Introduction Examples Hydrodynamics Computation Current and future work Unstable Periodic Orbits as a Unifying Principle in the Presentation of Dynamical Systems in the Undergraduate Physics Curriculum Bruce M. Boghosian 1 Hui Tang 1 Aaron Brown 1 Spencer Smith 2 Luis Fazendeiro 3 Peter Coveney 3 1 Department of Mathematics, Tufts University 2 Department of Physics, Tufts University 3 Centre for Computational Science, University College London APS March Meeting, 16 March 2009
Transcript
Introduction Examples Hydrodynamics Computation Current and future work
Unstable Periodic Orbits as a Unifying
Principle in the Presentation of DynamicalSystems in the Undergraduate Physics
Curriculum
Bruce M. Boghosian1 Hui Tang1 Aaron Brown1
Spencer Smith2 Luis Fazendeiro3 Peter Coveney3
1Department of Mathematics, Tufts University2Department of Physics, Tufts University
3Centre for Computational Science, University College London
APS March Meeting, 16 March 2009
Introduction Examples Hydrodynamics Computation Current and future work
Introduction Examples Hydrodynamics Computation Current and future work
More lessons
Lessons learned
Very simple dynamical systems can exhibit both periodicityand chaos
State space is generally replete with UPOs
Introduction Examples Hydrodynamics Computation Current and future work
Logistic map
Logistic map
State space is again [0, 1]
One-parameter family of maps:
xn+1 = fλ(xn) := 4λxn (1 − xn) .
Pitchfork bifurcation, stable and unstable periodic orbits
Easy programming exercise for students
Introduction Examples Hydrodynamics Computation Current and future work
Logistic map
Periodic Orbits of Logistic Map
Finding period-two orbits:
Solve: x = fλ(fλ(x))
Minimize: F (x , y) = [x − fλ(y)]2+ [y − fλ(x)]
2
Finding period-three orbits, etc:
Solve: x = fλ(fλ(fλ(x)))Minimize:F (x , y , z) = [x − fλ(y)]
2+ [y − fλ(z)]
2+ [z − fλ(x)]
2
Introduction Examples Hydrodynamics Computation Current and future work
Logistic map
Cantor Set
Logistic map with λ > 1
Interval leaves state space in one iteration
Preimages of interval leave in two iterations, etc.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
xn
x n+
1
Introduction Examples Hydrodynamics Computation Current and future work
Still more lessons
Lessons Learned
The notions of transient behavior and attracting set
Period doubling route to chaos
Pitchfork bifurcation
UPOs discovered by either root finding or minimization
Cantor-set nature of state space when λ > 1
Introduction Examples Hydrodynamics Computation Current and future work
Lorenz attractor
Lorenz attractor
Continuous time dynamical system on the state space R3
Dynamical equations:
x = σ(y − x)y = −xz + Rx − y
z = xy − bz
0 5 10 15 20 25
-20
-10
0
10
20
30
40
t
x, y, z
Attracting set has periodic orbits
Introduction Examples Hydrodynamics Computation Current and future work
Lorenz attractor
Unstable Periodic Orbits of Lorenz Attractor
Correspond to any binary sequence (e.g., 11110)Dense in the attractorSystem is hyperbolic
S = 50. eT = 4.41957 eF = 0.155848 e
-20
0
20x
-20
0
20y
0
20
40
60
z
-20
0
20x
-20
0
20y
If you know all UPOs with period < T , you can makestatistical predictions of any observable (DZF formalism).UPOs and their properties can be tabulated, stored, and madeavailable in a curated database.
Introduction Examples Hydrodynamics Computation Current and future work
Lorenz attractor
These can be tabulated. . .
Viswanath, Nonlinearity 16 (2003) 1035-1056
Out[36]=
T = 1.55865
-100
10
-20
-10
010
20
20
30
40
T = 2.30591
-100
10
-20
-100
10
10
20
30
40
T = 3.02358
-100
10
-100
1020
10
20
30
40
T = 3.72564
-100
10
-20-10
010
10
20
30
40
T = 3.82026
-100
10
-20
-100
1020
10
20
30
40
T = 3.86954
-100
10
-20
-100
1020
10
20
30
40
T = 4.53411
-100
10
-20
-100
1020
10
20
30
40
T = 4.59381
-100
10
-20
-100
1020
10
20
30
40
T = 4.61181
-100
10
-20
-100
10
10
20
30
40
Tabulated up to ∼ 20 symbols
Introduction Examples Hydrodynamics Computation Current and future work
More lessons
Lessons Learned
These observations work for continuous-time dynamicalsystems
The same labeling of orbits used in the Bernoulli map worksfor the Lorenz attractor
Symbolic dynamics
Introduction Examples Hydrodynamics Computation Current and future work
Laminar and periodic flow
Stable periodic orbits
Laminar (stationary) flow is a fixed point in function space
von Karman vortex street is a closed periodic orbit in functionspace
Introduction Examples Hydrodynamics Computation Current and future work
Turbulent flow
Turbulent flow in two dimensions
Reference: N.T. Ouellette, J.P. Gollub, “Curvature Fields, Topology, and the
Dynamics of Spatiotemporal Chaos,” Phys. Rev. Lett. 99 (2007) 194502.
Periodic flow in square domain
Periodic forceF = A sin (2πmx) sin (2πny) ex + A cos (2πmx) cos (2πny) ey
Flow closely follows F for low Re
Turbulent for high Re
Introduction Examples Hydrodynamics Computation Current and future work
Yet more lessons
Lessons Learned
All this can be made to work for dynamical systems oninfinite-dimensional state-spaces
Introduction Examples Hydrodynamics Computation Current and future work
Chaos & turbulence
Unstable Periodic Orbits (UPOs)
Attracting sets in a wide variety of dynamical systems arereplete with periodic orbits
If the dynamics are hyperbolic, the UPOs are unstable
The UPOs are dense in the attracting set
The UPOs are countable and have measure zero in theattracting set
In spite of zero measure, UPOs are exceedingly important, asaverages over the natural measure can be derived from them
“The skeleton of chaos” (Cvitanovic)
Introduction Examples Hydrodynamics Computation Current and future work
Chaos & turbulence
Attracting sets and turbulent averages
The driven Navier-Stokes equations in the turbulent regimedescribe nonlinear dynamics in an infinite-dimensional(function) space
These dynamics possess an attracting set
The attracting set is finite-dimensional, and its dimensiongrows as a power law in Reynolds number (Constantin, Foias,Manley, Temam, 1985)
Long-time averages over this attracting set impart a “naturalmeasure” to it
The problem of turbulence is that of extracting averages ofobservables over this natural measure
Introduction Examples Hydrodynamics Computation Current and future work
Shooting method
Computing UPOs I: Shooting Method
Begin on surface of codimension one in function space
Evolve NS equations until return to that surface
Use Newton-Raphson to close the gap
Serial in time
Constructed to obey equations of motion, but not periodic
Introduction Examples Hydrodynamics Computation Current and future work
Relaxation method
Computing UPOs II: Relaxation Method
Begin with periodic orbit that is smooth
Relax to solution of NS equations using variational principle
∆([f ],T ) =1
2
T−1X
t=0
X
r
X
j
˛
˛
˛
˛
fj (r, t + 1) − fj(r, t) −1
τ
h
feqj
(r, t) − fj (r, t)i
˛
˛
˛
˛
2
Constructed to be periodic, but not obey equations of motion
Conjugate-gradient algorithm
Higher-order differencing needed
Local spline fitting to orbit needed
Enormous amounts of memory are needed
Introduction Examples Hydrodynamics Computation Current and future work
General HPC tools
Work in Progress I
Plot of DHt,TL
Buhl-KennelHMM
Clustering algs.
Statistical error ~ N-1�2
Conjugate gradient
Kennel-Mees
Improved accuracy
Time sequence
Initial guess for UPOs
UPOs
Ζ function
Monodromy matrix
Symbolic dynamics
Orr-Somerfeld equation
Pseudospectra
Dimension of attractor
Turbulent averages
Introduction Examples Hydrodynamics Computation Current and future work
Work in progress
Work in Progress II
LUPO: Laboratory for unstable periodic orbits
Shell models of turbulence (Tang, Boghosian)
2D Navier-Stokes (Latt, Smith, Boghosian)
3D Navier-Stokes (Faizendeiro, Coveney, Boghosian)
Introduction Examples Hydrodynamics Computation Current and future work
Conclusions
Conclusions
UPOs are a unifying concept in dynamical systems
Connects mathematical and physical way of understandingdynamical systems
Improved understanding of fluid UPOs may lead to newstatistical descriptions of turbulence
HPC is just at the point where this can be done forinfinite-dimensional systems
LUPO will give students a way to experiment with suchsystems
Creation of a UPO database will help spread and shareinformation about UPOs
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