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Some figures adapted from a 2004 Lecture by Larry Liebovitch, Ph.D.ChaosBIOL/CMSC 361: Emergence1/29/08
EmergenceNon-linearCoherenceDynamicSelf-OrganizationComplexityMacro-level Property (Structured)UnexpectedUnpredictable
Properties of ChaosDeterministicSmall Number of Variables Complex OutputBifurcationsAttractorsSensitive to Initial ConditionsForecasting Uncertainty grows ExponentiallyPhase Space is Low Dimensional
Deterministicpredict that valuethese valuesA future state fully determined by previous states Chaos: future states fully determined by initial state
RandomA process or system whose behavior is Stochastic;without definite aim, reason or pattern;Whose outcome is described by a probability distribution.
Probability: relative possibility that an event will occur
Stochastic: a non-deterministic process
Deterministic: a prior state fully determines the future state of the process or system
MeanVariancePower SpectrumRandomChaotic
RandomData 1x(n) = rand()
ChaosData 2
Properties of ChaosDeterministicSmall Number of Variables Complex OutputBifurcationsAttractorsSensitive to Initial ConditionsForecasting Uncertainty grows ExponentiallyPhase Space is Low Dimensional
Population Growth
Bifurcation DiagramChoose a constant starting value for x (x0=0.1)
Choose a starting value for the rate (r = 0)
Use the equation to compute successive values of x(n) from prior values up to x(1000)
Ignore the first 900 values of x; these are transient values (before system stablizes)
Plot x(901) to x(1000) on the Y-axis versus the current value for r
Change the value of r and repeat
ZeroSteadyChaos
Bifurcations
Attractors
Another Example
Lorenz: Convection1963. J. Atmos. Sci. 20:13-141COLDModelHOT
Lorenz EquationsX(t)Z(t)Y(t)
Lorenz EquationsX = speed, direction of the convection circulationX > 0 clockwiseX < 0 counterclockwiseY = temperature difference between rising and falling fluidZ = rate of temperature change through fluid column
Lorenz EquationsPhase SpaceZXY
Sensitivity to Initial ConditionsX(t)X= 1.00001Initial Condition:differentsameX(t)X= 1.00
Lorenz AttractorsX < 0X > 0cylinder of air rotating counter-clockwisecylinder of air rotating clockwise
Why an Attractor?Trajectories from outside:pulled TOWARDS itwhy its called an attractorstarting away:
Why Strange?strangenot strange
Lorenz Strange AttractorTrajectories on the attractor:pushed APART from each othersensitivity to initial conditionsstarting on:
Properties of ChaosDeterministicSmall Number of Variables Complex OutputBifurcationsAttractorsSensitive to Initial ConditionsForecasting Uncertainty grows ExponentiallyPhase Space is Low Dimensional
Sensitivitynearly identicalinitial valuesvery differentfinal valuesor...very different behaviors...
Sensitivitysmall change in a parameterone patternanother pattern
Non-Chaotic Systemsystem outputcontrol parameter
Chaotic Systemsystem outputcontrol parameter
Clockwork UniverseInitial Conditions X(t0), Y(t0), Z(t0)...Cancomputeall futureX(t), Y(t), Z(t)...Equations
Chaotic UniverseInitial Conditions X(t0), Y(t0), Z(t0)...sensitivityto initial conditionsCan notcomputeall futureX(t), Y(t), Z(t)...Equations
Shadowing Theorem: Non-ChaoticIf 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.CalculatedTrue
Shadowing Theorem: ChaoticThere is an INFINITE number of trajectories. Were on an exact trajectory, just not on the one we thought we were on.CalculatedExpected
Deterministic: ChaoticX(n+1) = f {X(n)}Accuracy of values computed for X(n):3.455 3.45? 3.4?? 3.??? ? ? ?
Why?Sensitivity to initial conditions means that the conditions of an experiment can be quite similar, but that the results can be quite different a little initial error has a large impact!
4. We are on a real trajectory.3. Pulled backtowards the attractor.2. Error pushesus offthe attractor.1. We start here.Trajectorythat we actuallycompute.Trajectory that we are trying to compute.
Properties of ChaosDeterministicSmall Number of Variables Complex OutputBifurcationsAttractorsSensitive to Initial ConditionsForecasting Uncertainty grows ExponentiallyPhase Space is Low Dimensional
Mechanism?ChanceDeterminism
Datax(t)t?
Mechanism?Chanced(phase space set)Determinismd(phase space set) = lowDatax(t)t?
ProcedureDetermine topological properties of the objectFractal Dimension
High Fractal Dimension RandomLow Fractal Dimension ChaoticFractal Dimension does not equalFractal Dimension
ProcedureDetermine the Topological Properties of this ObjectEspecially, the fractal dimension.
High Fractal Dimension = Random = chance Low Fractal Dimension = Chaos = deterministic
Fractal DimensionA measure of self similarityXtimed
Fractal 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)
Mechanism that generated the experimental data.DeterministicRandomd = lowd The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.