Deign of Compound Hyperchaotic Sytem with Application in Secure Data Tranmiion Sytem D. Chantov Key Word. Lyapunov exponent; hyperchaotic ytem; chaotic ynchronization; chaotic witching. Abtract. In thi paper an approach for deigning hyperchaotic ytem with two or more poitive Lyapunov exponent on the bai of low-order chaotic model i propoed. The motivation i that hyperchaotic ytem are preferable for ue in chaotic data protection ytem due to their more complex behaviour but very few uch model are known. The propoed hyperchaotic ytem i baed on the Van der Pol Duffing third-order chaotic model. After proving that the deigned ytem i hyperchaotic a table ynchronization cheme between two uch ytem i propoed. To facilitate the tability analyi a novel linear-nolinear decompoition ynchronization approach i ued. A chaotic witching ecure data tranmiion ytem i projected on the bai of the propoed hyperchaotic ytem. The imulation experiment with a binary data ignal how that the ignal can be reliably protected when it i maked with a hyperchaotic ignal on it way from the tranmitter to the receiver. I. Introduction The ytem with chaotic behaviour are a pecific cla of nonlinear ytem which are characterized with great enitivity to initial condition a poitive Lyapunov exponent and a trange attractor in the phae pace. Mot of the known chaotic model are fully determinitic ytem but their behaviour i random-like. Thi pecific feature of the chaotic ytem make them appropriate for implementation in different type of ecure data tranmiion ytem. The common principle of uch ytem i to build two chaotic ytem in the tranmitter and in the receiver and to deign a table ynchronization cheme between them. After the ytem are ynchronized the chaotic ynchronizing ignal can be ued to bear and hide the data ignal. The hyperchaotic ytem are imilar to the chaotic ytem with the difference that they poe two or more poitive Lyapunov exponent determining that the etting apart of the nearby orbit intrinic to the chaotic ytem take place in two or more direction of the phae pace. Thu the dynamic of the hyperchaotic ytem i more complex and if uch ytem are ued in chaotic data protection ytem a higher degree of data protection can be achieved. The problem i that very few hyperchaotic model are known and not all of them are uitable for uch purpoe. In thi paper a method for deigning hyperchaotic ytem with two or more poitive Lyapunov exponent i propoed. The method et on creating a chain of two or more low-order chaotic ytem coupled in uch way that their dynamic i hyperchaotic. A an example two Van der Pol Duffing thirdorder chaotic ytem are coupled uing the partial replacement ynchronization principle in uch way that the dynamic of the econd ytem i chaotic but it i different from that of the firt ytem and at the ame time it i dependent on it. Then the two coupled ytem can be viewed a one compound hyperchaotic ytem. To prove that the compound ixth-order ytem i hyperchaotic it Lyapunov exponent are calculated and viualized. A the main aim of creating compound hyperchaotic ytem i to ue them in data protection ytem a table ynchronization cheme between two compound Van der Pol Duffing hyperchaotic ytem i deigned. The ynchronization problem itelf i not an eay tak a no univeral ynchronization method uitable for all chaotic ytem exit. To facilitate the tability analyi which i crucial for chaotic ynchronization problem a novel linear-nonlinear decompoition technique with additional feedback coupling i propoed. In thi way the ynchronization tability i proved with a linear error ytem which i a great advantage over other known ynchronization method. After deigning a ynchronization cheme between two compound Van der Pol Duffing ytem and proving it tability a chaotic witching ecure data tranmiion ytem i deigned. The ecure tranmiion of a binary information ignal i imulated. The advantage of thi approach i that the information ignal i not directly tranmitted and the ole tranmitted ignal between the tranmitter and the receiver of the communication ytem i the chaotic ynchronization ignal. II. Deign of Compound Hyperchaotic Sytem on the Bai of Low-order Chaotic Model The Lyapunov exponent [] give a quantitative aement of the rate of chaoticity of a given nonlinear ytem. or a n-th order ytem n Lyapunov exponent exit which give the rate of eparation of infiniteimally cloe trajectorie in all n direction of the phae pace. The condition λ λ... λn hold for all chaotic ytem. If the maximum Lyapunov exponent λ = λmax > the ytem i chaotic. If there are two or more poitive Lyapunov exponent the ytem i called hyperchaotic and i characterized by more complex behavior. A all chaotic ytem ome poe poitive and negative Lyapunov exponent their behaviour in the phae pace i characterized by a trange attractor formed of a virtually infinite number of untable periodic orbit confined cloe to each other. Every two nearby orbit eparate exponentially but after a finite time interval they again come nearer. A typical chaotic attractor 8 9
(Lorenz attractor []) i hown on figure a. igure b how the principle of eparation of nearby orbit in the i-th direction of the phae pace correponding to the poitive Lyapunov exponent λ max. The phae pace of a third-order chaotic ytem with three different initial condition confined in an infiniteimal phere with diameter d i hown on figure c. x according to the PR method replace it correponding variable x t only at one poition in the Slave ytem model. By the ynchronization problem the tak i to find uch coupling that: lim e ( = lim x ( = () ( ) where e ( i called error or difference function. x λ i > t t x i-th direction of the phae pace x d t igure. a the Lorenz chaotic attractor; b i-th direction of the phae pace correponding to λ i = λ max > ; c eparation of the nearby orbit of third-order chaotic ytem Due to the exponential eparation of the nearby trajectorie the initial condition phere with diameter d change it hape to an ellipoid (for third-order model) with diameter d i along the principal axe. The i-th Lyapunov exponent correponding to the i-th direction i then calculated by: di ( () λ i = lim log. t d ( t ) Mot of the known model of chaotic ytem are of thirdorder i.e. with one poitive Lyapunov exponent. Very few higherorder chaotic ytem with two or more poitive Lyapunov exponent are known. Then if for a given problem uually in the field of ecure communication with chaotic data protection a ytem with two or more poitive Lyapunov exponent i needed to achieve a higher level of ecurity and the available hyperchaotic model can not be ued for ome reaon one can try to artificially yntheize a hyperchaotic ytem built from two or more low-order chaotic model. Without lo of generality only the cae for yntheizing a hyperchaotic ytem from two identical low-order chaotic model will be conidered. In chaotic ynchronization problem the tak i to find a proper coupling between two (identical) chaotic ytem uch that the error function between them tend to zero and they began to evolve identically in the phae pace even when they are tarted from different initial condition. Uually the coupling i unidirectional and by the partial replacement ynchronization method (PR method) [] the ytem are written in the form: () Mater x & = f ( x Slave x = f( x x i n n & where x R and x R are the tate vector of the two ytem called Mater and Slave repectively and ( x x i f ( x f =. The Mater ytem variable i If one can find a coupling with which the ytem are not ynchronized and the dynamic of the Slave ytem i chaotic but i ubordinated to the Mater ytem and i different from the baic ytem chaotic dynamic one can aume the two coupled ytem () a one higher-order chaotic model. To prove if it i hyperchaotic one mut find the Lyapunov exponent of the combined n-th order ytem called a compound chaotic ytem. A chaotic ytem are nonlinear ytem no univeral chaotic ynchronization method exit and the ynchronization tak i individually olved for every particular pair of chaotic ytem. The PR method offer different poibilitie to couple the two ytem and can be further expanded with uing not one but two or more variable for the coupling or with uing the coupling variable to ubtitute it counterpart not at one but at two or more poition in the Slave ytem model. The ame hold when the tak i to deign a compound chaotic ytem one mut ytematically tet different type of coupling between the two ytem and try to find uch which will guarantee hyperchaotic behaviour of the compound ytem. No common rule for deigning the coupling can be formulated neither for ynchronization problem nor for one for deigning compound chaotic ytem. That i why thee tak are olved individually each time for a given chaotic model and for a particular problem. III. Compound Hyperchaotic Sytem Baed on the Van der Pol Duffing Model The Van der Pol Duffing (VPD) ytem i a model of a chaotic electrical ocillator [67] which i decribed with the differential equation: 9 9
λ λ λ igure. VPD ytem: a chaotic attractor; b time erie of x ( ) ; c Lyapunov exponent t () ( x = β x = x x àx x ) x where ν = à =. β =. A thi model will be ued only a a reliable third-order chao generator the principal electrical cheme and the phyical equivalent of the tate variable and ytem parameter will not be dicued here. The VPD ytem chaotic attractor obtained by imulation with MATLAB i hown on figure a. igure b how the typical chaotic time erie of one of the tate variable the behaviour of the other two variable i imilar. The Lyapunov exponent are calculated according to Eq. () with a MATLAB program from [8] and are viualized on figure c. The mean value of the Lyapunov exponent for calculated after excluding the initial tranient proce are λ = λmax =. 7 λ =. and λ =.9667. To deign a compound ytem with hyperchaotic behaviour different coupling combination between two VPD ytem according to the partial replacement principle () are examined. After ome reearch and auming that the variable of the Slave ytem are denoted with: [ x x ] T [ x x x ] T x = = x 6 compound VPD ytem with hyperchaotic dynamic wa deigned: () 6 ( x = β x ( x = x = β x = x x x x x 6 àx x ) à x x ) the following where the coupling variable i x in the firt equation of the Slave ytem model. Thi ynchronization cheme doe not yield ynchronization i.e. the ynchronization condition () doe not hold. Optionally the parameter of the econd ubytem of () may not be identical to thoe of the firt ubytem and for the experiment they are changed to: ν = à =. β =. The attractor of the econd ubytem in the phae ubpace ( x x x6 ) i hown on figure a. It i obviou that thi attractor i different from the baic VPD attractor from figure. a. In thi way by applying a proper coupling cheme the econd ubytem change it behaviour. It i again chaotic but of different type and i dependent on the firt ubytem. Thu one can view the two coupled ytem a one compound ixth-order chaotic ytem with complex behaviour. To give a better glance at the new 6-th order attractor it projection in two of ubpace of the combined phae pace are hown on figure b and c. The error ytem between the two ubytem of Eq. () i alo chaotic figure d how the attractor in the error phae pace where: ei = xi xi. The dynamic of the calculated Lyapunov exponent of the compound VPD ytem i hown on figure e. The mean value of the Lyapunov exponent for calculated after excluding the initial tranient proce are λ = λmax =. 8 λ =. λ =. 67 λ =. 98 λ =.78 and λ 6 =. 87 x ( ) =...... or better viualization purpoe the dynamic of λ and λ 6 i not hown on the figure. It i clear that the compound VPD ytem ha two poitive Lyapunov exponent which prove that it i hyperchaotic. To give better repreentativene of the reult further experiment with changing the imulation tep time olver type and initial condition are conducted. The reult how that the maximum two Lyapunov exponent alway remain poitive. Experiment with mot of the other poible coupling PR cheme how that the two ytem either ynchronize or the econd ytem become unchaotic i.e. the compound ytem i not hyperchaotic and poee only one poitive Lyapunov exponent. Thereby by deigning and teting the coupling cheme () and proving that it poee two poitive Lyapunov exponent it equation can be aumed a a reliable hyperchao generator which can be ued in data protection ytem with chaotic making of the information ignal. To do thi firt a table and. The initial condition are: [ ] T 9
λ λ λ λ d. e. igure. Compound VPD ytem: a attractor of the econd ubytem; b attractor in the x x ) ubpace; ( x6 c attractor in the ( x x x ) ubpace; d attractor of the error ytem; e dynamic of Lyapunov exponent fat ynchronization cheme between two hyperchaotic ytem ha to be deigned. IV. Synchronization between Two Compound VPD Sytem with Modified Linear-nonlinear Decompoition Method To achieve table ynchronization between two chaotic (or hyperchaotic) ytem a proper coupling between the ytem ha to be deigned. The problem i that no univeral ynchronization method exit and for every particular ynchronization problem one ha to conitently tet variou method with plenty of coupling variant. One further problem concern the proving of ynchronization tability which in ome cae e.g. in chaotic communication ytem i a crucial tak. In [9] a chaotic ynchronization method with proving the ynchronization tability from a linear error ytem i propoed. Thi method aume that a chaotic ytem can be decompoed to a linear and a nonlinear part in the following way: (6) x &( = f ( x = g( ) + h( where g ( ) = A i the linear part of the ytem and h( i the nonlinear part. If (6) i the Mater ytem in a chaotic ynchronization cheme the Slave ytem i deigned a a copy of the linear part of Eq. (6) which i driven by the nonlinear part h ( : (7) x & ( = f ( x x = g( ) + h( t ) where g ( x ( ) = A x ( i the linear part of the Slave ytem. The error ytem of the ynchronization cheme i obtained by ubtracting Eq. (7) from Eq. (6): (8) e &( = ( ( = A( x ( ) = Àe(. It i evident that the error ytem i linear. To proof the ynchronization tability one ha only to find the eigenvalue of the contant A matrix which i a far eaier tak than proving the ynchronization tability from a nonlinear error ytem a i the uual cae with the mot of the known ynchronization method. If all eigenvalue are with negative real part according to the tability criterion for linear ytem the error ytem i table and uch i the ynchronization cheme []. The main limitation of thi method which will be referred a LN method further in thi article i that for a particular chaotic ytem there i only one variant for ytem decompoition (to a linear and nonlinear par and for mot of the known chaotic model thi decompoition doe not yield table ynchronization. To correct thi limitation and at the ame time to retain the main advantage of the LN method of obtaining a linear error ytem the following modified linear-nonlinear decompoition ynchronization method i propoed. Let the Mater and the Slave ytem of a ynchronization cheme are defined with the equation: (9) x &( = À + A + h( ; () x & ( = A ( ) ( ) ( ( ) ) ( ( ) x t + A x t + h x t t + α Å x t x ( ) where the linear part of the Mater ytem i further decompoed to two part decribed with the A and the A matrice and the A -part i alo ued for the coupling. urthermore an additional linear feedback-type coupling with coupling gain α and coupling matrix E i introduced in the Slave ytem. The E matrix contain only zero and one and determine the exact 9
place of introduction of the feedback coupling in the Slave ytem model and which tate variable are ued for the coupling. The error ytem obtained by ubtracting Eq. () from Eq. (9) i: () e& ( = ( ( = ( À α E) e(. Apparently the error ytem i linear and the tability of the ynchronization cheme (9) () i proved by the eigenvalue of the contant ( À α E) matrix i.e. the main advantage of the LN method i retained. At the ame time for every particular chaotic ytem which can be initially decompoed to a linear and nonlinear part the modified method allow to deign and tet a large number of coupling cheme. Thu by conitently examining variou additional decompoition of the linear part and variou feedback coupling one can poibly find the bet olution for the given ynchronization problem. The teting of the modified method on ome of the mot popular chaotic model (Chua [] Hide [] Roler [] Shimiru&Morioka [] and other ytem) how that for every chaotic model the modified method offer many coupling which guarantee table ynchronization. Moreover for each of the above mentioned ytem the application of the modified method give ome cheme with fater ynchronization than the fatet poible ynchronization achieved by the application of the mot popular ynchronization method. In the cae of ynchronization between two compound hyperchaotic VPD ytem the baic LN method i examined firt. Eq. () i conidered to decribe the Mater ytem of the ynchronization cheme. It can be decompoed to linear and nonlinear part according to Eq. (6) with: () va g( ) = A = va a v b v b x x x x x x6 ; x h ( t ) t ) = x The eigenvalue of the A matrix however are: ρ = 7. ρ = ρ =. ±.6 j ρ 6 =.6 ± 6.7 j. A ρ i poitive a ynchronization cheme of the type of Eq. (6) and (7) between two compound VPD ytem could not yield table ynchronization. The imulation with MATLAB confirm that concluion. A the baic LN method could not be ued to ynchronize to compound VPD ytem many different coupling cheme with the application of the modified linear-nonlinear decompoition method (Eq. (9) and ()) are teted. The aim i to find a cheme with the hortet poible tranient proce before complete ynchronization between the two ytem i achieved in order to make poible a fater data tranmiion rate when thi ynchronization cheme i ued in a data protection ytem. After ytematically teting the different additional decompoition of the linear part according to Eq. (9) the different feedback coupling according to Eq. () and the combination between them the following ynchronization cheme with fat. ynchronization wa found: () Mater ( x àx x ) = x x x = β x ( x àx x ) Slave = x x x6 = β x 6 ( x àx x ) ( = x x x + α x x ) = β x ( x à x x) = x x x6 = β x 6 where the A matrix from Eq. (9) which pecifie the additional linear coupling contain only one nonzero element A =ν.a the E matrix from Eq. () which pecifie the feedback coupling contain alo only one nonzero element E = and the feedback gain i α =. Chooing the right value of the feedback gain i a delicate problem a uually the increaing of the value of α lead to fater ynchronization. However a can be een from Eq. () the coupling concept i uch that the eventual noie added to the driving ignal from the Mater ytem on it way to the Slave ytem i increaed α-time and then i fed into the Slave ytem. So large value for α can diturb the ynchronization or even the two ytem can deynchronize. After conitently teting different value for α and after taking into conideration the above mentioned limitation it wa found that the optimum value for the coupling gain i α =. The ( À α E) matrix for the coupling cheme () i: () v α À b α E) = v a a v b ( with eigenvalue: ρ ρ =. ±.6 j ρ 6 =. ± 9. j. The cae when the real part of the maximum eigenvalue i equal to zero correpond to the o called Marginal ynchronization []. Thi pecial type of ynchronization i characterized in general by: lim e ( = c i i where c i are contant depending on initial condition. Uually not all error function are contant when marginal ynchronization occur. It i poible ome of the error function to be equal to zero a with identical ynchronization lim ( = or lim ( = l( e j e k where l( i a non-chaotic function. The fact that the ynchronization cheme () yield marginal intead of identical ynchronization i not a problem a can be een in the following ection. The experiment how that for thi particular cheme lim e ( = the error function e ( e ( and e ( tend to different contant after the tranient proce and e ( and e 6( t ) are non-chaotic function. A the ynchronization cheme () ue only the x x and x Mater ytem variable for the coupling only the error function e ( e ( t ) and e ( are of interet a only they are acceible for obervation in the Slave ytem. The dynamic 9
igure. Synchronization between compound VPD ytem: a error function e ( = x ( ; b error function e = x ( x ( ) ; c error function e = x ( x ( ) ( t of thee error function i hown on figure. The tranient proce of e ( and e ( i about. imulation econd. or comparion purpoe ome of the mot popular ynchronization method a the Pecora&Carroll method [] the partial replacement method [] and the tandard feedbackbaed method [78] are alo teted over the compound VPD ytem. The fatet poible ynchronization achieved by the firt two method i with tranient of about namely time lower than the cheme (). The fatet achieved ynchronization with the feedback method i with about tranient i.e. % lower than (). To give a better view of the ynchronization behaviour between the Mater and the Slave ytem the dynamic of the pair x ) ( t x ( x ( and x ( x ( i hown on figure. ( t are deigned. Some of them are completely oftware-baed and are intended for data protection in local area network or in Internet [] other are for communication over radio-channel with chaotic ytem built a electronic circuit [] third are for very fat (in the gigabit per econd rate) communication over commercial fibre-optic channel where the chaotic ytem are emiconductor laer []. The common principle of all chaotic-baed ecure communication ytem i to build ynchronized chaotic ytem in the tranmitter and in the receiver and to ue the noie-like chaotic ynchronization ignal to hide the tranmitted information ignal. The chaotic making technique implie the direct addition of the information ignal to the chaotic ynchronization ignal but it could not alway guarantee the deired rate of data protection []. The chaotic modulation technique i imilar to the chaotic making but the information ignal i added not igure. Synchronization between compound VPD ytem: a time erie of x ) ( t ; b time erie of x x ( ) ; c time erie of x x ( ) ( t ( t V. Secure Data Tranmiion with Chaotic Switching on the Bai of Synchronized Compound VPD Sytem Short after the firt chaotic ynchronization cheme were propoed [] attempt to ue thi phenomenon in ecure communication ytem were made. Several completely different approache for data protection with chaotic ignal are propoed o far e.g. chaotic making [9] chaotic modulation [] and chaotic witching [8]. Alo practical realization of chaotic data protection ytem of completely different apect directly to the ynchronization ignal but i omehow integrated in the Mater ytem model []. The leat popular ynchronization technique i the chaotic witching but with proper tuning it can guarantee a very high level of data protection. The core of the chaotic witching data protection method i in encoding a binary information ignal in term of different attractor of the Mater ytem model which exit for different value of ome of the ytem parameter(). In i aumed that the zero of the information ignal will correpond to the value p i of the choen parameter and the one to the value p i (or contrariwie). If the value for p i and p i are carefully 9
choen the moment of witching between the two attractor remain undetected in the time erie of the coupling chaotic ignal() between the Mater and the Slave ytem. The value of the correponding parameter in the Slave ytem i equal to p i during the whole data tranmiion proce. Then the Mater and the Slave ytem are ynchronized when a zero i tranmitted and deynchronized when a one i tranmitted (to achieve ynchronization the parameter of the two ytem mut coincide with ome tolerance within -% maximum difference). A the ole acceible ignal between the Mater and the Slave ytem repectively between the tranmitter and the receiver of the communication ytem are the chaotic coupling ignal the information ignal i reliably protected. On the other hand a after achieving identical ynchronization the Slave ytem i in fact an oberver-ytem of the Mater ytem After ome reearch with the compound hyperchaotic VPD ytem the parameter β i elected for the chaotic witching procedure. The value for p i which will correpond to the zero of the binary ignal i choen to coincide with the nominal value of the parameter β =. The elected value for p i i β =. irt the ynchronization cheme i imulated with β = in the Mater ytem and β = in the Slave ytem to enure that the two ytem will deynchronize when tranmitting. The error function e ( and e ( which are of main interet for the particular compound VPD ynchronization cheme are hown on figure 6a and 6b. It i obviou that the two ytem are not ynchronized. At the ame time the ytem igure 6. Lack of ynchronization for b = : a error function e ( ) ; b error function e ( ) ; c projection of the hyperchaotic attractor in the ( x x x) ubpace t t ( x i ( = xi ( ) the obervation of one of the acceible error function e i ( in the receiver allow the recovering of the information ignal. Thu by deigning a imple filter in the receiver to track the change in the behaviour of the error function the information ignal can be eaily recovered. Some requirement have to be kept when deigning a chaotic witching data protection ytem:. A table chaotic ynchronization cheme ha to be deigned. A the proving of ynchronization tability i not alway an eay tak the modified linear-nonlinear decompoition approach which facilitate the tability analyi can be ued.. The deigned ynchronization cheme hould be of hort tranient to allow fater data tranmiion.. The two value of the parameter p i mut be elected in uch way that the witching between the attractor to remain undetected from the tranition between the different untable periodic orbit within the baic chaotic attractor.. It i important to chooe carefully the duration of a ingle-bit tranmiion t BIT. To make the principle of chaotic witching poible t BIT mut be greater than the ynchronization tranient t TR to allow the ytem to ynchronize when tranmitting the zero of the information ignal. attractor for β = occupie the ame area of the tate pace a the baic attractor for β =. Thi can be een by comparing the projection of the attractor of the compound VPD ytem for β = in the tate ubpace ( x x x) hown on figure 6c with the ame projection of the baic ytem for β = hown on figure a. Thi hold for all other projection of the 6-th order tate pace of the compound VPD ytem. The attractor are almot the ame yet the ytem deynchronize. The tranition between the attractor when the chaotic witching technique i applied will therefore remain undetected from the tranition between the untable periodic orbit of the baic compound VPD attractor. After electing the proper value for β the duration of a ingle-bit tranmiion ha to be pecified. A the ynchronization tranient for the cheme () i about. t BIT mut be greater than thi value. To make thing more clearer t BIT i et at. It may eem that for tranmiion of a ingle bit i an enormou time but thi i imulation time. Moreover parameter-caling technique exit with which the peed of a ytem evolution repectively the ynchronization tranient can be ignificantly hortened. Let the binary equence of the information ignal begin with:.... It modulate the parameter β a decribed above. In order to tart the chaotic witching 9
igure 7. Chaotic witching: a modulated information ignal; b error function e ( ) ; c error function e ( t procedure the two ytem are firt ynchronized and then the modulator in the tranmitter begin to alter the choen parameter in term of the binary ignal. To recover the binary ignal in the receiver a imple filter to track the change in one of the acceible error function i needed. The binary ignal encoded with the two value of the parameter β i hown on figure 7a. igure 7b how the error function e ( t ) oberved at the receiver. The time erie of the error function i divided to equal-length window. The length of the window i t BIT. If in a particular window the error function i chaotic the tranmitted bit i. If in a window the error function tend to zero after a tranient of. the tranmitted bit i. In the cae of everal conecutive zero the ynchronization i retained until the next (window -6 -). One very important feature of the chaotic ynchronization phenomenon which make the chaotic witching technique poible i that the length of the tranient doe not depend on initial condition. Thi enure that after each - tranition the tranient will be exactly.. A during the tranition the behaviour of the error function i chaotic after each - witching the ynchronization i tarted with different initial condition. igure 7c how the error function e (. It can alo be ued for ignal recovering but a the ynchronization cheme i with marginal ynchronization and normally thi error function tend to a nonzero contant (figure. a) the recovering procedure i lightly different. The one are clearly identifiable by the chaotic behaviour of the error function. A marginal ynchronization depend on the initial condition after each tranition from to the error function e ( i tabilized at different contant. However the recontruction of the information ignal i alo poible from thi error function only the filter mut be lightly different. To prove that the information ignal i reliably protected on it way to the receiver the time erie of the ynchronization variable x ( t ) x ( and x ( during the data tranmiion with chaotic witching are compared with thoe of a normal ynchronization procedure without data tranmiion. The reult are hown on figure 8. Although the time erie of the coupling variable are different for ynchronization with and without data tranmiion the witching moment are completely unidentifiable by the obervation of the time erie of d. e. f. igure 8. Comparion of the coupling chaotic ignal with and without chaotic witching: a c e x i ( without data tranmiion; b d f x i ( with data tranmiion 9
coupling variable x i (. Thi i achieved by the proper election of the value of the modulated parameter uch that the tranition between the attractor are imilar to the tranition between the trajectorie within the attractor. VI. Concluion The contribution of the preent work can be formulated a follow:. A method for deign of high-order compound chaotic ytem with hyperchaotic behaviour i propoed. A very few model of hyperchaotic ytem are known and ince on the other hand thee ytem poe more complex behaviour than the regular chaotic ytem with only one poitive Lyapunov exponent which can be of advantage in chaotic data protection ytem the propoed approach offer an eay tool for artificial deign of hyperchaotic ytem.. A novel linear-nonlinear decompoition ynchronization technique i propoed which facilitate the tability analyi and offer a great variety of coupling variant for each particular chaotic model. The econd advantage of the propoed method i that it allow to deign fater ynchronization cheme compared to the variant of the mot of the known method. The only limitation of the method concern the poibility of a particular chaotic model to be decompoed to linear and nonlinear part.. A principle chaotic witching ecure data tranfer ytem baed on compound hyperchaotic VPD ytem i deigned. The witching parameter and it two value are choen in uch way that a reliable protection of the data ignal i aured. Reference. Tuffilaro N. T. Abbot J. Reilly. An Experimental Approach to Nonlinear Dynamic and Chao. Addion-Weley 99.. Panchev S. Chao Theory. Prof. Marin Drinov Sofia 996 (in Bulgarian).. Lorenz E. Determinitic Nonperiodic low. Journal of the Atmopheric Science 96 -.. Guemez J. M. Matia. Modified Method for Synchronizing and Cacading Chaotic Sytem. Phyical Review E 99-8.. Pecora L. et. al. undamental of Synchronization in Chaotic Sytem Concept and Application. Chao 7 () 997 -. 6. Gome M. G. King. Bitable Chao. II. Bifurcation Analiy. Phyical Review A 6 99 No.6 -. 7. King. G. T. Saito. Bitable Chao. I. Unfolding the Cup. Phyical Review A 6 99 No.6 9-99. 8. Govorukhin V. Calculation Lyapunov Exponent for ODE. MATLAB Central ile Exchange http://www.mathwork.com/matlabcentral/ fileexchange/. 9. Yu H. L. Yanzhu. Chaotic Synchronization Baed on Stability Criterion of Linear Sytem. Phyic Letter A Iue 9-98.. Guemez J. C. Martin M. Matia. Approach to the Chaotic Synchronized State of ome Driving Method. Phyical Review E 997 No. -.. Chua L. C. Wu A. Huang G. Zhong. A Univeral Circuit for Studying and Generating Chao Part I: Route to Chao. IEEE Tranaction on Circuit and Sytem-I 99 No. 7-7.. Hide R. A. Skeldon D. Acheon. A Study of Two Novel Self-exciting Single-dik Homopolar Dynamo: Theory. Proc. R. Soc. London A 69-9.. Roler O. An Equation for Continuou Chao. Phyic Letter 7A 976 No. 97-98.. Shimizu T. N. Morioka. On the Bifurcation of Symmetric Limit Cycle to an Aymmetric one in a Simple Model. Phyic Letter 76A 98 -.. Shahverdiev E. Marginal Hyperchao Synchronization with a Single Driving Variable. arxiv:chao-dyn/988 998 -. 6. Pecora L. T. Carroll. Synchronization in Chaotic Sytem. Phyical Review Letter 6 99 No.8 8-8. 7. Boccaletti S. et. al. The Synchronization of Chaotic Sytem. Phyic Report 66 -. 8. Ogorzalek M. Taming Chao Part I: Synchronization. IEEE Tranaction on Circuit and Sytem-I 99 No. 69-699. 9. Cuomo K. A. Oppenheim. Circuit Implementation of Synchronized Chao with Application to Communication. Phyical Review Letter 7 99 No. 6-68.. He R. P. Vaidya. Implementation of Chaotic Cryptography with Chaotic Synchronization. Phyical Review E 7 998 No. -.. Chen G. et.al. Chao-Baed Still Image Encryption; Chao-Baed Secure Voice Communication; Chao-Baed Optical Communication Sytem Centre for Chao Control and Synchronization Hong Kong http://www.ee.cityu.edu.hk/ccc.. Argyri A. et.al. Chao-baed Communication at High Bit Rate Uing Commercial ibre-optic Link. Nature 7 Nov. -6.. Guojie H.. Zhengjin M. Ruiling. Choen Ciphertext Attack on Chao Communication Baed on Chaotic Synchronization. IEEE Tranaction on Circuit and Sytem-I No. 7-79.. Kocarev L. U. Parlitz. General Approach for Chaotic Synchronization with Application to Communication. Phyical Review Letter 7 99 No. 8-. Manucript received on 7..8 Dragomir Chantov wa born in 97 in Sofia Bulgaria. He graduated the Technical Univerity of Gabrovo Bulgaria peciality Sytem and Control and received M.Sc. degree in 996. In 6 he received Ph.D. degree in the field of Automation Sytem and Control. rom 998 he work in the Technical Univerity of Gabrovo Department of Automation Information and Control Technic a aitant prof. and enior aitant prof. Hi reearch interet are in the field of nonlinear control ytem and in particular in ytem with chaotic behaviour. Contact: Technucal Univerity of Gabrovo Department of Automation Information and Control Technic Hadzhi Dimitar Str. Gabrovo Bulgaria e-mail:dchantov@yahoo.com 6 9