Science 249:499-504,Conclusion: measles: chaotic chickenpox: noisy yearly cycle,EpidemicsConclusion:,time series: voltageKaplan and Cohen 1990 Circ. Res. 67:886-892,normal,fibrillation death,D = 1chaos,D = random,Phase spaceV(t), V(t+ t),ElectrocardiogramECG: Electrical
48、 recording of the muscle activity of the heart.,8,time series: voltagenormalfibr,time series: voltageBabloyantz and Destexhe 1988 Biol. Cybern. 58:203-211,normal,D = 6chaos,ElectrocardiogramECG: Electrical recording of the muscle activity of the heart.,time series: voltagenormalD =,Electrocardiogram
49、ECG: Electrical recording of the muscle activity of the heart.,time series: time between heartbeatsBabloyantz and Destexhe 1988 Biol. Cybern. 58:203-211,normal,D = 6chaos,fibrillation death,D = 4chaos,induced arrhythmias,D = 3chaos,Evans, Khan, Garfinkel, Kass, Albano, and Diamond 1989 Circ. Suppl.
50、80:II-134,Zbilut, Mayer-Kress, Sobotka, OToole and Thomas 1989 Biol. Cybern, 61:371-381,Electrocardiogramtime series:,ElectroencephalogramEEG: Electrical recording of the nerve activity of the brain.Mayer-Kress and Layne 1987 Ann. N.Y. Acad. Sci. 504:62-78,time series: V(t),phase space:,D=8 chaos,V(
