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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




                                              – sinks                                   – saddles



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
       Traditionally, the Schur decomposition
                
          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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