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.