Analytical Hopf Bifurcation and Stability Analysis of T System

Transcription

1 Commun. Theor. Phys Vol. 55, No. 4, April 15, 011 Anlyticl Hopf Bifurction nd Stbility Anlysis of T System Robert A. Vn Gorder nd S. Roy Choudhury Deprtment of Mthemtics University of Centrl Florid, P.O. Box , Orlndo, FL , USA Received November 15, 010 Abstrct Complex dynmics re studied in the T system, three-dimensionl utonomous nonliner system. In prticulr, we perform n extended Hopf bifurction nlysis of the system. The periodic orbit immeditely following the Hopf bifurction is constructed nlyticlly for the T system using the method of multiple scles, nd the stbility of such orbits is nlyzed. Such nlyticl results complement the numericl results present in the literture. The nlyticl results in the post-bifurction regime re verified nd extended vi numericl simultions, s well s by the use of stndrd power spectr, utocorreltion functions, nd frctl dimensions dignostics. We find tht the T system exhibits interesting behviors in mny prmeter regimes. PACS numbers: , 0.30.Oz, 0.70.Wz Key words: extended Hopf bifurction nlysis, method of multiple scles, T system, stbility nlysis 1 Introduction In this pper, we study Hopf bifurctions nd the resulting post-bifurction dynmics in the T system, three-dimensionl utonomous nonliner system introduced recently by Tign [1 ] which hs potentil ppliction in secure communictions. In prticulr, systems with sensitivity to the initil conditions cn prove useful in secure communictions see, e.g., Refs. [3 6] nd the references therein for discussion of the pplictions for such systems. The study of the T system itself hs been n re of interest in the recent literture, s it is three-dimensionl system, which exhibits wide vriety of behviors in vrious prmeter regimes. To this end, we remrk tht, very recently, Tign nd Dumitru [10] nd Tign nd Constntinescu [11] hve studied the heteroclinic orbits in the T system, nd tht their nlysis prllels our own in the heteroclinic cse. The present uthors hve recently generlized this nlysis nd extended it to homoclinic orbits nd the resulting Shil nikov-type chos, [1] s well s employed the new technique of competitive modes nlysis to identify nd ctegorize possible chotic prmeter regimes in the multi-prmeter spce for this system. [13] The present pper cn thus be viewed s complementry to these studies. Here we perform generlized Hopf bifurction nlysis [7 8] of the system. An nlysis of the T system in this light ws previously performed by Jing et l. in [9], who show tht Hopf bifurction does occur in the T system, while lso providing some numericl results. In the present pper, such results re extended, s we derive nlyticl expressions for the periodic orbits resulting from E-mil: rv@knights.ucf.edu E-mil: choudhur@cs.ucf.edu c 011 Chinese Physicl Society nd IOP Publishing Ltd this Hopf bifurction by employing the method of multiple scles. Furthermore, we then nlyze the stbility of the periodic orbits. The T system is n excellent exmple of reltively simple model exhibiting rich vriety of behviors, due to the three free model prmeters. In ddition to the bove mentioned works, Wu et l., [14] Li et l., [15] nd Yong et l. [16] hve ll studied the T system. Due to the forementioned potentil ppliction in secure communiction systems, mongst others, such models serve s gret motivtion for future work in the re of dynmicl systems nd chos. Formultion nd Locl Stbility The T system is given by ẋ = y x, ẏ = c x xz, ż = xy bz, 1, b, nd c re rel vlued prmeters see [1 ]. Jing et l. [9] show tht if b 0, b/c < 0 then 1 hs only one equilibrium, E 0 = 0, 0, 0 while if b/c > 0 then 1 hs the three equilibri E 0 = 0, 0, 0, E + = x 0, y 0, z 0, nd E = x 0, y 0, z 0, x 0 = y 0 = b/c nd z 0 = c /. Furthermore, when b = 0, then 1 hs only one equilibrium, E 0 = 0, 0, 0. The locl stbility nlysis ws considered in [1] [13], [10] nd else. We summrize some of the key points here, s they will be useful in our computtions. The Jcobin mtrices for the T system 1 t ech of E 0 nd E ± re given by 0 DF E0 = c b

2 610 Communictions in Theoreticl Physics Vol DF E± = c z 0 0 x 0 ±x 0 ±x 0 b, respectively we hve used the fct tht x 0 = y 0. The chrcteristic polynomils re then J E0 λ = detλi DF E0 = λ + bλ + λ c, 3 J E± λ = detλi DF E± = λ bλ + bcλ + bc. 4 Regrding stbility of the equilibrium E 0, from 3 the three relevnt eigenvlues re given by λ 0 = b, λ ± = ± 1 + 4c. 5 As mentioned in Jing et l., [9] the equilibrium E 0 is symptoticlly stble if > 0, b > 0, nd c <, while E 0 is unstble if < 0 or b < 0 or c > > 0. When sgn = sgnb nd 4c < 3, we find tht λ 0 R nd λ ± re complex conjugtes, so E 0 is sddle focus we shll mke prticulr use of this lter. When < 0, b < 0, nd 3/4 > c >, we find tht λ 0 > 0, λ ± > 0 so E 0 is n unstble node. Finlly when < 0, b < 0, nd c <, λ 0 > 0, λ + > 0, nd λ < 0, thus E 0 is sddle node. Regrding stbility of the equilibr E ±, let us mke the chnge of vribles η = λ + 1/3 + b. Then, the chrcteristic polynomil for the equilibri E ± is trnsformed to J E± η = η 3 + bc b η b3 + bc 1 3 bc + b = η 3 + pη + q. 6 Define the quntity by q p 3. = When > 0, J E± η = 0 hs the unique rel root η 0 nd complex conjugte roots η ± given by η 0 = 3 q + + q 3, 8 η ± = 1 3 q q 3 ± i 3 q + 3 q, 9 i = 1. Then, when > 0 the chrcteristic polynomil for the equilibri E ±, 4, hs roots λ 0 = η b, λ ± = η ± 1 + b When > 0 nd b > 0 we hve tht λ 0 < 0. Menwhile, when 0 < 3 + b < 3 q + q 3, 11 the rel prt of the complex conjugte roots λ ± is positive. Thus, when > 0, b > 0, nd 11 hold, E + is sddle focus. The chrcteristic polynomil is the sme for E +, so E is lso sddle focus whenever E + is. Note tht, while E nd E + re sddle foci t the sme time, E 0 nd E + or E 0 nd E cn never be sddle foci t the sme time. 3 Anlyticl Hopf Bifurction Anlysis of T System Here we employ the method of multiple scles to construct the periodic orbits rising through the Hopf bifurction of the fixed points of the T system 1, nlyticlly. Such results complement the numericl results presented in, for instnce, Ref. [9] It will be helpful to recll the Hopf bifurction theorem see, e.g., [7 8]. Also note tht it is sufficiently generl to consider the Hopf bifurction nlysis bout E + since by symmetry the results cn be pplied to E, s well. We remrk tht E 0 cnnot hve Hopf bifurction. Indeed, if E 0 hs Hopf bifurction, then 3 hs the solutions λ ± = ±iω for some ω > 0. Now, λ ± will be determined explicitly from 3 by λ ± = ± 1 c. 1 Yet, if 3 hs two such purely complex roots, then = 0. This in turn implies tht ω = 0, contrdiction to ω > 0. Hence, there cn be no Hopf bifurction t the equilibrium E Hopf Bifurction Anlysis bout E + Consider the equilibrium E + nd let us ssume tht the solutions to the T system 1 undergo Hopf bifurction on some submnifold in prmeter spce corresponding to fixed c = c 0. Then, the chrcteristic polynomil 4 hs roots λ 0 R nd λ ± = ±iω ω R. Then, it must be the cse tht J E± λ = λ λ 0 λ + ω Clerly, = λ 3 λ 0 λ + ω λ λ 0 ω = λ bλ + bcλ + bc. 13 λ 0 = + b, ω = bc, λ 0 ω = bc. 14 As ω R, we require bc > 0. In order to stisfy the finl condition, we must hve tht c = c 0 = b. 15 Thus, the region of prmeter spce stisfying bc > 0 permits Hopf bifurction t c = c 0. Furthermore, from 15 nd the condition bc > 0, we cn only hve Hopf bifurction if sgnb = sgn b or, in other words, if either of the inequlities 0 < b < or < b < 0 hold. We shll now show tht, in neighborhood of c 0, sy ǫ + c 0, ǫ + c 0 ǫ > 0, the T system undergoes

3 No. 4 Communictions in Theoreticl Physics 611 Hopf bifurction t c = c 0, nd furthermore tht such bifurction is subcriticl. Let us note tht, in generl, λ = λc. Then, from the chrcteristic polynomil 4, define the reltion fλc, c =λc bλc + bcλc + bc. 16 Then, root λc of the chrcteristic polynomil 4 stisfies the reltion fλc, c = Differentition of 17 with respect to c yields df dλ dλ dc + df dc = 0, 18 which in turn implies tht dλc dc = df df 1 dc dλ bλc + b = 3λc + + bλc + bc. 19 Tking the root λc = λ + c, evluting t c = c 0 so tht λ + c 0 = iω, nd then substituting this bck into 19, we find tht dλ+ c ib bc 0 + b = dc c=c 0 bc 0 i + b. 0 bc 0 Algebric mnipultion of 0 gives d Re λ+ c = bbγ, dc c=c 0 bc 0 γ = 4b c > 0. 1 b bc 0 Yet, sgnb = sgn b for ny vlues of nd b dmitting Hopf bifurction, so the bove quntity is positive. Hence, by 1, we hve tht dre λ+ c > 0. dc c=c 0 As such, the requirements for the Hopf bifurction theorem re stisfied. The T system therefore undergoes Hopf bifurction t E + when c = c 0, nd periodic solutions will exist in neighborhood of the point c 0 provided tht < b < 0 holds. To put things into perspective, we hve obtined submnifold M of the prmeter spce R 3 of dimension on which the corresponding solutions to the T system 1 undergo bifurctions resulting in chnge in solution behvior. Explicitly, this submnifold M is given by M = M + M, 3 the union of two disjoint connected submnifolds of R 3, M + = {, b, c R 3 0 < b <, c = / b }, 4 M = {, b, c R 3 < b < 0, c = / b }. 5 As we pss from beneth to bove M in the sense tht we increse c, while holding nd b fixed, we hve shown tht the solutions to the T system undergo bifurction. Note tht only the submnifold M + corresponds to the Hopf bifurction, s + b > 0. In the cse of the other submnifold, M, the solutions to the T system my exhibit oscilltions bout trjectory which is unstble, s + B < 0. Both behviors re depicted in subsequent sections. We remrk tht the nlysis performed here bout the equilibrium E + will pply to E s well, by symmetry. Thus, the T system therefore undergoes Hopf bifurction t E when c = c 0, nd periodic solutions will exist in neighborhood of the point c 0 provided tht either 0 < b < or < b < 0 hold. 3. Anlyticl Construction of Periodic Orbits bout E ± Here we employ the method of multiple scles to construct nlyticl pproximtions to the periodic orbits resulting from the Hopf bifurction of the equilibrium point in the T system. In prticulr, the limit cycle will be determined by expnding bout the equilibrium point using progressively slower time scles. As the Hopf bifurction is of interest here, we shll restrict our ttention to the cse + b > 0 corresponding the the mnifold M + defined in 4. First we consider limit cycles bout the equilibrium E +. We shll ssume solution xt, yt, zt to 1 of the form 3 xt = x 0 + ε n X n T 0, T 1, T +, 6 yt = y 0 + xt = z 0 + n=1 3 ε n Y n T 0, T 1, T +, 7 n=1 3 ε n Z n T 0, T 1, T +, 8 n=1 T n = ε n t nd ε > 0 is smll non-dimensionl prmeter. The stndrd time derivtive becomes d dt = D 0 + εd 1 + ε D +, 9 D n = / T n. Furthermore, the dely prmeter c = c 0 + ε c c 0 is s given in 15 will be used to study the periodic orbits rising from n order ε perturbtion in c bout the bifurction point c = c 0. Substituting the definitions 6 9 into 1 nd equting like powers of ε, we obtin the system of nine equtions L k X n, Y n, Z n = S k,n, 30 for k = 1,, 3 nd n = 1,, 3, the liner opertors L k re defined by L 1 X n, Y n, Z n = D 0 X n + X n Y n, 31 L X n, Y n, Z n = c X n z 0 X n + x 0 Z n D 0 Y n, 3

4 61 Communictions in Theoreticl Physics Vol. 55 L 3 X n, Y n, Z n = y 0 X n + x 0 Y n bz n D 0 Z n, 33 nd the source terms S n,k re defined by S 1,1 S 1, S 1,3 = 0, 34 S,1 = D 1 X 1, 35 S, = D 1 Y 1 X 1 Z 1 cx 0, 36 S,3 = D 1 Z 1 X 1 Y 1, 37 S 3,1 = D X 1 D 1 X, 38 S 3, = D Y 1 + D 1 Y X 1 Z X Z 1 cx 0, 39 S 3,3 = D Z 1 + D 1 Z X 1 Y X Y Observe tht from the definition of L 1 in 31, we my set k = 1 in 30 to obtin Y n = D 0 X n 1 S 1,n, 41 for ll n = 1,, 3. Likewise, from the definition of L in 3, we my set k = in 30 to obtin Z n = 1 D x 0 D 0 X n 1 S,n, 4 x 0 for ll n = 1,, 3. Then, replcing Y n nd Z n with 41 nd 4, respectively, nd setting k = 3 in 30, we find tht the definition of L 3 in 33 implies tht X n stisfies the liner ODE in T 0 = t given by PX n = Q n, 43 we define the liner differentil opertor P by P = x 0 + x D 0 b + D x 0 D 0 = x 0 + b + b D 0 + x 0 D0 + 1 x 0 D0 3, x 0 44 nd the source terms Q n re defined by Q n = x 0 S 1,n 1 x 0 b + D 0 S,n + S 3,n, 45 for n = 1,, 3. By Eq. 34, S 1,k = 0 for ll k = 1,, 3, so we my ssume solution for X 1 of the form X 1 T 0, T 1, T =αt 1, T e iωt + βt 1, T e iωt + γt 1, T e +bt, 46 β = ᾱ is the complex conjugte of α since λ + = λ nd X 1 is rel-vlued function. Now, the α nd β modes correspond to the center mnifold the eigenvlues λ ± = ±iω re purely imginry nd the Hopf bifurction occurs, while γ corresponds to the ttrctive direction or, the stble mnifold. As we desire to construct the periodic orbits, which lie on the center mnifold, we set γ 0 in 46. Employing the solution form 46 in 41 nd 4, we find tht the other first-order fields re given by Y 1 T 0, T 1, T = 1 + i ω αt 1, T e iωt + 1 i ω βt 1, T e iωt, 47 ω ω Z 1 T 0, T 1, T = + i αt 1, T e iωt x 0 x 0 ω ω + i βt 1, T e iωt. 48 x 0 x 0 Next, let us ssume prticulr solution for the secondorder field X of the form X T 0, T 1, T = R 0 T 1, T + R 1 T 1, T e iωt + R T 1, T e iωt. 49 Employing 43, it is immedite tht α = β = T 1 T 1 Furthermore, we find tht R 0 T 1, T = Γ 1,0 αβ, 51 R 1 T 1, T = Γ 1,1 α, 5 R T 1, T = Γ 1, β = R 1 T 1, T, 53 Γ 1,0 = bω 3 x 0 3 x 0, 54 Γ 1,1 = 3 x 0 bω ω x 0 + 4ω + 4bω + ω 3 + ω x 0 bω 8ω 3 + bω x 0 x 0 + 4ω + 4bω + 8ω 3 + bω Γ 1, = Γ 1,1. + i ω3 + ω x 0 bω x 0 + 4ω + 4bω 3 x 0 bω ω 8ω 3 + bω x 0 x 0 + 4ω + 4bω + 8ω 3 + bω, 55 By use of 49 in 41 nd 4, nd reclling tht 50, we find tht the other second-order fields re given by Y T 0, T 1, T =Γ 1,0 αβ i ω Γ 1,1 α e iωt + 1 i ω Γ 1, β e iωt, ω θ = Z T 0, T 1, T = ω 3 x αβ + θ α e iωt + θ β e iωt + c 0, 58 x 0 ReΓ 1,1 ω x 0 ImΓ 1,1 + ω 3 x 0 4ω + i ImΓ 1,1 + ω ReΓ 1,1 + x 0 x 0 ω x 0. 59

5 No. 4 Communictions in Theoreticl Physics 613 Finlly, ssume prticulr solution for the thirdorder field X 3 of the form X 3 T 0, T 1, T = R 0 T 1, T + R 1 T 1, T e 3iωt + R T 1, T e 3iωt. 60 Employing 43, we find tht α = Λ 3 α β β + Λ 1 cα, = T T Λ 3 β α + Λ 1 cβ, 61 we define the constnts Λ 1 nd Λ 3 by x Λ 1 = 0b + b bω + ω x 4 0 x 0 b + b + b ω 4bω + 4ω x +i 0ω + ωb ωb x 4 0 x 0 b+b +b ω 4bω +4ω, 6 Λ 3,1 + Λ 3, + iλ 3,3 Λ 3,4 Λ 3 = x 0 x 0 + b + bω ω, 63 Λ 3,1 = bθ 3 x x 3 0 Γ 1,1 + 4 x 3 0 Γ 1,0 x 0 bγ 1,0 ω bω x 0 bγ 1,1 ω x 0 Γ 1,0 ω + x 0 ω Γ 1,1 x 0 + b, 64 Λ 3, = 3 x 3 0 Γ 1,0ω + x 0 Γ 1,0 ω x 0 ωθ + x 0 Γ 1,1 ω 3 x 0 bγ 1,0 ω + 3 x 3 0Γ 1,1 ω + ω 3 + x 0 bγ 1,1 ωbω ω, 65 Λ 3,3 = 3 x 3 0 Γ 1,0ω + x 0 Γ 1,0 ω x 0 ωθ + x 0 Γ 1,1 ω 3 x 0 bγ 1,0 ω + 3 x 3 0Γ 1,1 ω + ω 3 + x 0 bγ 1,1 ω x 0 + b, 66 Λ 3,4 = bθ 3 x x 3 0 Γ 1,1 + 4 x 3 0 Γ 1,0 x 0 bγ 1,0 ω bω x 0 bγ 1,1 ω x 0 Γ 1,0 ω + x 0 ω Γ 1,1 bω ω. 67 Furthermore, we find tht R 0 T 1, T = Γ,0, 68 R 1 T 1, T = Γ,1 α 3, 69 R T 1, T = Γ, β 3 = R 1 T 1, T, 70 Γ,0 = b c x 0, Γ,1 = 4Γ 1,1 3 x 0 bγ 1,1ω 6Γ 1,1 ω bθ x 0 18bω + 18ω 4 x 0 x 0 18bω + 18ω 4 x ω 3 + 3bω Γ, = Γ,1. + 6Γ 1,1ω 3 + 6Γ 1,1 ω x 0 bγ 1,1 ω + 6θ ω x 0 54ω 3 + 3bω x 0 18bω + 18ω 4 x ω 3 + 3bω + i 6Γ 1,1ω 3 + 6Γ 1,1 ω x 0 bγ 1,1 ω + 6θ ω x 0 18bω + 18ω 4 x 0 x 0 18bω + 18ω 4 x ω 3 + 3bω i 4Γ 1,1 3 x 0 bγ 1,1 ω 6Γ 1,1 ω bθ x 0 54ω 3 + 3bω x 0 18bω + 18ω 4 x ω 3 + 3bω 71, 7 73 By use of 49 in 41 nd 4, we find tht the other third-order fields re given by Y 3 T 0, T 1, T = Γ,0 αβ i 3ω Γ,1 α 3 e 3iωt + 1 i 3ω Γ, β 3 e 3iωt, 74 Z 3 T 0, T 1, T = c + c α + β + θ 3 α 3 e 3iωt x θ 3 β 3 e 3iωt + θ 4 α β + θ 4 β α, 9Γ,1 ω θ 3 = + Γ 1,1ω x 0 3 x + θ 0 x 0 3Γ,1 ω + i + Γ 1,1ω x 0 x, 76 0 θ 4 = Γ 1,0 + Γ 1,1 ω 3 x + ω 0 4 x 3 + θ 0 x 0 ω + iγ 1,0 Γ 1,1 x Now, we determine α nd, β = ᾱ. Writing α = 1/Ae iθ, plcing this into 61, nd seprting the expression into rel nd imginry prts, we find da = 1 dt 4 Re Λ 3A 3 + Re Λ 1 ca, 78 dθ = 1 dt 4 ImΛ 3A + ImΛ 1 c. 79 The fixed points of 78, A 0 = 0 nd A ± = ± ReΛ 1, 80 ReΛ 3 c give the mplitude of the solution α = 1/Ae iθ with A ± corresponding to the bifurcting periodic orbits. Clerly, A ± re rel vlued for Re Λ 1 Re Λ 3 c < 0 sgnλ 3 = sgnλ 1 sgn c. 81 Furthermore, the Jcobin J of the right hnd side of 78 is given by J = 3 4 Re Λ 3A + c Re Λ 1, 8

6 614 Communictions in Theoreticl Physics Vol. 55 so, t A ± we see tht J A± = c Re Λ 1 sgnj A± = sgnre Λ If J A± > 0, then the Hopf bifurction t c = c 0 is subcriticl, while if J A± < 0, then the Hopf bifurction t c = c 0 is supercriticl. By 78, 79, nd 80, the periodic orbits 6 8 up to first-order re given by xt = x 0 + A + cos[ωε c ω]t, 84 yt = y 0 + A + cos[ωε c ω]t Ωε c ω + A + sin[ωε c ω]t, 85 Ωε c ω A+ zt = z 0 + cos[ωε c ω]t x 0 Ωε c ω A+ + sin[ωε c ω]t, 86 x 0 A + is s defined in 80 nd the constnt Ω is given by Ω = Re Λ 1 ImΛ 3 + ImΛ Re Λ 3 If J A± > 0 J A± is s defined in 83, then the periodic orbits re unstble. Similrly, if J A± < 0, then the periodic orbits re stble. 3.3 Numericl Results Here we fix the vlues of the model prmeters, nd obtin numericl results to better view the behvior of the obtined periodic orbits ner Hopf bifurction. We tke x0 = 10, y0 = 10, nd z0 = 0 in ll cses considered, s these vlues re good representtives llowing us to study the overll behvior of the system ner Hopf bifurction. Next, we consider the cse when c = 16 = c 0. In Fig., we tke = 0, b =, c = 16 = c 0 nd observe the behvior of the solutions to the T system 1 in phse spce when the system undergoes Hopf bifurction. Observe tht the trjectories pproch the periodic orbits centered round E + s were constructed bove, before moving off t lrge time towrd n ttrctor t E. Fig. Phse portrit for the T system when = 4, b =, c = 16, x0 = 10, y0 = 10, nd z0 = 0. Here we observe the behvior of the T system t the Hopf bifurction point c = c 0. The trjectories remin close to the periodic orbit surrounding the ttrctor E + before becoming unstble n moving off towrd the ttrctor t E. Fig. 3 Autocorreltion function for the time series for the T system when = 4, b =, c = 16, x0 = 10, y0 = 10, nd z0 = 0. Fig. 1 Phse portrit for the T system when = 4, b =, c = 10, x0 = 10, y0 = 10, nd z0 = 0. Here we observe the behvior of the T system before Hopf bifurction occurs c < c 0. Notice tht the orbits decy to the equilibrium point E +. In Fig. 1, we fix = 4 nd b = so tht c 0 = 16, gin, c 0 is s defined in 15 nd plot the obtined solutions to the T system 1 for c = 10 < 16 = c 0. Here we see tht the orbit is stble s expected for c < c 0. Fig. 4 Plot of converged frctl dimension for the T system given by the best fit horizontl line when = 4, b =, c = 16, x0 = 10, y0 = 10, nd z0 = 0. The best-fit horizontl stright line revels converged frctl dimension of bout.35.

7 No. 4 Communictions in Theoreticl Physics 615 Figures 3 nd 4 show the utocorreltion function nd converged frctl dimensions corresponding to Fig.. Detils of the numericl lgorithms used my be found in [17 18]. The best-fit horizontl stright line in Fig. 4 yields converged frctl dimension of bout.35. more rpidly the solutions to the T system will brek orbit of the periodic solution round E + nd pproch the ttrctor t E. Similr result hold on the cse tht E nd E + re reversed. In the event tht + b 0, we no longer hve the Hopf bifurction t c = c 0 s discussed bove. However, the trnsition of c pst c 0 still results in oscilltions. As + b = 6 < 0, we hve tht λ 0 > 0 nd thus these solutions re unstble, s is seen in Fig. 6. Fig. 5 Phse portrit for the T system when = 4, b =, c = 17, x0 = 10, y0 = 10, nd z0 = 0. Here we observe the behvior of the T system fter Hopf bifurction occurs c > c 0. As in Fig., the trjectories remin close to the periodic orbit surrounding the ttrctor E + before becoming unstble n moving off towrd the ttrctor t E. Here, the orbits move wy sooner, due to the increse in c pst the point of Hopf bifurction, c 0. Fig. 7 Phse portrit for the T system when = 4, b =, c = 16. A homoclinic orbit, not of the type stisfying Hopf bifurction theorem, is evident. From the time series plots given in Fig. 6, we see tht there re oscilltions bout exponentilly incresing solutions; the precise mplitude of such oscilltions will depend on the prmeter vlues selected. Generlly, the mplitude of the solutions will increse while the period decreses, s t. When we hve < b < 0, there is still interesting behvior observed in the solutions for the T system. Tking = 4, b =, c = 16, we obtin the orbit given in Fig. 7. Fig. 6 Time series for the T system when = 4, b =, c = 16, x0 = 10, y0 = 10, nd z0 = 0. Here we observe the unstble oscillting solutions to the T system t the Hopf bifurction point c = c 0, when < b < 0 Recll tht < b < 0 implies tht the eigenvlue λ 0 > 0, which leds to the observed instbility. Finlly, we consider the cse when = 4, b =, nd c = 17 > 16 = c 0 in Fig. 5. We see tht the orbit is unstble s expected for c > c 0, nd eventully the trjectory will tend towrd the chotic ttrctor seen in the cse of the Smle horseshoe chos, considered bove. The further one increses c from c 0 in some neighborhood of c 0, the 4 Conclusions We hve conducted comprehensive nlysis of Hopf bifurctions nd the subsequent post-bifurction dynmics in the T system, three-dimensionl utonomous nonliner system. We find tht Hopf bifurction occurs t the prmeter vlue c = c 0, c 0 is in generl function of the other model prmeters, nd b, provided certin conditions on the model prmeters hold. Additionlly, we were ble to derive nlyticl expressions for the periodic orbits resulting from this Hopf bifurction by employing the method of multiple scles. The constructed periodic orbits, centered round either E + or E, strongly gree with the numericl results, highlighted in Figs. 1 3.

8 616 Communictions in Theoreticl Physics Vol. 55 Furthermore, we re ble to deduce the stbility of the periodic orbits, bsed only on the prmeter vlues selected, in the stndrd wy. These nlyticl predictions re verified nd extended numericlly, both by direct numericl simultions, s well s the use of stndrd numericl dignostics. As third order dynmicl system exhibiting rich vriety of behviors, the T system 1 serves s n excellent motivtion for future work. In ddition to potentil pplictions in secure communictions, the bundnce of behviors pprent in such simple model further suggests tht the study of chotic behvior in dynmicl systems is vitl in understnding systems of nonliner differentil equtions of mny shpes nd forms. References [1] G. Tign, Sci. Bull Politehnic Univ. Timisor [] G. Tign, In: Proceedings of the 3rd Interntionl Colloquium, Mthemtics in Engineering nd Numericl Physics, Buchrest, Romni [3] G. Alvrez, S. Li, F. Montoy, G. Pstor, nd M. Romer, Chos, Solitons nd Frctls [4] L.M. Pecor nd T.L. Crroll, Phys. Rev. Lett [5] L.M. Pecor nd T.L. Crroll, Phys. Rev. A [6] J.C. Sprott, Phys. Rev. E R647. [7] J.E. Mrsden nd M. McCrcken, The Hopf Bifurction nd Its Applictions, Springer-Verlg, New York [8] S. Krise nd S. Roy Choudhury, Chos, Solitons nd Frctls [9] B. Jing, X. Hn, nd Q. Bi, Nonliner Anlysis: Rel World Appliction [10] G. Tign nd O. Dumitru, Chos, Solitons nd Frctls [11] G. Tign nd D. Constntinescu, Chos, Solitons nd Frctls [1] R.A. Vn Gorder nd S. Roy Choudhury, ASME Journl of Computtionl nd Nonliner Dynmics [13] R.A. Vn Gorder nd S. Roy Choudhury, Interntionl Journl of Bifurction nd Chos [14] Y. Wu, X. Zhou, J. Chen, nd B. Hui, Chos, Solitons nd Frctls [15] X.F. Li, Y.D. Chu, J.G. Zhng, nd Y.X. Chng, Chos, Solitons nd Frctls [16] C. Yong nd Y. Zhen-Y, Comm. Theor. Phys [17] A.H. Nyfeh nd B. Blchndrn, Applied Nonliner Dynmics, Wiley, New York [18] S. Roy Choudhury, Chos, Solitons nd Frctls