Analysis of Stability and Chaos concerning Difference Equations

_________________________________

*Department of Mathematical Sciences, Faculty of Science and Engineering, Doshisha University, Kyotanabe, Kyoto, 610-0321, Japan, TEL:+81-774-65-6702, E-mail: ssaito@mail.doshisha.ac.jp.

Analysis of Stability and Chaos concerning Difference Equations

Seiji SAITO*

(Received 10 May, 2010)

In this paper we deal with difference equations arising from economics models with the Walras’ law, mathematical models in biology, red blood density models and the Lotka-Volterra’s equations. Moreover we discuss the stability and chaos of the systems by applying the Liapunov’s second method and chaos criteria.

aadifference equationȈstabilityȈchaosȈLiapunov’ s second methodȈordinary differential equation

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(

)

max[x_k(n)+vE_k(n),0]

k=1 m

j #m

HǶ8Ȉņºj‡Œ“˜jŽ–1UąĆIJEba lIĂóZƘí8>ȉ EJſƠD, A>ȉ•’lƅǒH0-CĜöHJȈm¹I ǘ2,VȈŏÊn #<IƒčµŝZ ' xj(n)'Ȉ x_j(n)=1

j=1 m

m,Zǘ IǜǫǾǃǶņǾǃDzE´Ƭ DzIǶņE8ȈťI•’lËZªĆ9WȎ E_j(n)x_j(n)=0

j=1 m

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ņD,WȉŽHȈşėJȈab" E 8CťIǜǫǾǃǶņI³ZǕǒ8>Ȏ

E₁=x₁+1 ax₂.

}‰˜qa = 0.6E8Ȉ0 < b < 2 ZţǠȈƯǠ

Zx(n)E8CȈ25,000 ' n ' 30,000 Z”s

w9W(Fig 1.ÝŹ)ȉǞǬIÆŕºHTU:Ȉ-:

XJœÔŷ0.625Ȉ2êŕŷȈ4êŕŷȈ8êŕŷ ƣHÞŘ9W6EZşėJƘí8>ȉ

Fig 1. Stability orbits.

ŗõéDJȈ<I¢ĵZnjō9WœǷǁDŽ™Ţ

ŒƩƐŶǤąĆIJIĆƅZĬ>ȉ™ƺHťIm¾ ěÄŊƛĥZƵ/Wȉ

x(n+1)= f(n,x(n))ȉ (1) 66Dn = 0,1,2,… Hč8Ȉx(n) Jmť¾‚fw

’DȈI = [0,1] E8C x(n) I^mD,Wȉĥ(1)Ik êŕŷǻå P(k) ZťHĆƳ9WȎ

P(k) = {x1, x2, …, xk}ȉ

>?8Ȉxp+1 = f (n,xp), xp xq, x1 = f (n,xk), (1' p < q ' k ; n = 0,1,2,…)D,Wȉĥ(1)I êŕŷǻå P(k) JȈÆȁ1UÃƏ8C1UŐ4 JœąĆD,W2ȈŒƩƐHŶǤąĆEGWE3Ȉ P(k)JœǷǁDŽ™ŢŒƩƐŶǤąĆD,WE-.ȉ ěÄĥ(1)I<IŶǤĽÔHǶ9WĆƳJȈĞĭ ÄŊƛĥEçŢHĆƳ9W ^8,19,22)ȉ<IĆƳZǥ NW>QHȈZ+ ZǿǗŇņIǻåȈP(k) IǤ¼ Z > 0 , xp %P(k)E8C

S(P(k), ) = U1qk { y I^m: y – xq < } E04ȉ y Jy I^mI{’ˆD,Wȉ ǻåP(k)ExIǝǼZ

d(P(k), x) = inf{ y – x : y P(k) } Dǀ9ȉ

êŕŷǻåP(k)JȈ[EV-UA.FC](œǷǁDŽ™Ţ ŒƩƐèĦƐ)Ȉ9GY@Ȉ¬ĶIǁDŽ { Cq I^m : U1qQ Cq I^m } E¬ĶI > 0Hč8Ȉ ,Wƍã N0 % Z+ ET0 %Z+ ZEXKȈ¬ĶIǻå CqȈ

¬ĶIÆŕŷx0 % CqȈ¬ĶIÆŕŏÊn0 ( N0

1UÃƏ9W¬ĶILJ(ǞǬ) x(n; n0, x0) JȈŏÊn ( n0 + T0IE3ȈêŕŷǻåIǤ¼S(P(k), ) H Ăó9WȈ9GY@Ȉ

d(x(n; n0, x0), P(k)) < .

êŕŷǻåP(k) J[EV-US](™ŢŒƩƐąĆ) D,WEJȈ¬ĶI > 0 Hč8CȈ,WƍãN0

% Z+E > 0ZEXKȈ¬ĶIÆŕŷx0

S(P(k),)E¬ĶIÆŕŏÊn0 ( N01UÃƏ9 W¬ĶILJ(ǞǬ) x(n; n0, x0) JȈŏÊn ( n0 I E3ȈêŕŷǻåIǤ¼S(P(k), ) D,WȈ9G Y@Ȉ

d(x(n; n0, x0), P(k)) < .

êŕŷǻåP(k) J[EV-UAS.FC](œǷǁDŽ™ŢŒ ƩƐŶǤąĆ)D,WEJ[EV-UA.FC]Ȉ1BȈ

[EV- US]D,W6EZ-.ȉ

êŕŷǻåP(k)JȈżƅăH05W`z’e˜

ǶņZ™ƺÕ8>‘[{€ǶņZİƈ8CȈť IĆƅJnjōD3>^2,19)ȉ

ǿǗćņǻåR+ = { 0' x < +}E8Ȉ

CIP = {a: R+$R+JŦĆºȈƀƳÙǑøÐGǧ

ƮǶņ}ȈěǻåA – B = { x A : x B}E9Wȉ

ĆĆƅƅ 11ȉêŕŷǻåP(k) JȈťIř«(a), (b) Z P>9V: Z+I^m$R+2Ăó9XKȈ[EV-UAS.FC]

D,WȎ

(a) ¬ĶIr > 0 Hč8Ȉ,Wƍã N0 (0Ȉar,

br % CIP ZEXKȈ¬ĶIn ( N0 E

x I^m & S(P(k),r)Hč8C

ar(d(P(k),x)) 'V(n,x)' br(d(P(k),x)) 2ĹVƠBȏ

(b) _kV(n,x) = V(n+k, f ^k(n, x))&V(n,x)E04ȉ

¬ĶIr > 0 Hč8Ȉ,Wƍã N0 (0Ȉcr % CIP ZEXKȈ¬ĶIn ( N0 E

x I^m & S(P(k),r)Hč8C

_kV(n,x) ' &cr(d(P(k),x)) 2ĹVƠBȉ

(.-&.7J

şėJĥ(1)HǶ8Ȉa = 0.5IE3ņºj‡Œ

“˜jŽ–ZĬCȈbalIÃƃZƘí8>ȉţ ǠZ0 < b < 2 DȈƯǠZx(n)E8CȈ25,000 ' n ' 30,000 Z”sw9W(Fig 2. ÝŹ)ȉ ŗõéDJb =0.7E8CȈťĥZÁ±ƐHƵ/

W⁷⁾.

x(n+1)= f(x(n)), (2)

>?8Ȉ f(x)= 11x²25x+14

42x²59x+24, n = 0,1,2,…

Fig 2. Chaotic orbits.

Ȋť¾ÏăƨHǶ9W‘˜—˜fIĆƅJťI E0VD,WȎ

ĆĆƅ 22.(‘˜—˜f)ěÄŊƛĥ(2)JȈ3êŕ ŷZRBE3ȈǶņf IǿêŕŷIǿÐƤǻå S 2Ăó8ȈťI(I),(II)ZP>9Ȏ

(I)SI¬ĶIŷxEfI¬ĶIêŕŷpHč8, lim

n fⁿ(x) fⁿ(p)>0; (II) SI¬ĶI2ŷx, yHč8C,

lim

f

(x) f

(y) > 0

lim

n

f

(x) f

(y) = 0.

šljIǶ¶ĥ(I),(II)2ĹVƠBE3Ȉĥ(2)J‘

˜—˜fIbal(ZRB)E-.ȉ

ǧƮǶņy = f (x),0'x, y'1J3êŕŷZRCKȉ bal2Ăó9WȉćǹȈ0.6'x'0.61H0-CȈ 3êŕŷ2Ăó9W.<I>QHJȈťI2œƣĥ 2ĹVƠB6EZƘ;KT-Ȏ

{f(0.6)&0.6}{f(0.61)&0.61} < 0ȏ f (x) & x > 0 on[0.6, 0.61]ȉ

][2F!),-"(.%) 67=SC

ƸŸƋH05W[mdp^ˆjEh†|rJȈ ä)Ȉĉ (host)EŁĊƇ(Parasitoid)IǶ¶H, WȉĪƶJȈÌƶZȃ-đ49Ƕ¶H,W ^1,5,10). ǼŅŏǵt = 0,1,2,…Hč8ĉ I¹±ņ(,W- JċĢ)Z Nt, ŁĊƇIZ Pt Eǀ9(Fig 3.ZÝ

Ź)ȉ

NtȎž¨DIĉ IċĢ.

( ^{t = 0 , 1 , 2 , …} )

PtȎž¨DIŁĊƇIċĢ.

f

c

HostȎ

Eggs $Larvae$Pupae$Adults eggs

ParasitoidȎAdults $ Larvae

Fig 3. Host and parasitoid.

ŒÆȈyh’o–E‚]‘˜JȈĉ EŁĊ ƇIǮǩ9WIJ1ï?5E8…[o–ÄĝZ ªĆ8ȈťIěÄŊƛĥZĬ>ȉ

N_t+1 =

(

)

«EŨǡ8CȈťIŠv’H·Ŧ7X>^1,12). N_t+1=N_te^r1

N_t K aP_t

, P_t+1=N_t

(

)

>?8Ȉt = 0,1,2,…, K > 0J2ƝZEVO4Ɔ

÷1UŬOW}‰˜qD,Wȉ‚fw’ x = (N, P)^T, f(x)=(Ne

r1Nt

K aP

, N(1e^aP))^TE03, TJǟƱIĶë9Wȉĥ(3)JťĥEGWȎ

x

= f (x

) (t = 0,1, 2,...).

ĥ(4)IœÔŷ_xe=

(

)

P_e=r

(

)

a Ȉ_Ne= ^Pe 1e^aPe

ZP>9ȉ}‰˜qJ3Ɲr, q, a D,Wȉ }‰˜qIßVŊHTVȈ4 }q˜–2ĬU XWȉ

1) r = 0.5, q = 0.4, a = 0.2 (Fig 4.ÝŹ) 2) r = 2.0, q = 0.4, a = 0.2 (Fig 5.ÝŹ) 3) r = 2.2, q = 0.4, a = 0.2 (Fig 6.ÝŹ) 4) r = 2.65, q = 0.4, a = 0.2 (Fig 7.ÝŹ)

Fig 4. Case with r = 0.5, q = 0.4, a = 0.2.

Fig 5. Case with r = 2.0, q = 0.4, a = 0.2.

Fig 6. Case with r = 2.2, q = 0.4, a = 0.2.

Fig 7. Case with r = 2.65, q = 0.4, a = 0.2.

ĥ(3)IœÔŷ xe = (Ne, Pe)^T IąĆIJZƘ9H JȈťI™ŢŶǤąĆIJĆƅ2œÓD,W ⁶⁾ȉœ

Ôŷxe 2™ŢŶǤąĆD,WEJȈ

[ȇx = xe J™ŢèĦƐ[UA],9GY@Ȉ×Äď

I>02Ăó8Ȉ¬ĶI > 0Hč8Ȉ ,WŦ

ņT0 > 0 ZEXKȈ¬ĶIÆŕŷx0 : x0 – xe <

E¬ĶIÆŕŏÊn0 ( 01UÃƏ9W¬ĶI LJ(ǞǬ) x(n; n0, x0) JȈŏÊn ( n0 + T0IE3Ȉ x(n; n0, x0) & xe < ȏ

]ȇx = xe J™ŢąĆ[US],9GY@Ȉ¬ĶI

> 0 Hč8CȈ,W > 0ZEXKȈ¬ĶIÆŕ

ŷx0 : x0 – xe < E¬ĶIÆŕŏÊn0 ( 01 UÃƏ9W¬ĶILJx(n; n0, x0) JȈŏÊn ( n0

IE3Ȉ

x(n; n0, x0) & xe <

2ĹVƠB6EZ-.ȉǝǼd(x,xe) = x – xe

E9Wȉ

ĆĆƅ 33.(Halanay etc.) ĥ(4)JȈœÔŷ xe ZR@Ȉ

<IǤ¼ZBe = { x R^m : x– xe < c } (c > 0)ȉ ťIř«(i), (ii) ZP>9V: Be$R+2Ăó9XKȈ

œÔŷxeJ™ŢŶǤąĆD,WȎ (i) ,Wa, b % CIP ZEXKȈ a(d(x,xe)) 'V(x)' b(d(x,xe)) 2ĹVƠBȏ

(ii) V(x) = V(f(x)) & V(x)E04ȉ,Wc %

CIP ZEXKȈ

V(x) ' &c(d(x,xe)) 2ĹVƠBȉ

}q˜–Ȋȇ2™ŢŶǤąĆD,W1JŖ?Ƙ 7XC-G-ȉ‘[{€ǶņIŊŮZİƈ9X KȈ<I×Äř«(³/K_e^r_<(qr+1)e^qr)^ZP>9 }‰˜qr, q 2ūQUXWȉ

}q˜–ȋȇȈȌȇJä)Ȉ‘‡swi]f’Ȉ êŕLJIĂóZƘí9Wȉ¦ĪIƔƟ2ŕĩ7X Wȉ

/O4$*7J

·Ŧyh’o–—‚]‘˜Šv’(3)I}q˜–

ȍȇJȈbalD,WE¢ĵ7X> ¹⁾ȉŗõéD JȈťI™ƺՆ”sw^13,17)IĆƅZƈ-CȈć ǹHȍȇIE3Ȉbal2Əƃ8C-W6EZǥ NWȉ

ĆĆ ƅ (™ƺՆ”sw) ĥ(4)JȈœÔŷ xe

ZRBȉťIř«(i) – (iii)2ĹƠ9WE3Ȉĥ(4) J‘˜—˜fIbalD,WȎ

(i) Ƕņf (x)JȈxeIǤ¼ Be DǧƮĭÄáƷȏ

(ii) ‹h~ƾÅ Df (xe) Iòœº!J9NC

! >1ȏ

(iii) ,Wz Be JȈz xe D,WŇņMHB 3åĹ f^M(z) = xe, 1BȈƾÅĥDf ^M(z) 0.

}‰˜qr = 2.65, q = 0.4, a = 0.2IE3Ȉšlj

Iř«(i),(ii)JĈŎHĹƠ2ƗǎD3W²⁰⁾ȉ

Fig 8. The Graph of y = h(N).

ř«(iii)HB3Ȉz = (N,P)^T E04EȈ_xe=

(

)

TVȈťĥZĬWȎ _{Pe =NNee}^r(1

N K)

ȉ

6IĥILJNIĂóZƘ;KT-ȉâIǶņZ _h(N)=NNee^r(1

N K)

E03Ȉh(N) = PeJLJZRB6EJøŴǀE Pe

Iº1UƗǎD3W(Fig8.ÝŹ)ȉĆƅ4Iř«(iii)

Iř«IĹƠRƗǎD3Wȉ©š1UȈ™ƺՆ

”swIĆƅ1UȈĥ(3)J}q˜–ȍȇIE3Ȉ balD,Wȉ

^[ VPI8:%)

»ģG¥ǵIǛƽƄIċĢJȈŦĞGžĸD, WNJĈƦñIǵDúÔ9WȉƊIJDJ6IƦñJ (5.4±0.8)10⁶ ¹/m² D,VȈĀIJDJ(4.8±0.6) 10⁶ ¹/m²D,WȉǙƽƎIT.GöåȈ6I MWO-JŏHÎƐHúÕ9Wȉ

ŏÊnIǛƽƄIċĢZxnȈŏǵÖǵ[n,n+1]I ǵDƕù7XWċĢZdn = axnȈƽƥŸHŃÃ7X WċĢZ_pn=b(xn)^r_e^sxnE9WEȈǛƽƄċĢŠv

’JťĥEGW^4,14)ȉ

x_n+1=x_nax_n+b(x_n)^re^sxn (5)

(n=0,1,2,…). O>ȈťĥZ04ȉ

F(x)=(1a)x+b(x)^re^sx

ĥJȈa = 0.6, b = 1, r = 6, s = 1.5IE3Ȉ3

œÔŷx1 = 0 < x2 < x3ZRBFig 9.ÝŹȉ

Fig 9. The Graph y = F(x) has 3-fixed points.

œÔŷx1 J™ŢŶǤąĆD,WȉǛƽƄċ Ģ2Ȉn”§ǤHGWEn”HÞŘ9W1UȈ6 XJ¥±HÚǸGžĸZĶë9WȉĭÄIƭčº

F’(0) < 1 1URōU1D,Wȉ

œÔŷx2 DJȈF’(x2) > 1 IE3œąĆD,Wȉ

ǛƽƄċĢJȈx2§ǤDJn”HÞŘ9W1Ȉš

Ō9W1I-:X1D,Wȉ

œÔŷx3 DJȈF’(x3) > 1IE3œąĆD,Wȉ

ĥJ ť¾ÏăƨD,W>QHȈąĆ—œąĆ IJIÇĆJŨǡƐĈŎD,Wȉ

ÝƵňƁ ȇDJȈr =8, s =16, b =1.110⁶a IE3Ȉ êŕŷIĂóp1, p2, p3, p4, p5, p6 êŕŷǻå2 BZƘ8Ȉ‘˜—˜fIĆƅT VȈĥJbalD,W6EZǥNC-WFig

10. ÝŹȉ

Fig 10. The Graph y = F³(x) with a=0.78 indicates the Li-Yorke’s chaos.

ĥ2 êŕŷZRCKbalD,VȈħŸœ

ąĆDR,Wȉ ť¾ÏăƨIöåȈÇĆJĈŎ D,Wȉûť¾ÏăƨD,WEȈ êŕŷ2Ăó 8CRƑ@HȈbalD,WEJǷUG-ȉTA Cûť¾ÏăƨDJȈbalD,W6EEœąĆ D,W6EJÖÈ9N3E-/Wȉ

ŗõéDJȈûť¾ěÄŊƛĥHǶ9WœąĆ IJĆƅZnjō8>6EZǥNWȉťIǿƸÒƨě ÄŊƛĥZƵ/WȎ

x(n+1)= f(n,x(n)) (n=0,1,2,...)ȉ (6)

ĆĆ ƅ œąĆIJĆƅ ĥJœÔŷ xe fn, xen #ZRBȉœÔŷxeJťI ř«(a) - (c) ZP>9V: Z+Bc(xe)$R+2Ăó9 XKȈąĆDG-Ȉ9GY@œąĆD,W.>?8Ȉ Bc(xe) ={ x : x & xe < c }Ȉ

E(xe ;p) = { x: 0 < x & xe < p } (0 < p < c) E9Wȉ

(a) ,Wa% CIP ZEXKȈ¬ĶIn ( 0 E

x E(xe;p) Hč8C V(n,x)' a(d(x, xe)) 2ĹVƠBȏ

(b) V(n,x) = V(n+1, f (n, x)) & V(n,x)E04ȉ

,Wb % CIP ZEXKȈ¬ĶIn ( 0 E

x E(xe ;p) Hč8C

V(n,x) ( b(d(x, xe)) 2ĹVƠBȏ

(c) V(n, xe) = 0 (n ( 0).

njōJȈňƁ IĞĭÄŊƛĥHč9W‘[

{€IœąĆIJĆƅEçŢD,Wȉ

_[*-#)'93@L;

ľȃ—ǁľȃŠv’EJȈƢ™ťžƋüĺŏH [x‘[űDdžČ7X>ƖȄȅI¹±ņIêŕƐ GøŴHǶ8CƵŞ7X>Šv’D,Wȉ„’u

JȅȂI¹±ņZľȃƶEǁȃƶI£BIƲH Ä5Ȉ£ƶIǮǩJľȃƶIøÐEǁľȃƶIŴ ĐHĨȀ9WE8>ȉÙ¯ŏǵħ>VHǮǩ9W ņJ<X=XIƲIņIƞHŨ³8Ȉ<XJľȃ ƶHJœÉHȈǿľȃƶHJœÉH½4EªĆ8

+

=

=

dxy dt cx

dy

bxy dt ax

dx

Dǀ7XWŊƛĥZƵ/>ȉŗõéDJȈ6IŊ ƛĥZa]˜Ǥ®HTVěÄÕ8ȈťIěÄŊ ƛ ĥ 2 b a l D , W 6 E Z ǥ N W ^18ȇȉ

x_n+1=rx_n

(

)

y_n+1=rx_ny_n (n=0,1, 2,...) (7) ƈ-WĆƅJȈƢ3ơI™ƺՆ”swIĆƅD ,Wȉĥ(7)JȈťĥ

u= x

y , F(u)= rx

(

)

rxy

HTVȈ

u_n+1=F(u_n) (8) EGWȉ™ƺՆ”swIĆƅZİƈ9Wȉĥ IœÔŷx_e = F(x_e)Jx_e= 1

r,12 r

T( r > 2IJ:).

œÔŷx_e H05W‹h~ƾÅJ

DF x

( )

r2 1 D,VȈ<IòœºJ +1= 94r

2

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ťHȈ™ƺՆ”swIĆƅIř«ZP>9 ř«ZĎ4ȉM E8ȈF(z)x_eGWz = (x,y)^T 2Ăó9WrIƦñZūQWȉ

( )

=

r r rxy

y x rx

1 2 1

1 .

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2r ZĬWȉ

6IE3ȈzI‹h~ƾÅJ⎜DF(z)⎜㱠 ZP>9．

©šTVȈĥJr IE3balD,Wȉ

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5

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L. Edelstein-Keshet, Mathematical Models in Biology, Random House (1988).

2) S.N. Elaydi, Discrete Chaos 2nd, Chapman & Hall/CRC (2008).

v\[sfȈƒ˜ˆmȈ*ý±ÏăI}]ay[>@

š+ȈNjȎæƉȈjŒ‘–c˜€_[˜fŚ¤

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Ę°ĤȈņƅƇżăÀǴȈ ÃŻĒ

A. Halanay,V. RasvanStability and Stable Oscillations in Discrete Time Systems, Gordon and Breach Sci. Publ.. (2000).

ęÜƅijāȈµŝǑŇěÄŊƛĥHǶ9WbalƃǖI ĂóȈçįƙüăĚăǯƓǔĚăƚØŠǒň 8) V. Lakshmikantham, S. Leela , Differential and Integral Inequalities I, Academic Press(1969).

T. V. Li, J. A. Yorke, Amer. Math. Monthly, 82, 985 (1975).

J. D. Logan, W. R. Wolesensky, Mathematical Methods in Biology, Wiley(2009).

11) E. N. Lorenz, J. Atmospheric Sciences, 20, 130(963).

12)ŪàƴþȈ‘[{€IŊŮHTWņƅƇżăH05 W·Ŧyh’o–—‚]‘˜ěÄŊƛĥHǶ9WĆIJLJ śȈçįƙüăĚăǯ]–u‘k_–wĴõĚăƚØŠ ǒň

F. R. Marotto,J. Math. Anal. And Appl., 63, 199(1978) 14) M. Martelli, Discrete Dynamical Systems and Chaos, CRC Press, LLC, BOCA BARTON(1992).

şėǦÿȈşėǦÿƻ²ǻȊÔăƐƪųƅǒȈĕů őġ

şėǦÿȈşėǦÿƻ²ǻ •’lƅǒȈĕůőġ

şƉî¡ȈƇżŠv’IbalȈŔ¸

18) DZàĖ¥Ȉľȃ—ǁľȃŠv’Ia]˜Ǥ®HTW

ěÄŊƛĥIbalȈçįƙüăĚăǯ]–u‘k_–

wĴõĚăƚØŠǒň

S. Saito, Advanced Studies in Pure Mathematics 53, 301(2009).

20) ǣǂǢȈ™ƺՆ”swIĆƅZİƈHTWņƅƇż ăH05W·Ŧyh’o–—‚]‘˜ěÄŊƛĥIba lȈçįƙüăĚăǯ]–u‘k_–wĴõĚăƚØŠ ǒň

ĔŗƜȈĞĭÄŊƛĥIąĆIJćńÃŻ 22) T. Yoshizawa , Stability Theory by Liapunov's Second Method, Math. Soc. Japan(1966).