LDS

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NAME

lds - generate and display a lattice dynamical system

SYNOPSIS

lds [-2APcdmpq] [-W width] [-H height] [-w wide] [-h high] [-i init] [-n lambda ] [-b n ] [-F freq] [-B f | r | n ] [-C epsilon1 ] [-L epsilon2 ] [-R epsilon3 ] [-r delta ] [-E rho ] [-M omega ] [-T height] [-o fname]

DESCRIPTION

lds generates and displays a sequence of curves or cellular automata like figures which graphically represent the evolution of a Lattice Dynamical System (LDS). An LDS is an array of cells, each of which represents a dynamical system. Each cell is coupled to its nearest neighbors. The dynamical systems available in this software package are currently the logistic map, circle map and tent map. The state of each cell in the LDS is a real number in the unit interval. The evolution of the lattice is determined by an iteration thru the specified dynamic (e.g. the logistic equation) followed by a weighted averaging with its nearest neighbors.

By default, lds calculates and displays a 1-dimensional LDS. A 2-dimensional LDS can be specified on the command line (see the descriptions of the -h, -2 and -H options below). In the 2-D case, nearest neighbors are defined to be left, right, upper and lower (no diagonal neighbors). In the 1-D case, nearest neighbors are simply left and right.

Command line arguments and run-time keyboard input allow lds to simulate a wide variety of lattice dynamical systems. The user can specify the dynamic to be used, the non-linearity parameter, the strength of coupling, the initial conditions, the size of the array, the lenght of the run, whether and how to evolve connection strengths, and more. During the run, the display of an evolving 1-D LDS can either be CA-like with each generation being represented as a horizontal line evolving upward in the window or as points or a curve with X axis the lattice and Y-axis the cell states [0,1]. The display of an evolving 2-D LDS can either be CA-like with each generation being represented as a rectangle evolving upward in the window or as points or a curve with X axis the projected lattice and Y-axis the cell states [0,1]. In addition, graphical display of cell states can be toggled between actual state and phase difference with left neighbor.

Site and spatial histogram windows can also be displayed. The X-axis for the site histogram curve is the unit interval [0,1] while the Y-axis is the number of lattice sites which have taken on that value. The X-axis for the spatial histogram window is the projection of the lattice down onto its horizontal width while the Y-axis is the unit interval [0,1].

Lattice dynamical systems are also referred to as coupled map lattices.

OPTIONS

      -2
         Default to a two dimensional lattice. Defaults for width and height are now 256x256.
      -A
         Creates site histogram curve
      -p
         Indicates draw phase 1st differences
      -q
         Indicates not to draw "quilt" style. In quilt mode, the lattice is moved upward in the window each generation (if the window is larger than the lattice). If the "-q" option is specified, subseqeunt generations overwrite the previous one.
      -c
         Indicates draw curves
      -P
         Indicates draw points (when in curve drawing mode)
      -m
         Indicates monochrome plot
      -B [ f | r | b] Selects fixed zero, random, or specified fixed value of boundaries. Option -Bf sets boundaries to zero; -Br selects randomly fluctuating boundaries; and -Bb assigns value 0 < b < 1 to boundaries. The default boundary specification is periodic.
      -w n
         Indicates the lattice is n cells wide (default is full screen)
      -h n
         Indicates the lattice is n cells high (default is 1 in the absense of the -2 argument and 256 when -2 is specified)
      -W n
         Indicates the windows are n pixels wide (default is full screen)
      -H n
         Indicates the windows are n pixels high (default is full screen in the absense of the -2 argument and 256 when -2 is specified)
      -i [ p n | nnnn | r ] Selects initial condition. Option -ip n indicates periodic initial conditions with frequency n. Option -i nnnnn indicates center cells are given values 1/n. Option -ir selects random initial conditions (which is the default)
      -n [ r | l | p m | lambda ] Selects non-linearity parameter values (default is 3.7). Option -nr selects randomly assigned non-linearity parameters. Option -nl selects linearly assigned non-linearity parameters. Option -np m selects periodically assigned non-linearity parameters with frequency m. Option -n lambda selects non-linearity parameter lambda for all sites.
      -b n
         Begin graphing at generation n (default is 1)
      -F n
         Indicates display every n'th generation (1 is default)
      -C epsilon1
         Indicates weight of center cell (default 0.9) where 0 < epsilon1 < 1.
      -L epsilon2
         Indicates weight of left neighbor (default 0.05) where 0 < epsilon2 < 1.
      -R epsilon3
         Indicates weight of right neighbor (default 0.05) where 0 < epsilon3 < 1.
      -r delta Indicates range outside of which differences are graphed (0 < delta < 1) This value also serves as the determining distance over which connection strengths weaken rather than strengthen (when the -E flag is specified).
      -E rho Indicates the rate at which connection strengths evolve (0 < rho < 1). Connection strengths do not change if no -E flag is present.
      -M omega Selects circle map
      -T height Selects tent map
      -o fname
         Outputs graphed generations to file fname.

NOTES

In the absence of either the -M or -T arguments, the logistic map is used.

During display, use of the keys 123456789+-<>BDEFHILPQRSWXcdfhimpswxq? indicates:

           (1-9) Set the frequency to 1-9
           (+) Increment the frequency by 1
           (-) Decrement the frequency by 1
           (>) Double the frequency
           (<) Halve the frequency
           (B) Begin again
           (c) toggle curve/cell display
           (d) draw
           (D) Flush the drawing buffer
           (E) Erase each generation
           (f or F) Save the drawing window to a file
           (h) Display histogram curve
           (H) Histogram tracking toggled
           (I) Increment the stripe interval
           (i) Decrement the stripe interval
           (L) Lines drawn
           (m) multi-step
           (p) Toggle display of phase 1st differences
           (P) Points drawn
           (R) Spin the color wheel
           (s) single step
           (S) Spin the color wheel and increment the spin length
           (w) Decrement the color wheel index
           (W) Increment the color wheel index
           (x) Clear the window
           (X) Toggle complex dynamical systems mode
           (Q or q) quit

AUTHOR

lds was written by Ronald Record. Questions, suggestions, and comments may be directed via e-mail to rr@sco.com or ...uunet!sco!rr.

REFERENCES

The literature is sparse but includes the following excellent papers :

"Lyapunov Analysis and Information Flow in Coupled Map Lattices" by Kunihiko Kaneko, Physica 23D (1986) 436-447

"Spatiotemporal Chaos in One- and Two- Dimensional Coupled Map Lattices" by Kaneko

"Spatiotemporal Chaos and Noise" by Gottfried Mayer-Kress and Kaneko, J. Stat. Phys.(1988)

"Pattern Competition Intermittency and Selective Flicker Noise in Spatiotemporal Chaos" by Kaneko, Physics Letters A, V125, 1 (1987)

"Pattern Dynamics in Spatiotemporal Chaos" by Kaneko (1987)

"Phenomenology of Spatial-Temporal Chaos" by Jim Crutchfield and Kaneko, appearing as a chapter in "Directions in Chaos" edited by Hao Bai-lin, World Scientific Publishing (1987)

"Robust Space-Time Intermittency and 1/f Noise" by James Keeler and Doyne Farmer, Physica 23D (1986) 413-435

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Time: 04:06:23 GMT, December 02, 2024