NEW IDENTIFICATION AND CONTROL METHODS OF SINE-FUNCTION JULIA SETS

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1 Joural of Applied Aalysis ad Computatio Volume 5, Number 2, May 25, Website: doi:.948/252 NEW IDENTIFICATION AND CONTROL METHODS OF SINE-FUNCTION JULIA SETS Jie Su,2, Wei Qiao 2,3 ad Shutag Liu 3 Abstract I this paper, we propose two ew methods to realize driverespose system sychroizatio cotrol ad parameter idetificatio for two kids of sie-fuctio Julia sets. By meas of these two methods, the zero asymptotic slidig variables ad the stability theory i differece equatios are applied to cotrol the fractal idetificatio. Furthermore, the problem of sychroizatio cotrol is solved i the case of a drive system with ukow parameters, where the ukow parameters of the drive system ca be idetified i the asymptotic sychroizatio process. The results of simulatio examples demostrate the effectiveess of the ew methods. Keywords Julia set, sychroizatio, parameter idetificatio. MSC(2 34H5, 93C28.. Itroductio The fractals theory which describes fractal properties ad correspodig applicatios was proposed by Madelbort [9]. As a forefrot oliear sciece theory, the fractals has successfully explaied a lot of oliear pheomea (Madelbrot [] ad Liu et al. [7]. At preset, this theory has a variety of applicatios i the meteorology, cacer cell growth, image processig, geography ad so o (Wag et al. [2], Wag et al. [3], Yag et al. [2], Yu ad Che [22], Wag ad He [4], Dig ad Jiag [2]. I recet years, researchers have paid much attetio to fractal sets which come from a aalytic mappig iteratio o the complex plae. The aalytic mappig iteratio divides the complex plae ito two parts: Fatou set ad Julia set. It has bee foud that the geeralized Madelbrot-Julia (M-J sets which ejoy a importat role i the fractals with a fie ad complex structure. I 24 ad 27, Wag et al. [5, 6] studied the applicatio of M-J sets i physics. That is a typical Lagevi problem, i.e. the aalysis about the dyamics of a charged particle which is uder the cotiuous ifluece of a costat impulse at oe-dimesioal discrete-time poits i a double-well potetial ad a time-depedet magetic field. Nowadays, the M-J fractal system research is maily based o a complex mappig as a represetative of the polyomial fuctio f(z, c = z α + c, c C, but for a The correspodig author. address: qiaowei@sdu.edu.c (Wei Qiao School of Computer Sciece ad Techology, Shadog Uiversity, Jia, 25, P.R. Chia 2 School of Mechaical, Electrical ad Iformatio Egieerig, Shadog Uiversity at Weihai, Weihai, 26429, P.R. Chia 3 College of Cotrol Sciece ad Egieerig, Shadog Uiversity, Jia, 256, P.R. Chia The authors were supported by the Natioal Natural Sciece Foudatio of Chia (Grat Nos , 2794 ad

2 Sie-fuctio Julia sets 22 o-polyomial fuctio it is rarely studied (Fu et al. [3]. Moreover, may mathematicias are iterested i properties ad dimesios of the polyomial-fuctio Julia set i theoretical studies (Wu ad Che [7], Wag ad Shi [8], Gao [5], Wag ad Su [9], Ashish et al. [], Huag ad Wag [6]. Importatly, people ofte have actual requiremets for the o-liear domai-rage ad parameters of the fractal collectio. For example, some sigificat achievemets about the fractal sychroous cotrol have bee reported (Zhag et al. [23], Liu ad Liu [8], Wag ad Liu [2]. It is poited out that these methods are feasible oly i the case of kowig the drive system parameters. However, the drive system parameters are usually ukow, so sychroizatio cotrol of the drive-respose system is difficult to be solved by the existig methods. I this paper, for two kids of sie-fuctio Julia sets, we propose two ew methods to realize drive-respose system sychroizatio cotrol ad parameter idetificatio. By applyig the zero-asymptotical slidig variable discrete cotrol method ad the stability theory i differece equatios to sie-fuctio Julia sets, sychroous cotrol of the drive-respose system ad parameter idetificatio of the drive system are achieved simultaeously uder the coditios that the drive system has ukow parameters. 2. Desig of sychroizatio cotroller ad parameter idetifier for sie-fuctio Julia sets 2.. Prelimiaries A Julia set J(f is created by the iteratio of a complex variable fuctio f, which is defied to be the closure of the repellig periodic poits of f. For the complex polyomial f, its Julia set has the followig properties. (i J(f is oempty ad bouded; (ii J(f is fully ivariat, i.e., J(f = f(j(f = f (J(f; (iii J(f=J(f p, for ay positive iteger p; (iv If ω is a attractive fixed poit of f, the A(ω = J(f, where A(ω is the attractive domai of the attractive fixed poit ω. That is, A(ω = {z C : f k (z ω, (k }, i.e., ω. From the defiitio of Julia set ad its properties, we ca cotrol the trajectory of iterative poits ad achieve the objective of cotrollig Julia set. Now, we cosider two complex systems with the same structure: x + = f(x, a i, c, (2. y + = f(y, a i, c. (2.2 We desig a cotroller for system (2.2 i order to correlate with system (2., that is, y + = f(y, a i, c + u (x, y, a i, (2.3 where a i (i =, 2,... are give complex umbers, ad c is a ukow complex parameter which eeds to be idetified. The, system (2. is a drive system ad system (2.3 is a respose system. The cotroller u of system (2.3 cotrol drive system (2. ad respose system (2.3 to achieve asymptotic sychroizatio. If

3 222 J. Su, W. Qiao & S. Liu the parameters a i, c, c are give, their correspodig Julia sets are also idetified, which are writte as J, J respectively. If lim (J J J J = (2.4 the systems (2. ad (2.3 ca achieve sychroizatio. I fact, we kow that the orbits of Julia set J(f ad f are closely related, accordig to the defiitio of Julia set. Therefore, we desig the followig cotroller based o Julia sets sychroizatio as soo as their orbits sychroized. Whe, systems (2. ad (2.3 gradually sychroize with the same iitial iterative value, i the meatime, the ukow parameter c of system (2.3 is idetified, that is, c c Desig of sychroous cotroller ad parameter idetifier for sie-fuctio Julia sets For a class of sie-fuctio Julia sets, x + = c si(x, (2.5 where c is a complex costat, we discuss the sychroizatio problem ad idetificatio of parameter c. We ca obtai the Julia set by iterative calculatio of the poits i a bouded regio usig the above property (i. Therefore, suppose, i the bouded regio D, we oly cosider the iteratio of the mid-poit of D. Usig the above property (ii, we oly eed to calculate the poits, whose trajectories are all i the D, because if there is a to make f (z D, usig the property (ii, we kow z J. System (2.5 is the drive system, ad the other system with the same structure of sie-fuctio Julia sets is the respose system: y + = c si(y + u, (2.6 where u is the sychroous cotroller to be desiged. We assume that the drive system ad the respose system are ucertaity fractal sets, i.e., the parameter c is ukow. The, we desig a appropriate sychroous cotroller for the drive system i the case of the respose system with ukow parameters. Thus, we will realize system (2.5 ad system (2.6 completely fully sychroizig ad c is idetified at the same time. We take the error variables betwee the drive system ad the respose system as follows: Next, we desig the sychroous cotroller e ( = x y, (2.7 e 2 ( = c c. (2.8 u = x + c si(y + (k e (, (2.9 where the costat k satisfies < k < 2. It should be remarked that the cotroller u is ot a truly real-time cotroller, which eeds to be implemeted with oe-step

4 Sie-fuctio Julia sets 223 time delay, amely, the cotroller eeds to be kept for oe step by a zero-orderholder ad the takes the actual cotrol actio. We desig the parameter idetificatio law where the costat k 2 satisfies < k 2 < 2. c + = ( k 2 c + k 2 x + / si(x, (2. Lemma 2. (Furuta [4]. If σ( is a slidig variable of a sigle-iput ad sigleoutput discrete variable structure cotrol system, the the discrete cotrol variable satisfies σ( σ( < 2 [ σ(]2, σ(, (2. where σ( = σ( + σ( ad σ( + < σ(, i which σ( teds to zero whe. Theorem 2.. If the sychroous cotroller of system (2.6 is described by (2.9 ad its parameter idetifier is described by (2., the drive system (2.5 ad respose system (2.6 with arbitrary iitial value ca achieve global asymptotic sychroizatio, ad the parameter of sie-fuctio Julia sets ca be idetified. Proof. From the previous discussio, if systems (2.5 ad (2.6 realize track sychroizatio, the the Julia sets of systems (2.5 ad (2.6 realize global asymptotic sychroizatio with arbitrary iitial value. We oly prove that systems (2.5 ad (2.6 realize track sychroizatio, that is, x y (. This is equivalet to that e ( (. From (2.5-(2.7, we have e ( + e ( = c si(x c si(y u x + y. (2.2 Substitutig (2.9 ito (2.2, we obtai e ( + e ( =c si(x c si(y x + y (k e ( + x + c si(y = k e (. (2.3 Accordig to the slidig variable property, we take e ( as the slidig variable. If < k < 2 is satisfied, the it follows from (2.3 that That is, e ([e ( + e (] = k [e (] 2 < 2 k2 [e (] 2 = 2 [e ( + e (] 2. (2.4 e ([e ( + e (] < 2 [e ( + e (] 2. (2.5 Hece, we have e (+ < e ( accordig to Lemma 2., that is, e ( as. So, Julia sets of systems (2.5 ad (2.6 have achieved global asymptotic sychroizatio with arbitrary iitial value. Similarly, accordig to the slidig variable property ad (2., if e 2 ( is the slidig variable ad < k 2 < 2, we get c + c = c + c (c c = e 2 ( + e 2 ( = k 2 e 2 (. (2.6

5 224 J. Su, W. Qiao & S. Liu If < k 2 < 2 is satisfied, the it follows from (2.6 that e 2 ([e 2 ( + e 2 (] = k 2 [e 2 (] 2 < 2 k2 2[e 2 (] 2 = 2 [e 2( + e 2 (] 2. (2.7 That is, e 2 ([e 2 ( + e 2 (] < 2 [e 2( + e 2 (] 2. (2.8 Accordig to the Lemma 2., we obtai e 2 ( + < e 2 (, i.e., e 2 (. It esures that c asymptotically teds to the actual value of c as. Example 2.. Let c = i, c =.6 for drive system (2.5 ad respose system (2.6. After steps, the Julia sets of drive system (2.5 ad respose system (2.6 are show i Figure, where k =.8, k 2 =.5. (a (b (c (d (e (f Figure. The Julia sets of respose system (2.6 after (a 5 steps, (b 2 steps, (c 5 steps, (e 8 steps, ad the Julia sets of drive system (2.5 after (d 5 steps, (f 8 steps, where k =.8, k 2 =.5. The figures show that respose system (2.6 sychroizes drive system (2.5 after 5 steps. Moreover, respose system (2.6 keeps sychroous with drive system (2.5 after 8 steps. Besides, the idetificatio process of c i respose system (2.6 ad e ( chagig with are show i Figure 2. From the simulatio results, i the iterative process the real ad imagiary parts of c are stable at ad ad the sychroizatio error e ( teds to zero. Hece, respose system (2.6 ad drive system (2.5 achieve sychroizatio, where the ukow c of drive system (2.5 ca be idetified. As the values of k ad k 2 icrease from to 2, the chagig processes of c ad e ( are show i Figure 3. From the results, we kow that as the values of k ad k 2 icrease from to, the idetificatio process of c is gettig better ad e ( teds to more quickly. Whe the values of k ad k 2 cotiue to icrease from to 2, the idetificatio process of c is gettig worse, ad e ( teds to slower. So, whe the values

6 Sie-fuctio Julia sets Re c Im c e ( (a (b (c Figure 2. The (a real, (b imagiary parts of c of system (2.6 ad (c e ( betwee systems (2.5 ad (2.6 chagig with, where k =.8 ad k 2 = k=.9,k2=.9 k=.5,k2=.5 k=.9,k2=.9 k=.5,k2=.5 k=.,k2= k=.9,k2=.9 k=.5,k2=.5 k=.9,k2=.9 k=.5,k2=.5 k=.,k2= k=.,k2=. k=.5,k2=.5 k=.9,k2=.9 k=.5,k2=.5 k=.9,k2=.9.2 ReC.2 ImC.8 e ( (a (b (c Figure 3. The (a real, (b imagiary parts of c ad (c e ( chage with for differet k ad k 2. of k ad k 2 are ear, the idetificatio effect of c is the best, ad the speed of e ( tedig to gets the fast. That is to say, the speed of the respose system ad the drive systems achievig sychroizatio is the fastest. However, if k = 2.5, k 2 = 3, the idetificatio process of c of respose system (2.6 ad the chagig process of e ( are show i Figure 4. 8 x 3 x Re c Im c.5 e ( (a (b (c Figure 4. The (a real, (b imagiary parts of c of system (2.6 ad e ( chagig with, where k = 2.5 ad k 2 = 3. From Figure 4, we kow that e ( betwee the two sychroous systems ad the c of system (2.6 ted to ifiity gradually. Therefore, the sychroizatio of the two systems ca ot be achieved ad the ukow c of drive system (2.5 ca ot be idetified.

7 226 J. Su, W. Qiao & S. Liu 2.3. Desig of sychroous cotroller ad parameter idetifier for aother sie-fuctio Julia set I this sectio, we discuss the problem of sychroous cotroller ad parameter idetifier for aother sie-fuctio Julia set: z + = si(z 2 + c. I this case, the drive system ad the respose system ca be writte as x + = si(x 2 + c, (2.9 y + = si(y 2 + c + u. (2.2 The error betwee drive system (2.9 ad respose system (2.2 is e = x y. (2.2 I order to desig a adaptive sychroizatio cotroller for sie-fuctio Julia sets, the followig lemma [] is firstly itroduced. Lemma 2.2. Cosider a lier equatio x +2 x + + kx =, where x is complex. If the real k satisfies < k <, the the equatio root is stable as, that is, x. Theorem 2.2. If the sychroous cotroller of system (2.2 is desiged by ad if the parameter idetifier is u = si(x 2 si(y 2, (2.22 c + c = k 3 e, (2.23 where the costat k 3 satisfies < k 3 <, the drive system (2.9 ad respose system (2.2 with arbitrary iitial value ca achieve global asymptotic sychroizatio, ad the parameters of the sie-fuctio Julia set ca be idetified. Proof. If systems (2.9 ad (2.2 realize track sychroizatio with arbitrary iitial value, the the Julia sets of systems (2.9 ad (2.2 realize global asymptotic sychroizatio. We oly prove that systems (2.9 ad (2.2 realize track sychroizatio, that is, x y (. This is equivalet to that e (. From (2.9-(2.2, we obtai e + = x + y + Next, substitutig (2.22 ito (2.24, we have = si(x 2 + c si(y 2 c u. (2.24 e + = si(x 2 + c si(y 2 c si(x 2 + si(y 2. That is, e + = c c. (2.25

8 Sie-fuctio Julia sets 227 Hece, That is, e + e = c c (c c = c c = k 3 e. (2.26 e + e + k 3 e =. (2.27 If k 3 satisfies < k 3 <, the the equatio root is stable as accordig to Lemma 2.2, that is, e. So, the Julia sets of systems (2.9 ad (2.2 achieve global asymptotic sychroizatio with arbitrary iitial value. Example 2.2. Let c =.7 +.3i, c =, for drive system (2.9 ad respose system (2.2. After steps, the Julia set of drive system (2.9 ad respose system (2.2 are show i Figure 5, where k 3 =.. (a (b (c (d (e (f Figure 5. The Julia sets of drive system (2.9 after (a steps; ad the Julia sets of respose system (2.2 after (b 5 steps, (c 2 steps, (d 5 steps, (e 8 steps, (f steps; where k 3 =.. The figures show that respose system (2.2 sychroizes drive system (2.9 after steps. Besides, the idetificatio process of c of respose system (2.2 is show i Figure 6. The chagig process of the e betwee drive system (2.9 ad respose system (2.2 is show i Figure 7. From the simulatio results, the real ad imagiary parts of the c are stable at -.7 ad.3 i the iterative process ad the real ad imagiary parts of error e ted to zero. Hece, respose system (2.2 ad drive system (2.9 achieve sychroizatio, where the ukow c of drive system (2.9 has also bee idetified. As the value of k 3 icrease from to, the chagig process of c ad e are show i Figure 8 ad Figure 9. From the simulatio results, we kow that as the value of k 3 icrease from to.5, the idetificatio process of c is gettig better ad e teds to more quickly. Whe the value of k 3 cotiues to icrease from.5 to, the idetificatio process

9 228 J. Su, W. Qiao & S. Liu Re(c Im(c (a (b Figure 6. The (a real ad (b imagiary parts of c of system (2.2 chagig with, where k 3 = Re(e.8 Im(e (a (b Figure 7. The (a real ad (b imagiary parts of e betwee systems (2.9 ad (2.2 chagig with, where k 3 =...5 k=. k=.3 k=.5 k=.7 k= k=. k=.3 k=.5 k=.7 k=.9.4 Re(c.5 Im(c (a (b Figure 8. Idetificatio processes of (a real ad (b imagiary parts of c for differet k 3. of c is gettig worse, ad e teds to slower. Whe the value of k 3 is ear.3 the idetificatio effect of c is the best, ad the speed of e tedig to gets the fastest. Thus, the speed of the respose system ad the drive system achievig sychroizatio is the fastest. However, if k 3 =., the idetificatio process of e betwee drive system (2.9 ad respose system (2.2 is show i Figure. From Figure, we kow that e betwee the two sychroous systems teds to ifiity gradually. Therefore, sychroizatio of the two systems ca ot be

10 Sie-fuctio Julia sets k=. k=.3 k=.5 k=.7 k= k=. k=.3 k=.5 k=.7 k=.9.5. Re(e Im(e (a (b Figure 9. The (a real ad (b imagiary parts of e betwee systems (2.9 ad (2.2 chagig with for differet k Re(e 5 Im(e (a (b Figure. The (a real ad (b imagiary parts of e betwee systems (2.9 ad (2.2 chagig with, where k 3 =.. achieved ad the ukow c of drive system (2.9 ca ot be idetified. 3. Coclusios I this work, two ew methods are put forward to realize the sychroizatio cotrol of a drive-respose system ad parameter idetificatio of sie-fuctio Julia sets. The zero asymptotic property of the slidig variable of the discrete cotrol system ad the stability theory i differece equatios are applied to realize idetificatio ad cotrol of sie fuctio Julia sets. The, we successfully solved the problem of sychroizatio cotrol of the drive-respose system ad parameter idetificatio, i the case of the drive system havig ukow parameters. Meawhile, we desiged a adaptive sychroizatio cotroller ad parameter idetifier. These results are sigificat to future importat applicatios of Julia sets. Refereces [] M. R. Ashish, M. Rai ad R. Chugh, Julia sets ad Madelbrot sets i Noor orbit, Applied Mathematics ad Computatio, 228(24, [2] Y. T. Dig ad W. H. Jiag, Double Hopf bifurcatio ad chaos i Liu system

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