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. 1 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Chaos Lorenz model: sensitivity to initial conditions Chaotic systems are typified by: 25 20 15 ◮ Sensitivity to initial conditions 10 5 0 y ◮ Attractor with fractional -5 -10 -15 dimension -20 -25 0 2 4 6 8 10 12 14 16 18 20 Example: Lorenz model t ◮ dx / dt = σ ( y − z ) Lorenz model: chaotic attractor 50 ◮ dy / dt = x ( ρ − z ) − y 45 40 35 30 ◮ dz / dt = xy − β z 25 z 20 15 ◮ σ = 10 , β = 8 / 3 , ρ = 28 10 5 0 -20 -15 -10 -5 0 5 10 15 20 x 2 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited 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 3 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited 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 0.35 0.3 0.25 0.2 0.15 b 0.1 0.05 0 -0.05 0 100 200 300 400 500 600 700 800 900 t 4 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited 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 5 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Reaction-diffusion networks (RDN) The general form of RDN dynamics is y ( x )) + D d 2 d y ( x ) = � dt � F ( � dx 2 H � y ( x ) . Or, in discrete form N d � y i = � dt � F ( � y i ) + D G ij H � y j j = 1 where typically � N j = 1 G ij is the discrete Laplacian G ij = ∇ 2 ij = δ i , j − 1 − 2 δ ij + δ i , j + 1 . √ Effective system size is determined by N / D . 6 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Boundary conditions Periodic No-flux Shortcut 7 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Gray-Scott model [GS84] Phase portrait F a = 1 − a − µ ab 2 0.4 F b = µ ab 2 − φ b 0.35 0.3 � � 1 0 0.25 H = 0 1 0.2 b µ = 33 . 7 , φ = 2 . 8 0.15 0.1 ◮ Represents an open autocatalytic 0.05 0 reaction A + 2 B → 3 B and B → C 0 0.2 0.4 0.6 0.8 1 a 8 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Gray-Scott model [GS84] Space −→ Phase portrait 0.4 0.35 ←− Time 0.3 0.25 0.2 b 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 a 9 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity B¨ ar-Eiswirth model [BE93] ǫ ( 1 − a )( a − b + β F a = a ) Phase portrait α F b = f ( a ) − b 1 0 if a < 1 / 3 0.8 1 − 6 . 75 a ( a − 1 ) 2 f ( a ) = 1 / 3 ≤ a ≤ 1 if 1 if a > 1 0.6 � � 1 0 b H = 0 0 0.4 α = 0 . 84 , β = 0 . 07 , ǫ = 0 . 12 0.2 0 ◮ Describes a surface reaction model for 0 0.2 0.4 0.6 0.8 1 the oxidation of CO on Pt a 10 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity B¨ ar-Eiswirth model [BE93] Space −→ Phase portrait 1 0.8 ←− Time 0.6 b 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 a 11 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Wacker-Sch¨ oll model [WBS95] b − a Phase portrait F a = ( b − a ) 2 + 1 − τ a 12 11.5 F b = α ( j 0 − ( b − a )) 11 � � 1 0 H = 10.5 0 8 10 b α = 0 . 02 , τ = 0 . 05 , j 0 = 1 . 21 9.5 9 ◮ Describes charge transport in a 8.5 simplified model of layered 8 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 semiconductors a 12 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited Extensivity Wacker-Sch¨ oll model [WBS95] Space −→ Phase portrait 12 11.5 ←− Time 11 10.5 10 b 9.5 9 8.5 8 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 a 13 / 43
Introduction Chaos Lifetime of Transient Chaos Spatiotemporal Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Reaction-diffusion networks Conclusions Models Works Cited 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: √ ◮ D L ∝ N / D √ ◮ ln � T � ∝ N / D (these measures will be defined later on) 14 / 43
Introduction Lifetime of Transient Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Average Lifetime Conclusions Works Cited Transient Chaos Space Time (a) (b) (c) (d) (e) ◮ (a) Gray-Scott, N=210 ◮ (b) B¨ ar-Eiswirth, N=460 ◮ (c)-(e) Wacker-Sch¨ oll, N=500,460,420 15 / 43
Introduction Lifetime of Transient Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Average Lifetime Conclusions Works Cited Average Lifetime: Gray-Scott model 10 7 10 6 10 5 <T> 10 4 10 3 10 2 100 140 180 220 260 N ( + ) no-flux ( � ) periodic with shortcut of length 50 ( △ ) periodic with shortcut of length N / 2 ( � ) periodic 16 / 43
Introduction Lifetime of Transient Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Average Lifetime Conclusions Works Cited Average Lifetime: B¨ ar-Eiswirth model 10 6 10 5 <T> 10 4 10 3 10 2 180 220 260 300 340 380 420 460 N ( + ) no-flux 17 / 43
Introduction Lifetime of Transient Chaos Lyapunov Exponents Transient Chaos Intensive Quantities Average Lifetime Conclusions Works Cited Average Lifetime: Wacker-Sch¨ oll model 10 6 10 5 <T> 10 4 10 3 100 200 300 400 500 600 700 800 N ( + ) no-flux ( � ) periodic 18 / 43
Introduction Lyapunov Exponents Lifetime of Transient Chaos Lyapunov Exponent Computation Lyapunov Exponents Lyapunov Spectrum and Related Quantities Intensive Quantities Extensivity Conclusions Y-Intercept Works Cited Lyapunov Exponents ◮ Lyapunov exponents describe the rate at which small perturbations expand or contract y ′ ( t ) − � ◮ ǫ� v ( t ) = � y ( t ) where ǫ is infinitesimal ◮ The largest Lyapunov exponent is positive in chaotic systems Lorenz model: sensitivity to initial conditions 25 20 15 10 5 0 y -5 -10 -15 -20 -25 0 2 4 6 8 10 12 14 16 18 20 t 19 / 43
Introduction Lyapunov Exponents Lifetime of Transient Chaos Lyapunov Exponent Computation Lyapunov Exponents Lyapunov Spectrum and Related Quantities Intensive Quantities Extensivity Conclusions Y-Intercept Works Cited 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 20 / 43
Introduction Lyapunov Exponents Lifetime of Transient Chaos Lyapunov Exponent Computation Lyapunov Exponents Lyapunov Spectrum and Related Quantities Intensive Quantities Extensivity Conclusions Y-Intercept Works Cited First Lyapunov Exponent 21 / 43
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