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```					                                Locating periodic orbits in
high-dimensional systems by
stabilising transformations

Ruslan L Davidchack
Jonathan J Crofts
Department of Mathematics
University of Leicester

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

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
The problem of locating periodic orbits
Finding periodic orbits of a map  ( x) is equivalent to finding zeros of a
function g ( x)   p ( x)  x, x  R n
     For discrete systems, the period p = 1, 2, 3, …
     For flows: dx / dt  f ( x),                     x(t )   t ( x), period t   R 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.

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Stabilising transformations
To locate fixed points of  p ( x ), integrate the associated flow:
dx
:           Cg ( x)
ds
where C is an n  n constant orthogonal matrix.
     The map  p ( x ) and flow  have identical sets of fixed points.
     We can select such C that an unstable fixed point of  p ( x ) becomes
a stable fixed point of .
     The goal is to find a (small) set C  {C1 , C2 , ... , CM } such that every
unstable fixed point of  p ( x ) can be stabilised by at least one C C    .

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
“Global” convergence of associated flow
1D case:

g(x)

q-Newton
C=1
C = −1

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

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Stabilising transformations
Conjecture: (Schmelcher and Diakonos, 1997)
Any unstable fixed point of  ( x ) can be stabilised by at least one C
p

from the set of all orthogonal matrices with only 1 non-zero entries.

     C = SP, where S  diag(1,  1, ...) and P is a permutation matrix.
     Total number of such matrices: M = 2nn!
     The conjecture is easily verified for n  2 and appears to be true for
n > 2.

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Convergence basins of associated flows for Ikeda map: p = 5

C 11
0
0
                                    C 11
0
0                                
C  0 0
1
1

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Integrating the associated flow

     Schmelcher and Diakonos use the Euler method
xn1  xn  hCg ( xn )

     A modified semi-implicit Euler method (RLD and Y.-C. Lai, 1999)
xn1  xn  [  snC T  Gn ]1 g ( xn )

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)

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

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

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

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Understanding stabilising transformations
For n = 2:                              
Cs ,  s cos cos
s sin 
sin
                ,   s  1,      

Eigenvalues of Cs,G ( G = Dg(x*) ):

Re – solid
Im – dashed

s = −sign det G                                  s = sign det G

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

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Extension to higher-dimensional systems
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  ( x ) is likely to stabilise
p

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
LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Extension to higher-dimensional systems
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)

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

D p ( x*)  V V 1  G  V ( S   I )V 1  Q  (G TG )1/ 2 G 1  C  Q T

where S  diag(1,  1, ...,1,1)

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

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

Dimension: n = 4                                                                * − incomplete set of orbits

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

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Periodic orbits of three coupled Hénon maps
p    BW    BW-r   ST       MAX
x  j
n 1    a  ( ~nj ) 2  bxnj1
x                                j  1,2,3                       1      8      8        8      8
2     28     28     28       28
~ j  (1   ) x j  1  ( x j 1  x j 1 )
xn                                                                                 3      0      0        0      0
n    2      n        n
4     34     34     40       40

a  1.4, b  0.3,   0.15
5      0      0        0      0
6     74     74     72       74
7     28     28     28       28
Dimension: n = 6
8    271    271    285      286
9      –     63     64       66
10      –    565    563      568
BW – Biham-Wenzel (Politi Torcini, 1992)                                          11      –    272    277      278
BW-r – Biham-Wenzel (reduced)                                                     12      –   1972   1999     1999
13*     –      –   1079        –
ST – stabilising transformations
14*     –      –   6599        –
MAX – max. no. of detected orbits                                                 15*     –      –   5899        –

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Locating periodic orbits in flows
If a “good” Poincaré surface of section can be found, the problem is
reduced to that for the maps

Otherwise, we can work with the flow as well

Given dx/dt             f ( x), the associated flow is
defined as
dx
 Cg ( x)  C (  ( x)  x)
ds
d                                 | g |2
  f (  ( x))  g ( x)            ,  0
ds                                  

The dimension of the associated flow is n + 1

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Stabilising 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 U  R nd spanning
the unstable subspace at the seed x0, we can construct
C  I n  U (Cd  I d )U T
where Cd is a d  d matrix ( d                             n ) from the SD set

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Stabilising within unstable subspace

of D ( x0 ) is used to find U (e.g. for
continuation problems, Lust Roose
Spence Champneys 1995).

It appears better to use SVD:
D t ( t ( x0 ))  USV T

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Locating periodic orbits of Kuramoto-Sivashinsky equation
ut  (u 2 ) x  uxx  u xxxx , x  [ L/2, L/2]

or in Fourier space: u  F (u )
ˆ
duk /dt  (q 2  q 4 )uk  iqF (( F 1u ) 2 ), q 
ˆ                    ˆ               ˆ                              2 k
L

Restricting to odd solutions: u ( x, t )  u ( x, t )  uk  iak
ˆ
dak /dt  (q 2  q 4 )ak  q  am ak m
m

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Locating periodic orbits of Kuramoto-Sivashinsky equation

L = 51.3; n = 31; d = 3

Cd = I3                                                     Cd = − I3
LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Relative periodic orbits

u ( x  , )  u ( x,0),  [ L / 2, L / 2]

 i 2L k
uk ( ) e
ˆ                          uk (0)
ˆ

 i 2L k
g (u, , )  uk ( ) e
ˆ                                   uk (0)
ˆ

d     | g |2

ds      

Associated flow is n + 2 dimensional

LMS Durham Symposium • Dynamical Systems and Statistical Mechanics • 3 – 13 July 2006
Locating relative periodic orbits of KS equation

L = 22.0; n = 30; C = I

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

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