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Fractals[EscapeTime]

  

Lyapunov

  

Lyapunov fractal generator

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

Lyapunov( n, ab, xrng, yrng )

Lyapunov( n, ab, xrng, yrng, opts )

Parameters

n

-

: positive integer specifying the dimensions of the square Array output

ab

-

: {list,Vector}(identical(0,1)) data specifying the alternating conditions

xrng

-

: realcons..realcons the range for the horizontal domain

yrng

-

: realcons..realcons the range for the vertical domain

opts

-

: (optional) keyword options of the form opt=value where opt and value are described below

Options

• 

output : keyword option of the form output=value where value is a name or list of names denoting the returned Array image(s). The accepted names are color, or raw. The default value is color.

• 

numterms : keyword option of the form numterms=value where value is a positive integer specifying the number of terms used in the summation that approximates the Lyapunov exponent at each input point The default value is 1000.

• 

container : An n-by-n-by-2 Array with datatype=float[8] and order=Fortran_order used in-place to store the raw data.

Description

• 

The Lyapunov command generates Array images which provide a visualization of a bifurcational fractal that maps regions of chaos and stability given a binary-valued sequence. The particular binary pattern determines the value of r in the iterative formula,

zi+1=rizi1zi

• 

The binary pattern dictates whether r[i] is taken as the x or the y value of a given point in the real plane. The values of z and r are used in computing the Lyapunov exponent according to the following definition.

lambda=limn→infinityi=1nlnri12zin

• 

The above limit is approximated using the finite sum of numterms individual terms. In regions where lambda>0 the behavior is chaotic, while it is stable in regions where lambda<0.

• 

The 2-D grayscale Array image returned by supplying the option output=raw contains the computed values of lambda.

• 

The 3-D color Array image returned by supplying the option output=color contains data where the three layers corresponding to red, green, and blue have been computed using the grayscale raw results.

Examples

withFractals:-EscapeTime

BurningShip&comma;Colorize&comma;HSVColorize&comma;Julia&comma;LColorize&comma;Lyapunov&comma;Mandelbrot&comma;Newton

(5.1)

withImageTools&colon;

n200&colon;

LLyapunovn&comma;1&comma;0&comma;0&comma;1&comma;1&comma;0&comma;2.1..3.9&comma;2.1..3.9

L 1..200 x 1..200 x 1..3 ArrayData Type: float8Storage: rectangularOrder: C_order

(5.2)

EmbedL

LLyapunovn&comma;Vector1&comma;1&comma;0&comma;0&comma;0&comma;0&comma;0&comma;1&comma;1&comma;1&comma;2.1..3.9&comma;2.1..3.9&comma;output&equals;raw&colon;

RArray1..n&comma;1..n&comma;1..3&comma;datatype&equals;float&lsqb;8&rsqb;&comma;order&equals;C_order&colon;

LColorizeL&comma;50&comma;500&comma;300&comma;container&equals;R&colon;

EmbedFitIntensityL&comma;R

randomize&colon;

prand350..450&colon;

Q0&comma;0&comma;0&comma;0&comma;1&comma;1&comma;0&comma;0&comma;1&comma;0&comma;0&comma;1&comma;0&comma;1&comma;0&comma;0&comma;0&comma;1&comma;0&comma;1&comma;1&comma;0&comma;0&comma;1

Q0&comma;0&comma;0&comma;0&comma;1&comma;1&comma;0&comma;0&comma;1&comma;0&comma;0&comma;1&comma;0&comma;1&comma;0&comma;0&comma;0&comma;1&comma;0&comma;1&comma;1&comma;0&comma;0&comma;1

(5.3)

LseqseqLColorizeComplementLyapunov300&comma;Qi&plus;k12&comma;2.1..3.9&comma;2.1..3.9&comma;output&equals;raw&comma;p&comma;2000&comma;p&comma;i&equals;1..2&comma;k&equals;1..2&colon;

EmbedL

EmbedseqLyapunov200&comma;1&comma;0&comma;0&comma;1&comma;1&comma;0&comma;2.1..3.9&comma;2.1..3.9&comma;numterms&equals;i&plus;10&comma;i&equals;0..600&comma;200

Compatibility

• 

The Fractals:-EscapeTime:-Lyapunov command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

Array

ImageTools

ImageTools:-Embed

 


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