Chaos in Space and Time •  Chris Mehrvarzi, Alireza Sedighi, and Mu Xu •  Faculty Sponsor: Prof. M. Paul •  Midterm Presentation •  ESM 6984 SS: Frontiers in Dynamical Systems

Why study chaos in space and time?

Many phenomena in nature are systems far-from equilibrium and exhibit chaotic dynamics in both space and time

   

NASA  images  

Falkowski  Nature  (2012)   NASA  image  

Presentation Outline

Lattice map with “diffusive” coupling – Difference equations

Calculating Lyapunov vectors – System of ODEs

Transport in complex flow – Governing PDEs

Future work

Coupled Map Lattice 1D

)]())()((21[)( 11

)1( ni

ni

ni

ni

ni xfxfxfDxfx −++= −++

nix

f (xin ) = axi

n (1− xin )

λ = limn→∞

1n

ln $f (xin )

i=0

n−1

∑

])())()((21[)( )()(

11)(11

)()1( ni

ni

ni

ni

ni

ni

ni

ni

ni xxfxxfxxfDxxfx δδδδδ ʹ′−ʹ′+ʹ′+ʹ′= −−+++

nix 1−

nix 1+

Coupled Map Lattice 1D

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.3512

0.3514

0.3516

0.3518

0.352

0.3522

0.3524

0.3526

0.3528

0.353

0.3532

D

λ

Chaos⇒> 0λ

Lyapunov Vectors Covariant Lyapunov  

Vectors Pros: •  True direction in phase

space. •  Reflect the direction of

perturbation •  Test hyperbolicity Cons: •  Difficult to calculate •  Algorithm only recently

available(Ginelli (2007) and Pazo (2007))

Orthogonal Lyapunov Vectors

Pros: •  Easy to calculate •  Leading order Lyapunov

vector is in correct direction

•  Can calculate fractal dimension

Cons •  Lose all direction except

leading order

Lorenz System

𝑑𝑥/𝑑𝑡 =σ(𝑥−𝑦) 𝑑𝑦/𝑑𝑡 =𝑥(𝜌−𝑧)−𝑦 𝑑𝑧/𝑑𝑡 =𝑥𝑦−𝛽𝑧

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

t

�

�1�2�3

𝜎=10 𝜌=28  𝛽= 8/3   

Result

The direction of the second covariant Lyapunov vector and the direction of the tangent vector should be same.

Transport in Complex Flows

∂c∂t+ (u ⋅ ∇

)c = L∇

2cσ −1(∂

u∂t+ (u ⋅∇

)u) = −∇

p+∇2 u + RTz

(∂T∂t

+ (u ⋅∇)T ) =∇

2T

∇⋅u = 0

Boussinesq Equations Advection-Diffusion Equation

Ning  et  al.  (2009)  (a) (b)

L = Dκ

R = αgd3

νκΔT

Direct Numerical Simulations

100 101 102

100

101

102

t

M2

Le = 10−1

Le = 10−2

Le = 10−3

Slope = 1

[L]2∝D[T ]Diffusive  Transport  

Pr=  1   Pr=  1  

Future Work

•  Explore diagnostic tools for higher dimensional map lattice –  Fractal dimension –  Lyapunov spectrum –  Covariant Lyapunov vectors

•  Transport enhancement due to complex flow •  Active transport