DYNAMICAL ANALYSIS OF AN NON-ŠIL NIKOV CHAOTIC SYSTEM

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A OF THE ROMANIAN ACADEMY Volume 9 Numer /8 pp. 8 6 DYNAMICAL ANALYSIS OF AN NON-ŠIL NIKOV CHAOTIC SYSTEM Kui Bio DENG Xio Bo WEI Yi Yn KONG Chun Li LI Liuzhou Voctionl & Technicl College Liuzhou 5456 Chin Hunn Institute of Science nd Technology College of Physics nd Electronics Yueyng 446 Chin E-mil: kuiio_deng@63.com A system with non-existence of Šil nikov sense chos ws constructed in this pper. To further understnd the complex dynmics of the system some sic ehviors such s the lrgest Lypunov exponent ifurction digrm Poincré mpping were reveled y rigorous numericl simultions. Interestingly the chotic ttrctors cn coexist with qusi-periodicity ttrctors in presented system which hs rrely een reported in three-dimensionl utonomous systems. Next three-dimensionl (3D) sphered ultimte ound nd positive invrint set were derived for the positive vlues of its prmeters Then the stility of non-hyperolic equilirium is investigted. Finlly the existence of singulrly degenerte heteroclinic cycles for suitle choice of the prmeters is studied. Key words: Šil nikov sense cho degenerte heteroclinic cycles coexisting ttrctor.. INTRODUCTION Since the now-chotic ttrctor ws presented y Lorenz in 963 [] chos phenomenon hs received more nd more ttention from scientific community menwhile mny more chotic systems were developed fter Lorenz system such s Rössler system [] Chu system [3] Chen system [4] Lü system [5] Liu system [6] the Lorenz system fmily [7] the conjugte Lorenz-type system [8] nd so on. The ove chotic systems re ll elong to hyperolic type of chotic systems. The hyperolic type of chotic systems re clled Šil nikov sense chos which exists heteroclinic orit or homoclinic orit. According to one of the commonly greed-upon nlytic criteri for proving chos in utonomous systems in [9] which requires the systems to hve t lest one unstle sddle-focus equilirium point. So the chotic system without sddlefocus type of equilirium is clled non-existence of Šil nikov sense chos. Although most chotic systems elong to Šil nikov chos there re still mny non-šil nikov chos. For instnce clss of systems without equilirium points were proposed in [ 4] Yng found chotic system with one sddle nd two stle node-foci [5] Yng lso introduced chotic system with two stle node-foci [6] Wng reported chotic system with only one stle node-focus equilirium [7] nd ctlog of chotic systems with line equilirium were presented y Jfri in [8]. Lter litertures [9 ] introduced new clss of chotic systems with circulr equilirium. Recently Li constructed simple chotic system with non-hyperolic equiliri []. More recently Chen nd Hn reported n interesting three-dimensionl qudrtic chotic system with two equilirium points of sddle-focus type ut it does not elong to Šil nikov type chos in view of the lgeric condition given y Elhdj []. It is very importnt to note tht Wei nd Yng introduced the generlized Sprott C system with only two stle equiliri which revels the influence of initil condition on the dynmics of system with the fixed prmeters vlues [3]. Therefore it is interesting to sk whether or not there is nother 3D system with non-existence of Šil nikov chos whose dynmicl ehvior is closely relted to initil vlue. This prolem will e investigted in the present pper. A thorough study of such kind of chotic systems my e eneficil to understnd the complicted mechnisms of chos. For such kind of chotic systems there seems to e no study on chotic encryption or decryption so it my e more security. Motivted y the ove works we introduce non-existence of Šil nikov existence sense chos with different types of equiliri nd initil vlues. The system cn generte doule-scroll chotic ttrctor which cn coexist with qusi-periodicity ttrctor. First of ll we use Lypunov stility theory to estimte the precise ound of this chotic system. In ddition non-chotic ehvior in the system ws discussed.

2 Dynmicl nlysis of n non-šil nikov chotic system 9 Moreover the stility of nonhyperolic equilirium is studied. At lst the existence of singulrly degenerte heteroclinic cycles for suitle choice of the prmeters is reveled.. AN NON-ŠIL NIKOV CHAOTIC SYSTEM The proposed chotic system is expressed s x = ( x y) y = xz () z = c+ xy z where dot denotes derivtive with respect to time t c re rel numers. It is esy to see tht system () is c glolly uniformly nd symptoticl stle out its equilirium point S = ( ) if > nd c <. Actully this is cn e demonstrted y constructing Lypunov function in the following form c V ( xyz) = x [ y + ( z ) ] c which leds to c V ( x yz) = [ x ( z ) ]. c (3) nd y setting { yz) V ( x yz) } c x = = ( x yz) x = z = y R. ( Which does not contin nontrivil trjectory of system (). We cn derive from the Krsnoselskii theorem tht system () is glolly uniformly nd symptoticlly stle out the equilirium S. This mens ( c ) > > c<. tht the system () is not chotic in the prmeter region { } ().. Coexisting ttrctors with different types of equiliri nd different initil vlues c If c > system () hs three equiliri S = ( ) S = ( c c ) + nd S - = (- c c ). When c c S = ( ) is unique equilirium. Although the system hs the simple form it cn disply complicted nd unusul dynmicl ehviors. Next we investigte complex dynmics of system () in different initil vlues. () When prmeter = = c = 5.5 it is clerly tht system () exists three equiliri S (5.5) S ± ( ± ) And the corresponding eigenvlues re λ (S ) = λ (S ) =.464 λ 3(S ) = nd λ (S ) =.9694 λ (S ) = i λ (S ) =.53.79i. ± ± 3 ± Oviously system () hs one sddle nd two stle node-foci equilir nd system () is not under the clss of Šil nikov sense chos. () For initil vlues (. 9 ) the ttrctor is qusi-periodicity. The Lypunov exponents of system () re L = L = L 3 = () System () displys chotic ehvior with initil vlues (.. ). The Lypunov exponents of system () re found to e L =.44 L = L3 = 3. 3.Thus system () cn disply chotic ttrctor. () With = = c = 6 it is esy to see the system () hs three equiliri S ( 6)

3 Kui Bio DENG Xio Bo WEI Yi Yn KONG Chun Li LI. 3 S ± ( ± 6 6). Whose corresponding chrcteristic vlue re λ( S) = λ( S) =.464 λ3( S) = nd λ ( S ) = 3 λ ( S ) =.884i λ ( S ) =.884i. ± ± 3 ± () Under initil conditions (.3 ) the Lypunov exponents of system () L = L = L3 = Thus system () is qusi-periodicity. () For the initil vlues (9.3 ) the Lypunov exponents of system (.) re found to e L =.5 L = L3 = Therefore system () indeed exists chotic ttrctor with sddle nd two non-hyperolic equilir. The corresponding view of ttrctor is displyed in Fig. nd Fig.. Fig. ) Qusi-periodicity ttrctor with initil vlues (. 9); ) chotic ttrctor with initil vlues (..) for = = c = 5.5. Fig. ) For = = c = 6 ) initil vlues (.3 ) the qusi-periodicity ttrctor of system (); ) initil vlues (9.3 ) chotic ttrctor of system (). Remrk. From Fig. nd Fig. it cn e found tht when we fix system prmeters nd chnge initil vlues the dynmics properties of the system mke lrge vritions. Of prticulr interest is the fct tht chos coexists with qusi-periodicity ttrctors. Remrk. Equilirium point plys n importnt role in their properties however the whole structure of chotic ttrctors is completely determined y it. Šil nikov criteriis is sufficient ut certinly not necessry for emergence of chos... Poincré mpping ifurction digrm Lypunov exponent spectrum The numericl fetures of the new chotic ttrctor cn e further illustrted y the Poincré mpping re shown in Fig. 3 nd Fig. 4 respectively.

4 4 Dynmicl nlysis of n non-šil nikov chotic system Fig. 3 ) For = = c = 5.5 initil vlues (..) projection on x-z Poincré mp; ) under = = c = 6 initil vlues (9.3 ) projection on x-y Poincré mp. Fig. 4 Prmeters = = nd c (5. 6.) under initil condition (..): ) ifurction digrm; ) the lrgest Lypunov exponent versus the prmeter c..3. Symmetry invrince nd dissiptivity It is esy to see tht system () is invrint under the coordintes trnsformtion ( x yz) ( x y z) tht is the system hs rottion symmetry round the z xis. The divergence of flow of the dynmic system () is x y z V = + + = (4) x y z Hence system () is dissiptive for + >. Tht is to sy volume element V() t = V exp(( )) t ecomes smller in time t. This mens tht ech volume contining the trjectories of the dynmicl system shrinks to zero s t t n exponentil rte. As result system orits finlly re restricted to suset whose volume is zero nd their symptoticl ehvior nd fixed on n ttrctor..4. The oundedness of the solutions of the system () For the oundedness of system () we derive the following result. THEOREM. Suppose tht > nd >. Then ll orits of system () re trpped in ounded region including chotic ttrctors. Proof. For ny one solution x y z of system () define the following Lypunov function V ( x yz) = [ x + y + ( z + ) ] (5) Computing the derivtive of V with respect to time t long the solution of system () we otin

5 Kui Bio DENG Xio Bo WEI Yi Yn KONG Chun Li LI. 5 V ( x yz) = x = x + ( c ) z z c ( z ) + c ( c + ) + 4 Let e > e so sufficiently lrge tht for ll ( x y z) stisfying V ( x y z) = e with e > e one hs. (6) Hence on the surfce { } tht the { x yz) V ( x yz) e} c ( c + ) x + ( z ) >. (7) 4 ( x yz ) V ( xyz ) = e with e> e one hs V ( x y) z < which indictes ( is trpped region of ll solutions of system (). This completes the proof. For the numericl simultion of theorem s shown in Fig. 5. Fig. 5 The orits or system () for prmeters = = c = 5.5 which oviously lie in ounded sphere..5. Non-chotic ehvior in the system () It is strightforwrd to show tht the whole structure of chotic ttrctors is not completely determined y the knowledge of fixed points nd their properties. We show here there re some nonchotic prmeter regions. This implies tht the system () cnnot hve chotic solutions in some cses. Next we prove the following Theorem. THEOREM. If one of the following conditions is stisfied () = () < c < + > (3) > c > + <. Then the system () is not chotic. Proof. It is oviously tht system () is not chotic under ssumption (). Then we only consider the cse. According to the first eqution of () we cn get the x = x y = x + xz (8) nd Clculting the derivtive of eqution (8) we hve Sustituting z = c + xy z into () one gets Multiply oth sides of () y x xz = x + x (9) x = x + xz + xz () x = x + xz + cx + x y xz () we get the eqution

6 6 Dynmicl nlysis of n non-šil nikov chotic system 3 3 xx = xx + xxz + cx + x y x z. () From x = ( x y) one derives x x y =. (3) Sustituting expression (9) nd (3) into expression () one otins 3 4 ( + ) xx xx + x x + xx + xx = x + cx x. (4) Integrting eqution (4) we hve + 4 t 4 ( + ) xx x + x + xx + x = [( + ) x + cx x ]dt+ C. (5) 4 Here C is constnt nd t >. The left side of polynomil equtions (5) cn e simplified to 3 ( + ) 4 ( x + x )( x + y) + ( + ) x z ( x + y) + x + x. (6) 4 When conditions () or (3) of theorem re stisfied the (6) is monotone s function of time nd hs limit L R s t if L is finite then ny ttrctor for the eqution lies on the surfce (6) nd is not chotic y virtue of the Poincreé Bendixson theorem. If L = ± then t lest one of the three vriles is not ounded nd system () is not chotic. 3. THE STABILITY OF NONHYPERBOLIC EQUILIBRIUM According to the Routh-Hurwitz stility criterion the stility of hyperolic equilirium is esily otined. So we only consider the stility of nonhyperolic equilirium. If c = the system hs unique equilirium S = ( ). The Jcoin mtrix of the system () t the origin S A = with the chrcteristic eqution λ( λ + )( λ + ) =. (7) Oviously Eq. (7) hs three roots λ λ = nd λ = with the corresponding eigenvectors eing = T T T ( ) () () respectively. THEOREM 3. Assume tht > > nd c = then nonhyperolic equilirium S is symptoticlly stle. Proof. From the ove discussion one sees tht the equilirium S is nonhyperolic with three eigenvlues nd. Next we will investigte the stility of S y using the center mnifold theorem [4]. Let us introduce the trnsformtion x x y = y z z Trnsform system () into the following form x x ( x y) z ( ) y = y + x y z (8) ( ) z z x y x According to the center mnifold theorem the stility of S cn e determined y studying fist-order ordinry differentil equtions on center mnifold which cn e represented s grph over the vriles x s ellow 3

7 4 Kui Bio DENG Xio Bo WEI Yi Yn KONG Chun Li LI. 7 W withδ sufficiently smll. Assume tht c ( ( x y z ) 3 R y = h ( x ) z = h ( x ) x ) = h () = h () = < δ h () = () = h Thus the center mnifold must stisfy y = h ( x ) = x + x + O( ) z = h ( x ) = x + x + O( ). (9) x x h ( x )( x h ( x ) h ( x ) + h ( x ) ( x h ( x ) h ( x ) = h ( x )( x h ( x ) h ( x ) + h ( x ) x ( h ( x ) x ) =. 3 Sustituting expression (9) into Eq.() nd equting the coefficients of x nd x on oth side one otins tht By Solving equtions () we find + = + = = () =. () = = = = () nd hence 4 4 h ( x) = x + O( x ) h ( x ) = x + O( x ). (3) Sustituting expression (3) into expressions (9) nd (8) one hs the vector field reduced to the center mnifold x = x x + O( x ). (4) Therefore one cn deduce tht when > the equilirium S is symptoticlly stle. For the conclusions of theorem 3 refer to the simultion result in Fig. 6. Fig. 6 The phse portrit of system () with = = c =. 4. SINGULARLY DEGENERATE HETEROCLINIC CYCLES For = c = the system () ecomes the following form x = ( x y) y = xz z = xy (5)

8 8 Dynmicl nlysis of n non-šil nikov chotic system 5 which hs the line of equiliri ( z) z R. Notice one is considering >. By linerizing system (5) t the equilirium point ( z) one otins the Jcoin mtrix J = z whose chrcteristic eqution is 3 λ + λ zλ =. (6) Therefore the eigenvlues λ = 4z ± + With the corresponding eigenvectors given s follows λ 3 =. (7) + 4z v = ( ) z v + + 4z = ( ) v 3 = ( ). (8) z Providing z < the eigenvlues λ re complex with the negtive rel prt. Considering lso the 4 corresponding eigenvectors (8) this mens tht the solutions loclly spirling towrd the equilirium point Q = ( z) on surfce tngent to the plne spnned y the eigenvectors v hence in direction norml to the z-xis. When < z < the eigenvlues λ re rel nd negtive. Therefore trjectories move towrd to z- 4 xis without spirling. If z > the eigenvlues λ re rel with opposite signs. Then tking into ccount the eigenvlues v the system hs normlly hyperolic sddle t the point P = ( z). In the specific cse in which z = the equilirium point () is more degenerted hving two vnishing eigenvlues. By ove nlysis nd crefully numericl study (see Fig. 7) of the solutions of system () with = c = nd > hs een performed which clerly revels tht the system proposes n infinite set of singulrly degenerte heteroclinic cycles. Ech one of these cycles is formed y one of the one-dimensionl unstle mnifolds of the sddle P which connects P with normlly hyperolic focus Q s t +. As the system presents n infinite numer of normlly hyperolic sddles P nd foci Q there exists n infinite set of singulrly degenerte heteroclinic cycles. In Fig. 7 some of them re shown: for ech initil condition considered sufficiently close to the sddle P t the z-xis singulrly degenerte heteroclinic cycle is creted. According to Fig. 7 we lso oserve tht the sddles P nd the stle focus Q extend to infinity on the negtive nd positive prts of z-xis. Fig. 7 Singulrly degenerte heteroclinic cycles of system () with ( c) = ( ): ) initil vlue (.7. 7.) nd (.7. 7.); ) initil vlues (.. z()) nd (.. z()) where z() {6.. 8.}.

9 6 Kui Bio DENG Xio Bo WEI Yi Yn KONG Chun Li LI CONLUSIONS An simple system with one sddle nd two stle node-foci equiliri or with one sddle nd two nonhyperolic equiliri tht coexists chos nd qusi-periodic torus hs indroduced in this pper. A significnt dynmics of this system is closely relted to initil conditions. Aundnt nd complex dynmicl ehvors hs een completely nd thoroughly investigted. We hope tht the finding discussed in this pper cn provide some light for further exploiting the dynmics of non-šil nikov chotic systems. REFERENCES. E.N. LORENZ Deterministic nonperiodic flow Journl of Atmospheric Science pp O.E. RÖSSLER An eqution for continuous chos Physics Letters A 57 pp L.O. CHUA M. KOMURO T. MASTSUMOTO The doule scroll fmily IEEE Trnsctions on Circuits nd Systems I 33 pp G.R. CHEN T. UETA Yet nother chotic ttrctor Interntionl Journl of Bifurction nd Chos 9 pp J. H. LÜ G.R. CHEN A new chotic ttrctor coined Interntionl Journl of Bifurction nd Chos pp C.X. LIU T.LIU L.LIU K. LIU A new chotic ttrctor Chos Solitons nd Frctls pp S. CELIKOVSKI G.R. CHEN On the generlized Lorenz cnonicl form Chos Solitons nd Frctls 6 pp Q.G. YANG G.R. CHEN T.S. ZHOU A united Lorenz-type system nd its cnonicl form Interntionl Journl of Bifurction nd Chos 6 pp C.P. ŠIL NIKOV A contriution to the prolem of the structure of n extended neighorhood of rough equilirium stte of sddle-focus type Mthemtics of the USSR-Sornik pp S. JAFARI J.C. SPROTT S.M.R.H. GOLPAYEGANI Elementry qudrtic chotic flows with no equiliri Physics Letters A 377 pp J.C. SPROTT A dynmicl system with strnge ttrctor nd invrint tori Physics Letters A 378 pp Z.C.WEI Dynmicl ehviors of chotic system with no equiliri Physics Letters A 376 pp V.T. PHAM C. VOLOS S. Jfri Z.C Wei X. Wng Constructing novel no-equilirium chotic system Interntionl Journl of Bifurction nd Chos 4 pp F.R. TAHIR S. JAFARI V.T. PHAM C. VOLOS X. WANG A novel no-equilirium chotic system with multiwing utterfly ttrctors Interntionl Journl of Bifurction nd Chos 5 pp Q.G. YANG G.R. CHEN A chotic system with one sddle nd two stle node-foci Interntionl Journl of Bifurction nd Chos 8 pp Q.G.YANG Z.C. WEI G.R. CHEN An unusul 3D utonomous qudrtic chtoic system with two stle node-foci Interntionl Journl of Bifurction nd Chos pp X. WANG G.R. CHEN A chotic system with only one stle equilirium Communictions in nonliner Science nd Numericl Simultion 7 pp S. JAFARI J.C. SPROTT Simple chotic flows with line equilirium Chos Solitons nd Frctls 57 pp T. GOTTHANS J. PETRZELA New clss of chotic systems with circulr equilirium Nonliner Dynmics 8 pp S.T. KINGNI V.T. PHAM S. JAFARI G.R. KOL P. WOAFO Three-dimensionl choticutonomous system with circulr equilirium: nlysis circuit implementtion nd its frctionl-order form Circuits System nd Signl Processing 35 pp C.L. LI J.B XIONG A simple chotic system with non-hyperolic equiliri Optik 8 pp B.Y. CHEN X.Z. HAN Anlysis of dynmicl ehviors in continuous chotic system without Šil nikov orits Interntionl Journl of Bifurction nd Chos 5 pp Z.C. WEI Q.G. YANG Dynmicl nlysis of the generlized Sprott C system with only two stle equiliri Nonliner Dynmics 68 pp S. WIGGINS Introduction to Applied Nonliner Dynmicl Systems nd Chos Springer-New York 99. Received Mrch 6 7