Iteration dynamical systems of discrete Laplacians on the plane lattice : Its mathematical structure and computer simulations of designs(Recent Developments in Dynamical Systems)

(1)

Iteration dynamical systems

of

discrete

Laplacians

on

the plane lattice

(Its

mathematical

structure

and

computer simulations

of

designs)

Y.MAKINO1)

_C.HADLICH

2)_,_G.GUERLEBECK_2),_A.KIMURA_3), _and_O.SUZUKI4$)$*

1) _Department _of _home _economics, _Shimane _{Prefectural Shimane} _Womep’s _College,

Matsue-city,Shimane, Japan.

2) Institute ofMathematics$,$ Weimar InstituteofTechnology, Geschister-Scholl str.8,

Weimar, Germany

3)_Graduate_{School ofLiterature and Social}_Sciences, _Nihon _{University Setagaya-ku, Tokyo,}

Japan.

4)_Department_{of Computer}_Sciences _and_{System Analysis,} _Nihon _{Uni\mbox{\boldmath $\nu$}ersity,} _Sakurajosui;

Setagaya-ku, Tokyo, japan.

(Y.Makino) _{[email protected]}

(C.Hendlich)[email protected](G.Guerelebeck)fossi.uni-weimar.de

(A.Kimura)[email protected](O.Suzuki)[email protected]

Keywords: dynamicalsystem, discreteLaplacians, design-samplers

INTRODUCTION

Thisis

a

continuation

of

papers on iteration

dynamical sytemsofdiscreteLapalcian$s([2],[3])$

.

$\ln$this

paper

we are

concemed with(1)Mathematical structure ofiteration dynamical systemof

discreteLaplacians

on

theplane lattice and(2)$The$design-pattems produced by the dynamical

system.

At first

we

give

a

stability theorem for the dynamical system whoseLaplacian isdefinedby

even

neighborhoods.Next

we

are

concerned withcomputer simulations of designs. We

can

realize

many

kinds of designs and

we

can

give

a

classification of

designs

bythe

choices

of neighborhoods,

sources

andthe steps ofthe

iterations.

Finally

we

analyzethe

variations

of

pattern and

we

can

show that

we

supply thedesign-samplersusing

our

software.

ITERATION

DYNAMICAL

SYSTEM OF DISCRETE LAPLACIAN

Werecall thedefinition oftheiterationdynamical system of discreteLaplacians([l]).We

choose the plane lattice which is generatedby twofamiliesof lineswhich

are

orthogonaleach

other. We $identi\theta$

a

latticepointwith

a

cell obtained by the lattice. Wecall

a

setof cellswhich

are

attached with thereference cells

a

neighborhood $U_{p}$

.

We call neighborhood even(orodd)

ifthenumberofthe cellsiseven(respectively odd). Welist several examples of neighborhoods.

(1) _Evenneighborhoods

(2)

(2)Odd neighborhoods -N 禍「 $N$ $-N$王

$\ulcorner w-$ $\ulcorner E$

$-8W^{\ulcorner}8\ulcorner 8\mathbb{E}$

NWNES NNEESW NW E$S$

Wedenote

a

neighborhood $U_{p}$bythedirectionsN,NE,$E$,ES,S,SW,W,NW.Forexample

we

can

denote theNeumann neighborhood byN,E,$S$,W. Wetakethe

space

$F$ of {0,1} valued fUnctions

on

theplane lattice and$def_{1}ne$theLapacian operationby

$\Delta_{U_{\rho}}f(p)=\sum_{q\in U_{p}}(f(q)-f(p))$

Choosing

an

initial function $f_{0}\in F$,

we

define the dynamical system defined by the iteration of the

Laplacian:

$\{f_{n}\},f_{n}=\Delta_{U}f_{n-l}(n=1,2,\ldots)$

We call point $p\in L$

a source

ofthe dynam$i$cal system when $f_{n}(p)=$] forany $neN$

.

Then we

can obtain the designs of distribubons of$0$ and 1 on the lattice plane and we

can

get various kind of

designsbythechoiceofneighborhoods,

sources.

SOME BASIC PROPERTIES ON THE DYNAMICAL SYSTEMS

Here

we

recall

some

basicnotations

on

thedynamical systems and state assertions

on

mathematical

structures([1],[2]). At first we notice that we consider dynamical system$s$ under the periodic

condition. Namely, choosing

an

integer $M$, which is called the size,

we

consider the following

periodicfunctions:

$F(N)–\{f\epsilon F|F\{x*M,$ $y+M$) $– F(x)(m, n\epsilon l)|$

Choosing neighborhoodsunder theperiodic condition,

we

can

define the discreteLaplacian

and

we can

considerthe iterationdynamical system. Weprepareseveral basicnotations:

(1) A dynamical systemiscalledstable if

$3k\epsilon N$

S.$tf_{n}=f_{k}(^{\vee}n\geq k)$

(2)A dynamical

$s_{3_{n,1s.tf_{n^{-}}}^{stemisca11edperiodic,If}3V}$

(3)A point $p\in F$ iscalled

a

source

ofadynamical system,if $f_{n}(p)=1$ forany $n\in N$

.

We

can

state

some

basic properties

on

the dynamical systems:

(3)

lftheneighborhoodis even,

we

see

that the dynamical systemisstable with the stability speed

2

$p$

for Moor neigh., Hexagonal neigh., Neumanneigh.,and Sierpinski neigh.

Ifthe neighborhoodisodd,

we see

that thedynamicalsystemisperiodic, periodis

different depending theneighborhoods.

(2) In the

case

where$M$isodd,

we

may

expect the dynamical system isperiodicinthe

case

of

a

single

source.

We give the table ofperiods for smaller$M$:

We

can

provethefollowingassertion:

PROPOSITION

In the

case

where $M=2^{p}$_, _neighborhood _is _Sierpinski _type, _and it _has

one

point source, the

dynamicalsystemisstable with the stabilityspeed $2^{p}$

.

PROOF

We give

an

outline of the proof of Proposition inthe

case

$p=4$

.

Only by

an

observation in this

simplecase we mayunderstand that

our

assertion holds. The detail will be given in anotherpaper.

口

$arrow$ $\circ$

$O$

口

$\div$

$-\backslash$

Wenoticethefollowing facts:

(1) Puttingthe

source

atthe

upper

corner

intherightside,

_{we see}

that the Pascaltrianglemod2

appears

inthe

upper

trianglepart.

(2)Atthe 4 step,everyelement in the diagonal is 1.

(3) The lower triangleisfilled by$0$

.

(4)

SOFTWARE”DESIGNER KENTAURUS 2005”

We make

a

brief comment

on

our

soft

ware.

Wehave well developed software named “Designer

Kentaurus 2005”.The softwareis writtenby JAVA.

The software has threepanels:(l)Themainpaneldescribes the behavior ofthe dynamicalsystem,

and(2)$the$second panel describes the table of$neigborh\infty ds$and₍₃₎$the$third

one

describes the

sources.

CLASSIFICATION OF DESIGN-PATTERNS

We

can

classifythedesign-pattems choosing neigborhoods, sources,steps. Wenotice that

their characters depend $on$ oddness and

evenness

of neighborhoods strongly

as we

have

seen

in

the previous

paper[3]. We treatdynamical systemswith

a

single

source.

Design-pattern I

-even

neighborhood

–

We

can

observethe followingkindsofdesigns. The left sideofthe explanationsunderthe

picture$s$istype ofneighborhood, for example moor, Neumanndiag.Neumannetcand the

(5)

(4)$Bordered$ pattern (5)$Crystal$pattern (6)$Braid$ pattern

(7)Gasket

pattern

(6)

Design-pattern

$II$

-odd

neighborhood

with

plural

sources-We

can

obtain design pattems with different taste $:(1)stripe$ _pattern, (2)$mosaic$pattem,

(3)$checkpattem,(4)undulate$patternand(5)$line$pattern:

(1)Stripe pattern

$NE$_{SE SW}$W[4]$ step-127

(2) Mosaic

pattern

(7)

(4)

_{Undulate pattern}

NWNESESW [5) $step-120$ _NWNESESW [5] step-128 NWNESESW [5) $step-152$

(5)Line pattern

(6)

Chaotic pattern

We

can

make chaotic

pattemsand

designs

of periodic

characters choosing

plural

sources.

When

we

put

the

sources

without symmetries,

we may

obtain

chaotic patterns easily.

Table 2

examination

of pattems

$\Pi$(odd

neighborhood)

(8)

CONSTRUCTION OF

DESIGN-SAMPLERS

By

these

observations,

we

can

obtain

the

following

manual

of

con

structing designs.

Table

3 structure

of software

We

need

some

experiences

for

getting

desired

pattem$s$

.

We

will

give

a

manual

ofproducing

designs

systematically.

(9)

(1)We have given

a

theorem

on

the mathematical structureofiterationdynamical system

on our

dynamical system inthe simple$st$

case.

Namely

we

haveproved

a

stability

theorem for the Sierpinski’s neighborhood with

a

source.

We

may

expecttoobtain

analogous resultsforgeneral

even

neighborhoods.This will be given intheforthcoming

paper.

(2) Wenoticethat

our

dynamicalsystem inthe

case

of line lattice is identicalwith the

dynamicalsystem

#90

of

Wolffam’s

table of cellularautomata([7]). Hence

we

may

expectto

obtain

analogousresults

for

the plane

lattice.

(3)Wehaveproduced

many

kinds ofdesignsby$u$

se

of

our

simulators and makeanalysis

on

them.Oursimulations

are

defined by theiteration dynamical systemsand

we can

reproduce them

as

one

wantstoobtain them. By this fact

we

havegiven themanualof

producing designs followingthe consumer’s needs.

(4)The$evenness/oddness$_of_{neighborhoods give big}_{difference not only}intheirimpressions

given bythe designsbutalso intheirmathematical structures.This

can

beobserved inthe

psychological experiments

on

the visualimpressions([3]). Wenoticethatthis character plays

an

importantrolein the discussions

on

the differenceofJapanese and European

designs. Also

we

have$s$

een

that

we

have made the simulations ofthetime change of

numbers of familiesofextinctanimals by

use

of the

even

neighborhoods([4]). Here

we

want to express

our

stress

on

thefact that

our

simulations

may

expecttodescribethe

evolution ofthe universe. This topics willbe discussed inthe forthcomingpaper. References

[1] Y.Aiba, K.Maegaito, Y.Makino and O.Suzuki: Dynamical systems defined by iterations ofdiscrete Laplacians andtheir computer simulations, 1-8, Proc. ISSAC Int. Conf. (ICU Univ. 2004, Tokyo)

2005.

$[2]Y.Aiba$, K.Maegaito and O. $Suzuki:Iteration$ dynamical systems of discrete Laplacian

on

the plane lattice(I)(Basic propert$i$

es

and computer simulations), Proc. IKEM2006

(Weimar)2006

[3] A.Kimura,Y.Makino, K.Maegaito and $O.Suzuki:Iteration$ dynamical systems of discrete

Laplacian

on

the plane lattice(II)(The underlying factors determining the visual impressi

ons

$\hslash om$ design pattems), Proc. IKEM2006(Weimar)2006

$[4]K.Kosaka$

ans

$\ovalbox{\tt\small REJECT}:lteration$ dynamical systems of discrete Laplacian and evolution

of extinct animals,To appear in Proc. ISSAC Int. Cong.(Catania,Italy)

[5] Y.Makino, A.Kimura, K.Maegaito and O.Suzuki: Dynamical system defined by iteration of discrete

Laplacian(IV) (Productionofdesignsamplers),65-70,Proc.ofConf. ofAppliedMath.2004.

[6] S. Watanabe: Pattern recognition: human and mechanical.NewYork:$W\ddagger 11y$

.

1985.