CHAOSCHAOSStrange Attractors and Lorenz Strange Attractors and Lorenz

EquationsEquations

DefinitionsDefinitions

• ChaosChaos – study of dynamical systems (non-periodic – study of dynamical systems (non-periodic systems in motion) usually over time systems in motion) usually over time

• AttractorAttractor – a set of points in phase space toward – a set of points in phase space toward which neighboring points which neighboring points

asymptotically asymptotically approach within a basin of approach within a basin of attractionattraction

- an attractor can be a point, curve, - an attractor can be a point, curve, manifoldmanifold

or a complicated set of fractals known or a complicated set of fractals known as a as a strange attractor strange attractor

• Strange AttractorStrange Attractor – an attractor that exhibits – an attractor that exhibits sensitive dependence on sensitive dependence on

initial conditions; initial conditions; they usually have fractal they usually have fractal structure (infinite structure (infinite detail). detail).

– There are several types of strange attractors including:There are several types of strange attractors including:• Chua – used in electronic circuitry.Chua – used in electronic circuitry.

• Duffing – used in nonlinear oscillators.Duffing – used in nonlinear oscillators.

• RRössler – used in chemical kinetics.össler – used in chemical kinetics.

• Ikeda – involved in the turbulence of trails of smoke.Ikeda – involved in the turbulence of trails of smoke.

• Lorenz – used in atmospheric convection.Lorenz – used in atmospheric convection.

Definitions & Applications of Definitions & Applications of AttractorsAttractors

Applications of AttractorsApplications of Attractors

Chua Duffing Lorenz

Rössler Ikeda

Edward LorenzEdward Lorenz• an American mathematician an American mathematician

and meteorologist, and is the and meteorologist, and is the first contributor to the chaos first contributor to the chaos theory and inventor of the theory and inventor of the strange attractor notion in strange attractor notion in 1963.1963.

• Discovered that minute Discovered that minute variations in initial weather variations in initial weather parameters led to grossly parameters led to grossly divergent weather patternsdivergent weather patterns

• Coined the term “butterfly Coined the term “butterfly effect” effect”

• The Lorenz Attractor is based on The Lorenz Attractor is based on three differential equations, three three differential equations, three constants, and three initial constants, and three initial conditions. The attractor represents conditions. The attractor represents the behavior of gas at any given the behavior of gas at any given time, and its condition at any given time, and its condition at any given time depends upon its condition at time depends upon its condition at a previous time. a previous time.

Lorenz AttractorLorenz Attractor

• Although originally Although originally developed to study the developed to study the upper atmosphere, the upper atmosphere, the Lorenz equations have Lorenz equations have since been used in the since been used in the study of batteries, study of batteries, lasers, and even the lasers, and even the simple chaotic simple chaotic waterwheel.waterwheel.

The Lorenzian WaterwheelThe Lorenzian Waterwheel

Lorenz EquationsLorenz Equations

The VariablesThe Variables

• x – refers to the convective flow.x – refers to the convective flow.

• y – refers to the horizontal y – refers to the horizontal temperature temperature distribrution. distribrution.

• z – refers to the vertical temperature z – refers to the vertical temperature distribution. distribution.

The ConstantsThe Constants

• σσ – sigma refers to the ratio of – sigma refers to the ratio of viscoscity viscoscity to thermal to thermal conductivity.conductivity.

• ρρ – rho refers to the temperature – rho refers to the temperature difference between the top and difference between the top and

bottom of a given slice. bottom of a given slice.

• ββ – beta refers to the ratio of the – beta refers to the ratio of the width to width to the height. the height.

BehaviorBehavior

• Chaotic behavior can only be found in Chaotic behavior can only be found in systems of equations with three or more systems of equations with three or more variables.variables.

• Within the Lorenz system, there are three Within the Lorenz system, there are three things that make the system chaotic.things that make the system chaotic.– EquationsEquations– Initial ValuesInitial Values– ConstantsConstants

• If the If the ρρ constant is below a certain value, then there will constant is below a certain value, then there will be no chaotic behavior, and the graph will converge.be no chaotic behavior, and the graph will converge.

The Butterfly EffectThe Butterfly Effect

^ x solution with respect to time.

^ y solution with respect to time.

« z solution with respect to time.

Proof for Bounded SystemProof for Bounded System

• (x) dx/dt = (-(x) dx/dt = (-σσx+x+σσy)(x)y)(x)

• (y) dy/dt = ((y) dy/dt = (ρρx-y-xz) (y)x-y-xz) (y)

• (z(znn) dz) dznn/dt = (-/dt = (-ββ(z(znn++ρρ++σσ)+xy) (z)+xy) (znn))

• ^{z^{znn=z-=z-ρρ--σσ}}

• {z=z{z=znn++ρρ++σσ}}

• ½dx½dx22/dt = /dt = --σσxx22++σσxyxy

• ½dy½dy22/dt = /dt = ρρxy-y2-xy(zn+xy-y2-xy(zn+ρρ++σσ))

• ½dz½dznn22/dt = -/dt = -ββzznn

22--βρβρzznn--βσβσzznn+xyz+xyznn

Proof Cont.Proof Cont.

• ½d(x½d(x22+y+y22+z+znn22)/dt = -)/dt = -σσxx22-y-y22--ββzznn

22--ββzznn((ρρ++σσ))

• aa22+b+b22 ≥ 2ab ≥ 2ab

• 2ab » -z2ab » -znn(√(√ββ-1)(√1/-1)(√1/ββ-1)-1)ββ((ρρ++σσ))

• aa22+b+b2 2 » » zznn22((ββ-1)+(-1)+(ββ22((ρρ++σσ))22/4(/4(ββ-1))-1))

• ½d(x½d(x22+y+y22+z+znn22)/dt = -)/dt = -σσxx22-y-y22-z-znn

22++

• ((ββ22((ρρ++σσ))22/4(/4(ββ--1))1))

Proof Cont.Proof Cont.

• --σσxx22-y-y22-z-znn22 ≤ -(x ≤ -(x22+y+y22+z+znn

22)) | |σσ|>1|>1

• --σσxx22-y-y22-z-znn22 ≤ - ≤ -σσ(x(x22+y+y22+z+znn

22)) | |σσ|<1|<1

• R(t) = xR(t) = x22+y+y22+z+znn22

• d(R(t))/dt = -2R(t)+2(d(R(t))/dt = -2R(t)+2(((ββ22((ρρ++σσ))22/4(/4(ββ--1))1))

• R’+2R=R’+2R=εε

Proof FinaleProof Finale

• R = R = εεtete-2t-2t+ce+ce-2t-2t

• R » c R » c t = 0t = 0

• R » 0R » 0 t » t » infinityinfinity

The EndThe End