lms durham symposium dynamical systems and statistical mechanics 3 – 13 july 2006 locating...

LMS Durham Symposium LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006• Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006

Locating periodic orbits in Locating periodic orbits in high-dimensional systems byhigh-dimensional systems by

stabilising transformationsstabilising transformations

Ruslan L DavidchackJonathan J Crofts

Department of MathematicsUniversity of Leicester


OutlineOutline

The problem of locating periodic orbits

Stabilising transformations approach

Application to low-dimensional maps

Extension to higher-dimensional system

Locating periodic orbits in high-dimensional flows

Kuramoto-Sivashinsky equation

Periodic and relative periodic orbits


The problem of locating periodic orbitsThe problem of locating periodic orbitsFinding periodic orbits of a map is equivalent to finding zeros of a

function

For discrete systems, the period p = 1, 2, 3, …

For flows: period is unknown.

Typically, the number of periodic orbits grows exponentially with p.

Systematic detection of POs of increasing period requires a reliable recipe for selecting useful seeds and an efficient iterative scheme.

When the fixed point is stable, simply iterating the map will find it.

For unstable orbits the general strategy is to construct an alternative map (e.g. Newton-Raphson method) where the POs become stable.

The seeds are obtained from close returns or the knowledge of the structure of the system.

( ) ( ) ,p ng x x x x R

/ ( ), ( ) ( ),tdx dt f x x t x t R

( )x


Stabilising transformationsStabilising transformations

To locate fixed points of , integrate the associated flow:

)(: xCgds

dx

where C is an constant orthogonal matrix.

The map and flow have identical sets of fixed points.

We can select such C that an unstable fixed point of becomes

a stable fixed point of .

The goal is to find a (small) set such that every

unstable fixed point of can be stabilised by at least one .

( )p x

( )p x

( )p x

( )p x

1 2{ , , ... , }MC C CCC C

n n


““Global” convergence of associated flowGlobal” convergence of associated flow

g(x)

1D case:

q-Newton

C = 1

C = −1

Associated flow with C = 1 ( −1) converges to roots with g´(x) < 0 ( g´(x) > 0 )


Stabilising transformationsStabilising transformations

Conjecture: (Schmelcher and Diakonos, 1997) Any unstable fixed point of can be stabilised by at least one C from the set of all orthogonal matrices with only non-zero entries.

C = SP, where and P is a permutation matrix.

Total number of such matrices: M = 2nn!

The conjecture is easily verified for and appears to be true for n > 2.

diag( 1, 1, ...)S

1( )p x

2n


Convergence basins of associated flows for Ikeda map: Convergence basins of associated flows for Ikeda map: pp = 5 = 5

1 00 1C

– sinks – saddles

0 11 0C 0 1

1 0C


Integrating the associated flowIntegrating the associated flow

Schmelcher and Diakonos use the Euler method

A modified semi-implicit Euler method (RLD and Y.-C. Lai, 1999)

where is a parameter, sn = |g(xn)|, and Gn = Dg(xn) is the Jacobian

Standard (variable order, variable step-size) methods for solving stiff ODEs, e. g. ode15s in Matlab lsoda from netlib.org (odepack)

)(1 nnn xhCgxx

)(][ 11 nnnnn xgGCsxx

T


Seeding with periodic orbitsSeeding with periodic orbits

When the goal is to locate complete sets of periodic orbits for each p, the traditional seeding schemes are not satisfactory

Use already detected periodic orbits as seeds to locate new ones:

For simple maps (Hénon, Ikeda) it is sufficient to use orbits of period p – 1 to locate orbits of period p

In more complex cases (e.g. n > 2) it is necessary to use orbits of period q > p to locate complete sets of orbits for each p

Scalability: The seeding scheme can be set up and tuned at small periods and then applied to larger periods


Periodic orbits in the Ikeda attractorPeriodic orbits in the Ikeda attractor

p

np Np p–1lnNp

10 46 483 0.6180

11 76 837 0.6118

12 110 1383 0.6027

13 194 2 523 0.6026

14 317 4 511 0.6010

15 566 8 517 0.6033

16 950 15 327 0.6023

17 1 646 27 983 0.6023

18 2 799 50 667 0.6018

19 4 884 92 797 0.6020

20 8 404 168 575 0.6018

21 14 700 308 777 0.60192

22 25 550 562 939 0.60186

23 44 656 1 027 089 0.60184

24 78 077 1 875 231 0.60184(RLD Lai Bollt Dhamala, 2000)


Extension to higher-dimensional systemsExtension to higher-dimensional systems

The use of SD stabilising transformations becomes inefficient for larger n due their rapidly growing number (M = 2nn!)

Even though in practice it appears that only a small subset of these transformations is necessary to stabilise all periodic orbits of a given system, it is not clear how to identify this subset a priori

To efficiently extend this approach to higher dimensions, we need to find a smaller set of stabilising transformations


Understanding stabilising transformationsUnderstanding stabilising transformations

For n = 2:

Eigenvalues of Cs,G ( G = Dg(x*) ):

x* is stabilised if s = sign det G and | − | < /2

, ,cos sin 1,sin coss

s ssC

Re – solidIm – dashed

s = −sign det G s = sign det G



Observations:

Properties of stabilising transformations depend essentially on the eigenvectors and signs (but not the magnitudes) of unstable eigenvalues of G

Typically, eigenvectors of different periodic orbits are locally aligned

A matrix C that stabilises a given fixed point of is likely to stabilise neighbouring fixed points (for all p)

Within the seeding with periodic orbits scheme it should be possible to construct stabilising transformations based on the knowledge of already detected orbits

( )p x



Note that Cs,== −QT, where Q is the orthogonal matrix in the polar decomposition of G:

G = QB, where B = (GTG)1/2

As in the 2D case, a fixed point x* with G = Dg(x*) is stabilised when the matrix C is “close” to −QT, where

Q = (GTG)1/2 G −1.

This is a corollary of Lyapunov’s stability theorem (Crofts and RLD, 2006)

Orthogonal matrices A and B are ”close” if all eigenvalues of the product ATB have positive real parts (i.e. all principal rotations are smaller than /2)


New stabilising transformationsNew stabilising transformations

Using a fixed point x* as seed, construct matrix C = −QT which will stabilise all the fixed points in the neighbourhood of x* with similar invariant directions and signs of unstable eigenvalues.

To stabilise fixed points with other signs, construct stabilising transformations as follows:

where

For each seed there are a total of 2k transformations, where k is the number of unstable eigenvalues

diag( 1, 1, ...,1,1)S

1 1 1/ 2 1( *) ( ) ( )pD x V V G V S I V G G G C T TQ Q


Periodic orbits of kicked double rotor mapPeriodic orbits of kicked double rotor map

p

np Np p–1lnNp

1 12 12 2.4849

2 45 102 2.3125

3 152 468 2.0495

4 522 2 190 1.9229

5 2 200 11 012 1.8613

6 9 824 59 502 1.8323

7 46 900 328 312 1.8145

8* 229 082 1 834 566 1.8028

Period p + 1 periodic orbit points were used as seeds to complete detection of period p

* − incomplete set of orbitsDimension: n = 4


Periodic orbits of three coupled HPeriodic orbits of three coupled Héénon mapsnon maps

3,2,1)~( 12

1 jbxxax jn

jn

jn

)()1(~ 1121 j

nj

nj

nj

n xxxx

15.0,3.0,4.1 ba

Dimension: n = 6

p

BW BW-r ST MAX

1 8 8 8 8

2 28 28 28 28

3 0 0 0 0

4 34 34 40 40

5 0 0 0 0

6 74 74 72 74

7 28 28 28 28

8 271 271 285 286

9 – 63 64 66

10 – 565 563 568

11 – 272 277 278

12 – 1972 1999 1999

13* – – 1079 –

14* – – 6599 –

15* – – 5899 –

BW – Biham-Wenzel (Politi Torcini, 1992)

BW-r – Biham-Wenzel (reduced)

ST – stabilising transformations

MAX – max. no. of detected orbits


Otherwise, we can work with the flow as well

Given , the associated flow is defined as

The dimension of the associated flow is n + 1

Locating periodic orbits in flowsLocating periodic orbits in flowsIf a “good” Poincaré surface of section can be found, the problem is

reduced to that for the maps

( ) ( ( ) )dx

Cg x C x xds

2| |( ( )) ( ) , 0

d gf x g x

ds

/ ( )dx dt f x


Stabilising within unstable subspaceStabilising within unstable subspace

If we don’t have any periodic orbits to use as seeds, is it possible to construct stabilising transformations?

C = I is always a good starting point, since it stabilises all periodic orbits with eigenvalues whose real parts are smaller than one. In chaotic flows with few unstable directions there is a fair fraction of such orbits.

It is also possible to construct transformations within the unstable subspace only. If we can find the set of d vectors spanning the unstable subspace at the seed x0, we can construct

n dU R

( )n d dC I U C I U T

where Cd is a matrix ( ) from the SD setd d d n


Stabilising within unstable subspaceStabilising within unstable subspace

Traditionally, the Schur decomposition of is used to find U (e.g. for continuation problems, Lust Roose Spence Champneys 1995).

It appears better to use SVD:

0( )D x

0( ( ))t tD x USV T


Locating periodic orbits of Kuramoto-Sivashinsky equationLocating periodic orbits of Kuramoto-Sivashinsky equation

2( ) , [ /2, /2]t x xx xxxxu u u u x L L

or in Fourier space: 2 4 1 2 2ˆ ˆ ˆ/ ( ) i (( ) ), k

k k Ldu dt q q u qF F u q

ˆ ( )u F u

Restricting to odd solutions: ˆ( , ) ( , ) ik ku x t u x t u a 2 4/ ( )k k m k m

m

da dt q q a q a a


Cd = I3 Cd = − I3

L = 51.3; n = 31; d = 3

Locating periodic orbits of Kuramoto-Sivashinsky equationLocating periodic orbits of Kuramoto-Sivashinsky equation


Relative periodic orbitsRelative periodic orbits

]2/,2/[,)0,(),( LLxuxu

)0(ˆe)(ˆ2

kki

k uu L

)0(ˆe)(ˆ),,(2

kki

k uuug L

2| |d g

ds

Associated flow is n + 2 dimensional


Locating relative periodic orbits of KS equationLocating relative periodic orbits of KS equation

L = 22.0; n = 30; C = I


ConclusionsConclusions Within the stabilising transformations approach it is possible to

construct a scheme to systematically locate complete sets of periodic orbits in maps

Extension to high-dimensional flows with small dimension of unstable manifold allowed us to find periodic and relative periodic orbits in the Kuramoto-Sivashinsky equation

Construction of the systematic detection scheme for such flows is possible

Thank you!

lms durham symposium dynamical systems and statistical mechanics 3 – 13 july 2006 locating...

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