Fractals-and-Chaos-Simplified-for-the-Life-S分形与混沌简化生活的课件.ppt
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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
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