lyapunov               package:fractal               R Documentation

_L_o_c_a_l-_G_l_o_b_a_l _L_y_a_p_u_n_o_v _S_p_e_c_t_r_u_m _E_s_t_i_m_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Estimates the local Lyapunov exponents over a range of user
     supplied scales and dimensions. The local Lyapunov spectrum is
     calculated as follows:

     _1 A delayed embedding of the input time series is formed.

     _2 For each global reference point (specified by an intger index in
          the reference matrix) a local Lyapunov spectrum is
          calculated, one exponent for each dimension from 1 to
          'local.dimension' and for each (integer) scale specified by
          the 'scale' vector. As the scales grow larger, the Lyapunov
          exponent estimates tend toward asymptotic values
          corresponding to the global Lyapunov exponents. The details
          of how each local spectrum is estimated is given below.

     _3 The local spectra are then averaged over each global reference
          point to stabilize the results. .in -5

          Each local spectrum is obtained by estimating the eigenvalues
          of the so-called Oseledec matrix, which is formed through a
          matrix product of successive local Jacobians with the
          transpose of the Jacobians. The number of Jacobians in the
          product is equivalent to the scale. Each Jacobian is formed
          by fitting a local neighborhood of points (relative to a some
          reference point) with a multidimensional polynomial of order
          'polynomial.order'. The number of neighbors found for each
          reference point in the embedding is chosen to be twice the
          polynomial order for numerical stability. To further
          stabilize the results, a local Lyapunov spectrum is formed
          for each local 'reference' point.

_U_s_a_g_e:

     lyapunov(x, tlag=NULL, dimension=5, local.dimension=3,
         reference=NULL, n.reference=NULL, olag=2,
         sampling.interval=NULL, polynomial.order=3, metric=Inf, scale=NULL)

_A_r_g_u_m_e_n_t_s:

       x: a vector containing a uniformly-sampled real-valued time
          series.

dimension: an integer representing the embedding dimension. Default:
          '5'.

local.dimension: an integer representing the dimension (number of)
          local Lyapunov exponents to estimate. This value must be less
          than or equal to the embedding dimension. Default: '3'.

  metric: the metric used to define the distance between points in the
          embedding. Choices are limited to '1', '2', or 'Inf' which
          represent an L1, L2, and L-inf norm, respectively. Default:
          'Inf'.

n.reference: the number of neighbors to use in in developing the
          kd-tree (used as a quick means of finding nearest neighbors
          in the phase space). These neighbors are collected relative
          to the reference points. This value must be greater than 10.
          Default: 'min(as.integer(round(length(x)/20)), 100)'.

    olag: the number of points along the trajectory of the current
          point that must be exceeded in order for another point in the
          phase space to be considered a neighbor candidate. This
          argument is used to help attenuate temporal correlation in
          the the embedding which can lead to spuriously low
          correlation dimension estimates. The orbital lag must be
          positive or zero. Default: 'length(x)/10' or '500', whichever
          is smaller.

polynomial.order: the order of the polynomial to use in fitting data
          around reference points in the phase space. This poloynomial
          fit will be used to form the Jacobians which are in  turn
          used to calcualte the Lypaunov exponents. Default: '3'.

reference: a vector of integers representing the indices of global
          reference points to use in estimating the local Lyapunov
          spectrum. A local spectrum is estimated around each global
          reference point, and all the local spectra are then averaged
          to stabilize the results. These global reference points
          should be chosen such that they are far apart in time.
          Default: Five indices uniformly distributed on the interval
          [1,M], where M = Ne - max(scale) - n.reference - 2 and Ne is
          the number of embedding points.

sampling.interval: a numeric value representing the interval between
          samples in the input time series. Default: 'deltat(x)'.

   scale: a vector of integers defining the scales over which the local
          Lyapunov exponents are to be estimated. As this scale
          increases, one expects the local Lyapunov exponent estimates
          to converge towards the global estimates. All scales must be
          greater than one. Default:
          'as.integer(2\eqn{\mbox{\textasciicircum}}{^}(seq(min(floor(logb(scale.max,2))
          - 2 , 10)) - 1))' where scale.max = Ne - 2 - 'n.reference'.

    tlag: the time delay between coordinates. Default: the
          decorrelation time of the autocorrelation function.

_V_a_l_u_e:

     an object of class 'FNN'.

_S_3 _M_E_T_H_O_D_S:


     _p_l_o_t plots a summary of the results. Available options are:

          ... Additional plot arguments (set internally by the 'par'
               function). .in -5


          _p_r_i_n_t prints a summary of the results. Available options are:

               ... Additional print arguments used by the standard
                    'print' function. .in -5


               _s_u_m_m_a_r_y summarizes the results.

_R_e_f_e_r_e_n_c_e_s:

     P. Bryant, R. Brown, and H.D.I. Abarbanel (1990), Lyapunov
     exponents from observed time series, _Physical Review Letters_,
     *65*(13), 1523-1526.

     H.D.I. Abarbanel, R. Brown, J.J. Sidorowich, and L. Tsimring
     (1993), The analysis of observed chaotic data in physical systems,
     _Reviews of Modern Physics_, *65*(4), 1331-1392.

_S_e_e _A_l_s_o:

     'embedSeries', 'infoDim', 'corrDim', 'timeLag', 'FNN'.

_E_x_a_m_p_l_e_s:

     ## Calculate the local Lyapunov spectrum for the 
     ## beamchaos series 
     z <- lyapunov(beamchaos)

     ## print the results 
     print(z)

     ## summarize the results 
     summary(z)

     ## plot the results 
     plot(z)

