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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Transient spatiotemporal chaos is extensive in three reaction-diffusion networks Dan Stahlke March 24, 2010 Dan Stahlke and Renate

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited

Transient spatiotemporal chaos is extensive in three reaction-diffusion networks

Dan Stahlke March 24, 2010

Dan Stahlke and Renate Wackerbauer, Transient spatiotemporal chaos is extensive in three reaction-diffusion networks, Physical Review E, 80 (2009), no. 5, 056211.

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Chaos

Chaotic systems are typified by:

◮ Sensitivity to initial conditions ◮ Attractor with fractional

dimension Example: Lorenz model

◮ dx/dt = σ(y − z) ◮ dy/dt = x(ρ − z) − y ◮ dz/dt = xy − βz ◮ σ = 10, β = 8/3, ρ = 28

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  • 15
  • 10
  • 5

5 10 15 20 25 2 4 6 8 10 12 14 16 18 20 y t Lorenz model: sensitivity to initial conditions 5 10 15 20 25 30 35 40 45 50

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  • 5

5 10 15 20 z x Lorenz model: chaotic attractor 2 / 43

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Spatiotemporal Chaos

Some systems show disorder in both time and space

◮ Sensitivity to initial conditions ◮ No long-range spatial correlations

Examples:

◮ Turbulence ◮ Some chemical reactions ◮ Fibrillation in heart

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Transient Chaos

◮ In some systems, chaos suddenly collapses after a lengthy

chaotic interval

◮ In this case there is a chaotic saddle instead of a chaotic

attractor

  • 0.05

0.05 0.1 0.15 0.2 0.25 0.3 0.35 100 200 300 400 500 600 700 800 900 b t

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Reaction-diffusion networks (RDN)

◮ RDN are systems having a local reaction term and a diffusion

term

◮ The domain can be continuous or a discrete network of nodes ◮ Example: chemical reactions ◮ Example: animal populations

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Reaction-diffusion networks (RDN)

The general form of RDN dynamics is d dt y(x) = F( y(x)) + D d2 dx2 H y(x). Or, in discrete form d dt yi = F( yi) + D

N

  • j=1

GijH yj where typically N

j=1 Gij is the discrete Laplacian

Gij = ∇2

ij = δi,j−1 − 2δij + δi,j+1.

Effective system size is determined by N/

√

D.

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Boundary conditions

Periodic No-flux Shortcut

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Gray-Scott model [GS84]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.2 0.4 0.6 0.8 1 b a Phase portrait

Fa = 1 − a − µab2 Fb = µab2 − φb H =

  • 1

1

  • µ = 33.7, φ = 2.8

◮ Represents an open autocatalytic

reaction A + 2B → 3B and B → C

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Gray-Scott model [GS84]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.2 0.4 0.6 0.8 1 b a Phase portrait

Space −→ ←− Time

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

B¨ ar-Eiswirth model [BE93]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 b a Phase portrait

Fa = a ǫ (1 − a)(a − b + β α ) Fb = f(a) − b f(a) =          if a < 1/3 1 − 6.75a(a − 1)2 if 1/3 ≤ a ≤ 1 1 if a > 1 H =

  • 1
  • α = 0.84, β = 0.07, ǫ = 0.12

◮ Describes a surface reaction model for

the oxidation of CO on Pt

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

B¨ ar-Eiswirth model [BE93]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 b a Phase portrait

Space −→ ←− Time

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Wacker-Sch¨

  • ll model [WBS95]

8 8.5 9 9.5 10 10.5 11 11.5 12 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 b a Phase portrait

Fa = b − a

(b − a)2 + 1 − τa

Fb = α(j0 − (b − a)) H =

  • 1

8

  • α = 0.02, τ = 0.05, j0 = 1.21

◮ Describes charge transport in a

simplified model of layered semiconductors

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Wacker-Sch¨

  • ll model [WBS95]

8 8.5 9 9.5 10 10.5 11 11.5 12 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 b a Phase portrait

Space −→ ←− Time

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Chaos Spatiotemporal Chaos Transient Chaos Reaction-diffusion networks Models Extensivity

Extensivity

Extended chaotic systems that have no long-range interactions are expected to be uncorrelated at large length scales and therefore should behave as a sum of their parts [Rue82]. Therefore, it can be expected that:

◮ DL ∝ N/

√

D

◮ ln T ∝ N/

√

D (these measures will be defined later on)

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Transient Chaos Average Lifetime

Transient Chaos

Space Time

(a) (b) (c) (d) (e)

◮ (a) Gray-Scott, N=210 ◮ (b) B¨

ar-Eiswirth, N=460

◮ (c)-(e) Wacker-Sch¨

  • ll, N=500,460,420

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Transient Chaos Average Lifetime

Average Lifetime: Gray-Scott model

102 103 104 105 106 107 100 140 180 220 260 <T> N

(+) no-flux () periodic with shortcut of length 50 () periodic (△) periodic with shortcut of length N/2

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Transient Chaos Average Lifetime

Average Lifetime: B¨ ar-Eiswirth model

102 103 104 105 106 180 220 260 300 340 380 420 460 <T> N

(+) no-flux

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Transient Chaos Average Lifetime

Average Lifetime: Wacker-Sch¨

  • ll model

103 104 105 106 100 200 300 400 500 600 700 800 <T> N

(+) no-flux () periodic

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Lyapunov Exponents

◮ Lyapunov exponents describe the rate at which small

perturbations expand or contract

◮ ǫ

v(t) = y′(t) − y(t) where ǫ is infinitesimal

◮ The largest Lyapunov exponent is positive in chaotic systems

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5 10 15 20 25 2 4 6 8 10 12 14 16 18 20 y t Lorenz model: sensitivity to initial conditions

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Lyapunov Spectrum

◮ The number of Lyapunov exponents is equal to the number of

degrees of freedom.

◮ They describe rates of expansion of infinitesimal perturbation

vectors belonging to a sequence of nested linear subspaces

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

First Lyapunov Exponent

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Second Lyapunov Exponent

1 2 1 2 1 2 1 2 2 1 2 1 2 2 2 2 1 1

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Error Estimation

Convergence of Lyapunov exponent calculation is slow. Error is estimated to be the difference between the final value and the maximum deviation from this value during the last half of the simulation.

0.0914 0.0916 0.0918 0.0920 0.0922 100000 150000 200000 250000 300000 350000 400000 λ1 t

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Lyapunov Spectrum

  • 0.1

0.0 0.1 5 10 15 20 25 30 35 λi i Lyapunov spectrum, Gray-Scott, N=500

  • 0.1

0.0 0.1 10 20 30 40 50 60 70 λi i Lyapunov spectrum, Gray-Scott, N=1000

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Extensivity of Lyapunov Spectrum

0.028 0.030 0.032 0.034 0.036 0.038 0.040 1000 2000 3000 4000 5000 λ(N / 20 √ D) + 1 N

  • 0.1

0.0 0.1 5 10 15 20 25 30 35 λi i

  • 0.1

0.0 0.1 10 20 30 40 50 60 70 λi i

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Lyapunov Dimension

The Lyapunov dimension, also called the Kaplan-Yorke dimension, DL = j + λ1 + . . . + λj

|λj+1| ,

is conjectured to be equal to the information dimension for typical attractors [Ott02].

  • 0.1

0.0 0.1 5 10 15 20 25 30 35 λi i

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Sum of Positive Exponents

The sum of positive Lyapunov exponents, +, represents an upper bound for the Kolmogorov-Sinai entropy [Ott02].

  • 0.1

0.0 0.1 5 10 15 20 25 30 35 λi i

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Extensivity of Lyapunov Dimension DL

10 20 30 40 50 60 70 200 400 600 800 1000 1200 1400 DL N

() Gray-Scott, µ = 33.5 (×) B¨ ar-Eiswirth (+) Gray-Scott, µ = 33.7 (△) Wacker-Sch¨

  • ll

() Gray-Scott, µ = 33.9

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Extensivity of Sum of Positive Lyap. Exponents +

0.5 1 1.5 2 200 400 600 800 1000 1200 1400 Σ+ N

() Gray-Scott, µ = 33.5 (×) B¨ ar-Eiswirth (+) Gray-Scott, µ = 33.7 (△) Wacker-Sch¨

  • ll

() Gray-Scott, µ = 33.9

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Y-Intercept of DL vs. N

◮ The y-intercept of DL vs. N should be zero for systems with

periodic boundary conditions

◮ Why? 10 20 30 40 50 60 70 200 400 600 800 1000 1200 1400 DL N

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Lyapunov Exponents Lyapunov Exponent Computation Lyapunov Spectrum and Related Quantities Extensivity Y-Intercept

Y-Intercept of DL vs. N

◮ Take the linear ansatz DL(N) → aN + b as N → ∞ ◮ For large N, 2DL(N) = DL(2N) ◮ Therefore b = 0 ◮ The results mostly verify this hypothesis

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Intensive Quantities

An extensive quantity divided by size gives an intensive quantity.

◮ Lyapunov dimension density: [Gre99]

δD ≡ limN→∞ N−1DL

◮ Log-lifetime density:

δT ≡ limN→∞ N−1 ln T

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Intensive Quantities

So, what do these quantities mean? Consider transient chaos. Space −→ ←− Time Probability of collapse is PN/ξ, so lifetime takes the form [TL08] T ∼ P−N/ξ = e−(ln P) N

ξ ,

and log-lifetime density takes the form δT = − ln P ξ .

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Intensive Quantities

δT = − ln P ξ

◮ The quantity δT apparently has units of number of coins

tossed per unit length

◮ δT is computable whereas P and ξ are only defined intuitively

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

A New Quantity

δT has dimensions of coins tossed per unit length and δD has units

  • f active degrees of freedom (i.e. attractor dimension) per unit
  • length. Taking their ratio eliminates the length units:

σ ≡ δT/δD.

This quantity has units of coins tossed per active degree of freedom.

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

A New Quantity

And what does σ mean? For large N,

δT = N−1 ln T T−1 = e−NδT = e−NδDσ = e−DLσ = (e−σ)DL.

This leads to an intuitive argument for understanding the escape rate from the chaotic saddle.

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Escape Route

Each time the chaotic trajectory ”orbits” around the chaotic saddle, it has an opportunity of escaping into a non-chaotic state. Think of a ”hole” in the chaotic saddle.

  • 0.05

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 b1 a1

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Escape Route

T−1 = (e−σ)DL

◮ Ignoring the fact that the chaotic saddle has fractal dimension; ◮ Ignoring the fact that DL is only approximately equal to the

saddle dimension;

◮ Considering the saddle as being approximately a set product

  • f smaller saddles;

◮ Then (e−σ)DL is the volume of a hypercube of width e−σ and

dimension DL.

◮ So, can we find a feature in the chaotic saddle that is size

e−σ?

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Escape Route

Well, it’s not quite that easy.

T−1 = (e−σ)DL

◮ e−σ is actually the geometric mean of the hole’s widths along

each dimension

◮ The trajectory passes by certain areas more often than

  • thers, and this needs to be taken into account

◮ So, the interpretation is not so clear cut

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Intensive Quantities A New Quantity Escape Route

Escape Route

Gray-Scott B¨ ar-Eiswirth Wacker-Sch¨

  • ll

e−σ ≈ 0.28 e−σ ≈ 0.62 e−σ ≈ 0.85

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.2 0.4 0.6 0.8 1 b a Phase portrait 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 b a Phase portrait 8 8.5 9 9.5 10 10.5 11 11.5 12 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 b a Phase portrait

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Auxiliary Slides Conclusions

Discretization Error

◮ Effective system size is determined by N/

√

D

◮ Small N =⇒ more efficient computation ◮ Small D =⇒ more discretization error ◮ What is the limit? 20 30 40 50 60 70 80 90 100 2 4 6 8 10 12 14 16 DL D

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited Auxiliary Slides Conclusions

Conclusions

◮ ln T, DL, and + grow linearly with size ◮ DL and + are constant for N/

√

D fixed

◮ Boundary conditions affect x-intercept (but not slope) of ln T

and DL vs. N

◮ Y-intercept for DL vs. N should be zero for periodic boundary

conditions

◮ The quantity e−σ may relate to escape routes from the chaotic

saddle

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Introduction Lifetime of Transient Chaos Lyapunov Exponents Intensive Quantities Conclusions Works Cited

  • M. B¨

ar and M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable medium, Phys. Rev. E 48 (1993), no. 3, R1635–R1637. Henry S. Greenside, Spatiotemporal chaos in large systems: the scaling of complexity with size, Semi-analytic methods for the Navier-Stokes equations (Montreal, QC, 1995), CRM Proc. Lecture Notes, vol. 20, Amer. Math. Soc., Providence, RI, 1999, pp. 9–40. MR MR1686874 (2000b:37026) P . Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor : Oscillations and instabilities in the system a + 2b → 3b; b → c, Chem. Engin. Sci. 39 (1984), no. 6, 1087 – 1097. Edward Ott, Chaos in dynamical systems, 2 ed., Cambridge University Press, 9 2002. David Ruelle, Large volume limit of the distribution of characteristic exponents in turbulence, Comm. Math. Phys. 87 (1982), no. 2, 287–302. MR MR684105 (85c:76046)

  • T. T´

el and Y. C. Lai, Chaotic transients in spatially extended systems, Physics Reports 460 (2008), no. 6, 245 – 275.

  • A. Wacker, S. Bose, and E. Sch¨
  • ll, Transient spatio-temporal chaos in a reaction-diffusion model, Europhys. Lett.

31 (1995), no. 5-6, 257. 43 / 43