ANALYSIS OF THE CHAOS DYNAMICS IN (X n,x n+1) PLANE

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1 ANALYSIS OF THE CHAOS DYNAMICS IN (X,X ) PLANE Soegiao Soelisioo, The Houw Liog Badug Isiue of Techolog (ITB) Idoesia soegiao@sude.fi.ib.ac.id Absrac I he las decade, sudies of chaoic ssem are more ofe used for classical chaoic ssem ha for quaum chaoic ssem. There are ma was of observig he chaoic ssem such as aalzig he frequec wih Fourier rasform or aalze iiial codiio disace wih Liapuov epoe. This paper eplais damic chaoic process b observig rajecor of damic ssem i (, ) pla. Lab.Fisika Theor da Kompuasi, Fisika UNAIR, Phsics Deparme, ITB, Idoesia Lecurer ad Professor of Phsics, Phsics Deparme, ITB, Idoesia

2 INTRODUCTION This paper ries o eplai some resuls of researches i aswerig he quesio of wha is chaos?. Chaos as characerisic of a ssem is defied as he ssem ha is suscepible o iiial codiio so ha he resul is oo much upredicable. This research ries o defie he cause of chaoic ssem. To aswer he quesio, he ivesigaors cosider (X, X ) plae. (, ) plae is o help udersadig he behaviors of ssem. (See picure ). X (X, X ) PLANE IS TOOL TO INVESTIGATE THE ITERATION OF SYSTEM DYNAMICS For eplaaio coveiece, he damic graph from ha ieraig process is used. Cosider he ieraio process for he logisic mappig: X = AX ( X + () for X =, ad A = 3 Afer imes ieraio he values are obaied: Value,,48 3,7488 4, , , , , , , ,74548, , , , ,6649 7, , ,734349,6689 Wih ha above umber, i is difficul o udersad behavior ha ssem. Eplaaio wih ) Picure. visualizaio of damic ieraio i (X,X ) plae To clarif he coecig lie is draw bewee do ad add lie X =X ad lie of he fucio F() = 3 ( - ) Wih ha picure he damical of he ieraio is kow. Afer his he curve F() is called ieraed fucio curve ad lie X =X c is called reversal lie. GUIDANCE DYNAMICAL SYSTEM IN (X,X ) PLANE The research ries o udersad he behavior of chaoic ssem i (, ) plae ad discussio i his paper is o heoreical basis o eperimeal for ODE ad PDE ha is wellkow havig chaoic ssem The resul will esimae ha: The ssem damics will be guided o move oward he poi or he regio. Afer he ssem damics is moved io his poi or regio he he ssem damics cao move o aoher place. X

3 The poi or he regios ha arac he damic ssem are called aracor. The Ssem will move damicall ad coiuousl sice here s o poi or regio o arac. I his case, he ssem damics is called he chaoic ssem. The ssem damics will be guided o repel ad escape o ifii. This codiio is o good because o iformaio ca be obaied. The ssem damics will be guided o move i circular moio like plaear moio, graig elecro, ec. I reali, deails of ssem damics guidace are: curve cross he reversal lie, ad saisf > > 4. The damic ssem will be guided o repel he crossig poi if he ieraig fucio curve is crossig he reversal lie, ad saisf <. The ssem damics will be guided o move dow if he ieraig fucio curve is locaed uder reversal lie Tha rule of guidace ssem damics will be used o aalsis he logisic equaio (equaio ).. The ssem damics will be guided o move up if ieraig fucio curve is locaed upper reversal lie. < A < Case 3. The damic ssem will be guide o oward crossig poi if ieraed fucio Because ieraig fucio curve is uder he reversal lie, so he ssem damics will be guided o move dow oward he crossig poi (aracor) ha is (,).

4 < A < 3 Case: Because ieraig fucio curve is locaed above he reversal lie so ha he ssem damics will be guided o move up oward he crossig poi ha is called aracor. curve ha is locaed above reversal lie so ha ssem damics will be guided o move up ad secod ieraig fucio curve ha is crossig reversal lie ha ad < Tha ssem damics will be guided o repel crossig poi < A < 4 Case: 3< A < Case: I he of case he curve likes above, here are wo rule of guidace: firs ieraig fucio curve ha is locaed above reversal lie. The ssem damics will be guided o move up ad secod ieraig fucio curve ha is crossig reversal lie ad saisf > > Tha ssem damics will be guided o move forward o crossig poi. Tha appeared soluio is - /A < A < Case: I he case of above curve, here s o aracor o arac he ssem damics so ha ssem damics will keep o movig Usig guidace rule, ssem damics ca defie aracor ad chaos, ha is: Aracor is he poi or regio ha arac he ssem damics, if he ssem damics is rapped io his poi or regio, he ssem damics will ever move awhere else. Chaos is a codiio ha here s o aracor o arac he ssem damics, ssem will keep movig. Wih he above defiiio, equaio of chaoic ssem ca be cosruced ha is equaio wih o aracor. Eample : I he case of above curve, here are also wo rule of guidace: firs ieraig fucio f ( ) = ( )... for... <.4 (,5 )... if.. else

5 Solvig ODE usig Numerical Aalsis i Chaoic Ssem Time series he damical above is I aure, phsical pheomea should be formulaed usig differeial equaios. I is ofe difficul o solve differeial equaio usig aalical mehod. Aoher mehods are usig umerical compuaios ha use ieraios. As i is udersood ha ieraio ca ield chaoic soluio. Oe eample below ca be solved b usig aalical mehod ha geerae eac soluio. Usig some umerical mehod for he same problem produce chaos. Eample : f ( ) = (.+. )... for... >.3 (,5 )... if.. else Eample : Cosider he equaio d( ) = a. b. d () wih a ad b as cosa he eac soluio ca be obaied a. e a + b. e a. = a. (3) The above ODE ca be re-wrie b usig Euler umerical soluio as : h = a. b. (4) Time series he damical above is Equaio (4) also ca be re-wrie as = α. + β. + (5) wih : From he above eplaaio, i ca be show ha ieraio ca produce chaos. Therefore, oe should be careful whe usig ieraio process. α = a. h + β = b. h (6) if α =β = A he equaio (5) ca be wrie :

6 = A. ( ) + (7) Time series equaio (7) is he same as equaio (), ad i produces chaoic damics. alhough i is kow o have eac soluio (equaio 3). This umerical soluio is o iflueced b h value, because h is ol used for α ad β. Suppose i is chose a = 3 ad b = + a, wih h =., Usig he umerical Euler mehod Profile ((),()).75 YEPC.5 Y_ solusi umerik solusi aaliik Usig Ruge - Kua mehod Profile ((),()) fmrk( ) fe( ) YE k.5 YRK k.5.5,, XE k.5,, XRKk Time series.998 Time series VROXVLSHUVDPDDQ.8.75 YE Y_.5 YRK Y_ solusi umerik solusi aaliik solusi umerik solusi aaliik Usig predicor correcor mehod Profile ((),()) fepc( ) YEPC k.5.5,, XEPC k Wih ha codiio, ew problem arises, i.e. if oe differeial equaio ca geerae eac soluio ad some umerical mehod for he same problems produce chaos, he i is assumed ha he eac soluio is correc. Wha abou differeial equaios ha have o eac soluio ad he umerical mehod o solve hem produce chaos? Ca he observed ssem be called chaoic ssem? Oe of he eample is Lorez equaio.

7 Coclusio From above sud, i ca be eplaied ha ieraio mehod coribue o chaoic soluio Some umerical mehod coai problems: ieraio compuaio, error samplig frequec, roudig he resul. To overcome hose ieraio, error samplig ad roudig he resul problems, oe mus provide o ieraig mehod. I is kow ha eural ework, fuzz logic, cellular auomaa, ec are o-ieraig mehod such as Euler, Predicor-Correcor, ec. Neural ework, fuzz logic ad cellular auomaa are kow as compee uiversal approimaor as well as adapive mehod. Equaios from uiversal approimaio ca be used o solve umerical problem of ODE ad PDE. B miimizig he hree umerical problems oe ca be sure ha compuig umerical process do o cause chaoic ssem. Refereces [] DeVries, Paul L.,993, A Firs Course i Compuaioal Phsics. Joh Wile & Sos, Ic [] Fu, Li Mi, 994, Neural Nework i Compuer Ielligece. McGraw-Hill [3]. H Nagashima & Y Baba,999, Iroducio o Chaos, IOP Publishig Ld. [4] Lagaris, e al, 997, Arificial Neural Nework for solvig ordiar ad parial differeial equaio. ArXiv:phsics / 9753