Journal of Computational and Applied Mathematics

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1 Journal of Computational and Applied Mathematics 34 () 3 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage:.elsevier.com/locate/cam A hperchaotic sstem from the Rabinovich sstem Yongjian Liu ab Qigui Yang a Guoping Pang b a School of Mathematical Sciences South China Universit of echnolog Guanghou 64 PR China b Department of Mathematics and Computation Science Yulin Normal Universit Yulin 37 PR China a r t i c l e i n f o a b s t r a c t Article histor: Received August 9 Received in revised form December 9 Keords: Hperchaos Ultimate boundedness Lapunov exponents Bifurcation his paper presents a ne 4D hperchaotic sstem hich is constructed b a linear controller to the 3D Rabinovich chaotic sstem. Some complex dnamical behaviors such as boundedness chaos and hperchaos of the 4D autonomous sstem are investigated and analed. A theoretical and numerical stud indicates that chaos and hperchaos are produced ith the help of a Liénard-like oscillator motion around a hpersaddle stationar point at the origin. he corresponding bounded hperchaotic and chaotic attractors are first numericall verified through investigating phase trajectories Lapunov exponents bifurcation path and Poincaré projections. Finall to complete mathematical characteriations for 4D Hopf bifurcation are rigorousl derived and studied. 9 Elsevier B.V. All rights reserved.. Introduction Hperchaos is characteried as a chaotic sstem ith more than one positive exponent [] this implies that its dnamics are expended in several different directions simultaneousl. hus hperchaotic sstems have more complex dnamical behaviors than ordinar chaotic sstems. At the same time due to its theoretical and practical applications in technological fields such as secure communications lasers nonlinear circuits neural netorks generation control snchroniation hperchaos has recentl become a central topic in nonlinear sciences research (see e.g. [ 7] as ell as their references). On the one hand the ultimate boundedness of a chaotic sstem is ver important for the stud of the qualitative behavior of a chaotic or hperchaotic sstem. If one can sho that a chaotic or hperchaotic sstem under consideration has a globall attractive set one knos that the sstem cannot have equilibrium points periodic or quasi-periodic solutions or other chaotic or hperchaotic attractors existing outside the attractive set. his greatl simplifies the analsis of dnamics of the sstem. Hoever the estimate of the ultimate boundedness of a chaotic sstem is still a ver difficult task [8 ]. Due to their complexit the stud of the ultimate boundedness of hperchaotic sstems is a more difficult task. So far their ultimate boundedness has not been sstematicall studied [7]. On the other hand the hperchaos theor is still in its infanc. Ver little has been achieved on hperchaotic sstems. he dnamics of the hperchaotic sstems have not been completel understood b mathematicians until no. For example some dnamical behaviors such as boundedness Hopf bifurcation and chaotic propert of the 4D hperchaotic sstem are still at the explorator stage. herefore it is necessar to make a ne stud for the hperchaotic sstem. his situation motivates us to further stud the properties of chaos and hperchaos and some subtle characteristics of 4D Hopf bifurcation of the ne hperchaotic sstem hich is generated from the chaotic sstem so as to benefit more sstematic studies of 4D quadratic sstems and to reveal the true geometrical structures of the loer-dimensional chaotic and hperchaotic attractors. he Rabinovich differential sstem hich as firstl introduced in [] is defined b {ẋ = h ax + ẏ = hx b x ż = d + x (.) Corresponding author at: School of Mathematical Sciences South China Universit of echnolog Guanghou 64 PR China. address: liuongjianmaths@6.com (Y. Liu) /$ see front matter 9 Elsevier B.V. All rights reserved. doi:.6/j.cam.9..8

2 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 x (a) x space. (b) x space. x Fig.. Hperchaotic attractor of the Rabinovich sstem (.) ith a = 4 b = d = h = 6.7 and k =. here (x ) R 3. It has a chaotic attractor for some values of parameters for example a = 4 b = d = 4.84 h h for some h 4.9 and this attractor looks similar ith the chaotic attractor of the Loren sstem. As comparing ith the famous Loren sstem [34] the Rabinovich sstem resembles man of its properties and it as stated in the book [] that both the Rabinovich sstem and the Loren sstem can be considered as particular cases of the so-called generalied Loren sstem. here have been extensive investigations on dnamical behaviors of the Rabinovich sstem [67]. In this paper a ne hperchaotic is generated from the 3D Rabinovich sstem via adding a linear controller to it and its basic dnamics and properties are investigated such as the stabilit of the sstem and the geometr of the attractor. he corresponding bounded hperchaotic and chaotic attractor is first numericall verified through investigating phase trajectories Lapunov exponents bifurcation path and Poincaré projections. o complete mathematical characteriations for 4D Hopf bifurcation are also rigorousl derived and studied. he fact that chaos and hperchaos are created via a Liénardlike oscillator motion around a hpersaddle stationar point at the origin is shon b numerical experiments. he 4D sstem preserves some properties of the 3D sstem such as the -axis smmetr and the attractor s double-lobe structure. he rest of this paper is organied as follos. In Section the ne hperchaotic sstem is introduced and its boundedness is also proved. he stabilit of the sstem and the geometr of the attractor are discussed and the dnamical behaviors of this hperchaotic sstem such as Lapunov exponents fractal dimension and chaotic behaviors are also analed in Section 3. In Section 4 b using the normal form theor and smbolic computations to complete mathematical characteriations for the 4D Hopf bifurcations are derived and investigated. Finall conclusions are dran in Section.. A ne hperchaotic sstem.. Formulation of the sstem he proposed dnamical sstem is given b the folloing Rabinovich equations linearl extended to 4D: ẋ = h ax + ẏ = hx b x + ż = d + x ẇ = k here k is positive constant parameter determining the chaotic and hperchaotic behaviors and bifurcations of the sstem. hus the controller has made the chaotic sstem (.) a 4D hperchaotic sstem (.) hich has four Lapunov exponents. When (a b d h k) = (4 6.7 ) the four Lapunov exponents are λ LE =.366 λ LE =.8 λ LE3 =. λ LE4 = and the Lapunov dimension is D L = Moreover numerical simulations have verified that sstem (.) indeed has a hperchaotic attractor hen (a b d h k) = (4 6.7 ) as depicted in Fig.. Fig. shos the Poincaré mapping on the x plane and poer spectrum of the time series x(t) for this hperchaotic sstem... Boundedness heorem.. Suppose a > b > d > h > and k >. hen all orbits of sstem (.) including hperchaotic orbits are trapped in a bound region. (.)

3 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 3 Poer Spectrum.. x f Fig.. (a) Poincaré mapping on the x plane and (b) poer spectrum of time series x(t) for the hperchaotic Rabinovich sstem (.) ith a = 4 b = d = h = 6.7 and k =. Fig. 3. Graphical illustration of varing eigenvalues. Proof. Construct the folloing Lapunov function: V(x ) = x + + ( 3h) + k. (.) Along the orbits of sstem (.) one has V(x ) = ax b d + 3dh ( = ax b d 3h ) + 9dh 4. Let d > be sufficientl large so that for all (x ) satisfing V(x ) = d ith d > d one has ( ax + b + d 3h ) > 9dh 4. Consequentl on the surface {(x ) V (x ) = d } ith d > d one has V(x ) < hich implies that the set Ω = {(x ) V(x ) d } is a trapping region of all solutions of sstem (.). hus all orbits of sstem (.) are bounded. he proof is thus completed. Remark.. From boundedness and to positive Lapunov exponents it follos that sstem (.) ith (a b d h k) = (4 6.7 ) indeed has a hperchaos attractor.

4 4 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 3. Dnamical behaviors of the hperchaotic sstem his section further investigates the dnamical behaviors of the hperchaotic sstem (.) including dissipativit equilibria and stabilit structure of attractor Lapunov exponents and bifurcation diagrams. First Figs. 6 sho some tpical dnamical behaviors of the sstem. 3.. Dissipativit For sstem (.) it is noticed that V = ẋ + ẏ + ż + ẇ = (a + b + d). Obviousl hen sstem (.) can have x dissipative structure ith an exponential contraction rate: dv = (a + b + d)v. hat is a volume element V dt is contracted b the flo into a volume element V e (a+b+d)t in time t. his means that each volume containing the sstem orbit shrinks to ero as t at an exponential rate (a + b + d) hich is independent of x and. herefore all sstem orbits are ultimatel confined to some subset of ero volume and the asmptotic motion settles on some attractors. 3.. Equilibria It is clear that sstem (.) (and thus its solution) is invariant under the transformation (x ) ( x ). his means that an orbit that is not itself invariant under must have its tin orbit in the sense of this transformation. Sstem (.) has the origin as the onl stationar point for all positive parameters values. he Jacobian matrix of Eqs. (.) is given b: a h + h b x J = x d. (3.) k he characteristic equation J λi = at the origin can be ritten as (λ + d)[λ 3 + (a + b)λ + (ab h + k)λ + ak] = (3.) hich gives λ = d and (λ) = λ 3 + (a + b)λ + (ab h + k)λ + ak =. (3.3) Let A = a + b B = ab + k h and C = ak. hen according to the Routh Hurit criterion the real parts of all the roots λ in (λ) = are negative if and onl if A > C > and AB C >. From these inequalities one obtains a > b > k > and h < ab + bk. a+b Based on the above discussion the folloing propert is verified. heorem 3.. Let a > b > d > and k >. hen sstem (.) has a unique equilibrium O( ). Furthermore the necessar and sufficient condition for equilibrium O to be local asmptotical stable is h < h ab + bk a+b. heorem 3.. Let a > b > d > k > and h < 8 ab. hen the equilibrium O of sstem (.) is globall uniforml and 9 asmptotical stable. Moreover sstem (.) is neither chaotic nor hperchaotic. Proof. Define the folloing Lapunov function V(x ) = (x + ) + + k. From a > b > and h < 8 ab one reduces that matrix 9 [ ] a 3h/ 3h/ b is positive matrix. Its time derivative along the orbit of sstem (.) is V(x ) = ax b + 3hx d < (3.4) and b setting { (x ) V(x ) = } = {(x ) x = = = R} hich does not contain a nontrivial trajector of sstem (.). he Krasnoselskii theorem implies that sstem (.) is globall uniforml and asmptoticall stable about the origin. Furthermore sstem (.) has neither chaotic attractor nor hperchaotic attractor. he proof is thus completed. For h < h the solutions of Eq. (3.) are three roots ith negative real part λ λ λ 3 and λ 4 here λ 3 = d. At the critical point h = h the roots λ and λ both disappear and give birth to a pair of purel imaginar. R(λ ) = dr(λ ) dh h= h = (a+b)3/ (a b+ab +bk) / > λ (a+b) 3 +ak = (a + b) <.

5 c Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 a b d Fig. 4. (a) Solution and nullcline of sstem (3.) for a = 4 b = d = k = / and h = 3. (b) (d) Solutions and nullcline of sstem (.) for a = 4 b = d = k = /: (b) h =.; (c) h = 3.4; (d) h =. Normaliation: = h (a )/[ad(h ab)] = 7ah (a )/[4d(h ab) 3 ]. his result indicates in accordance ith the Hopf theorem [8] the birth of a limit ccle at h hich gros in sie ith h and has initial period given b = π/i(λ ) here I(λ ) = [ak(a b+ab +bk)] /. he basic role of this limit ccle on the (a+b) 3 +ak sstem dnamics ill be explained in Section 3.3. he limit ccle persists even hen the parameter h goes beond a certain critical point h = h here the imaginar part of λ disappears and its real part remains positive and bifurcates giving rise to a ne real eigenvalue λ 4 > as shon in Fig. 3. Note that for h > h the origin is a hpersaddle stationar point ith λ > λ 4 > > λ 3 > λ. Instead of calculating the roots exactl one can get simpler relations b observing that the cubic polnomial in Eq. (3.3) can be ritten as λ[λ + (a + b)λ + (ab h )] + (λ a)k. he bracketed expression is just the polnomial appearing in the characteristic equation of the 3D Rabinovich sstem hich has the roots λ = (/){ [(a b) + 4h ] / (a + b)} and λ = (/){[(a b) + 4h ] / (a + b)}. One can rite the folloing identit: λ(λ λ )(λ λ ) + (λ a)k (λ λ )(λ λ )(λ λ 4 ) =. After identifing coefficients of equal poers of λ and arranging the resulting equations one get the folloing approximate solutions: λ λ λ λ (λ a)k λ (λ λ ) (λ a)k λ (λ λ ).

6 6 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 he third eigenvalue is λ 3 = λ 3 = d and the fourth one introduced b the ne variable is given b: λ 4 = ak ak λ λ ab h. As an example for a = 4 b = d = h = 6.7 and k = one has: λ = 9.4 and λ = 4.4. he exact and approximate values for sstem (.) are λ = λ = and λ 4 =..9. he proximit of the 3D and 4D eigenvalues is the reason for some common features shared b the to sstems. It also makes clear the main role of λ 4 in the hperchaotic behavior. Indeed for the above parameters the eigenvectors corresponding respectivel to λ λ λ 3 and λ 4 are: v = [ ] v = [.6.7.3] v 3 = [ ] v 4 = [.6..98]. he related 3D eigenvectors are v = [ ] v = [ ] v = [ 3 ]. Note that v 4 lies almost entirel along the axis and the (x ) subspace preserves the eigenvector structure of the original 3D Rabinovich sstem Structure of the attractor o identif the nature of the limit ccle described so far one first observes that it occurs next to the intersection of the hpersurface x = h/a + /a (given b dx/dt = ) and the hpersurface = x/d (given b d/dt = ). Using these relations in sstem (.) one obtains the reduced sstem d dt = b + d = k dt dh ad h 3 (ad ) hich is a generalied Liénard equation. he nontrivial solutions of sstem (3.) converge to a limit ccle corresponding to a clockise motion around the origin over the humps of the nullcline (found b setting d/dt = ) as illustrated in Fig. 4(a). his result indicates that the presence of the extra variable caused the incorporation of a Liénard-like dnamics into the 4D Rabinovich sstem. he nullcline is a useful reference curve for studing the geometric structure of the attractor. In the 4D space it is the locus of points satisfing dx/dt = d/dt = d/dt = described b the equations = b = x/b x = dh/(ad ). dh ad + h 3 (ad ) Consider Fig. 4(b) (d). In the case of small k and h the solutions of sstem (.) remain ver close to the lateral branches of the nullcline. Numerical calculations using the Jacobian matrix (3.) at points along these branches give to negative real eigenvalues and a complex conjugate pair ith negative real part this explains the damping oscillations along the orbit shon in Fig. 4(b). One of the real eigenvectors is practicall tangential to the nullcline and orthogonal to the eigenspace spanned b the other three eigenvectors. For large enough h the real part of the complex eigenvalues changes from negative to positive. Accordingl the limit ccle carries a contracting expanding spiral that hirls transversel around the lateral branches of the nullcline [Fig. 4(c)]. he trajector is sitched from the half-space < to the half-space > and vice versa as a result of the underling Liénard-like dnamics combined ith the spiral-like flo thus forming the to lobes of the 4D attractor. Fig. shos a 3D vie of the trajector shon in Fig. 4(c). For h even higher the attractor goes flattened and confined close to the hpersaddle stationar point at the origin giving rise to chaos and hperchaos [Fig. 4(d)]. In the case of large k the trajector reaches the neighborhood of the origin at smaller h values. For this reason sstem (.) usuall displas chaotic behavior in the lo h region here the related 3D sstem is stable. (3.) 3.4. Small k disturbance It ma be interesting to consider the 3D Rabinovich sstem as arising from the 4D sstem in the limit k ith initial condition () =. In this case the 4D Rabinovich sstem ith (a b d h k) = (4 6.7 ) has four Lapunov exponents λ LE =.467 λ LE3 =. λ LE4 = he small k disturbs hich leads that λ LE or λ LE3 leaves naught and becomes positive number or negative number. Fig. 6 shos that the ver small positive k causes λ LE from ero to positive value and transits 4D chaotic sstem into 4D hperchaotic sstem. 3.. Lapunov exponents and bifurcation diagrams In the to sections belo some properties of the ne 4D sstem are discussed ith k varing. And the simulation results are further obtained b using Matlab ools.

7 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () Fig.. Solution of sstem (.) ith a = 4 b = d = k = / and h = 3.4. Normaliation: = h (a )/[ad(h ab)] = 7ah (a )/[4d(h ab) 3 ] =. 4 Lapunov exponents k Fig. 6. Lapunov exponents of sstem (.) ith a = 4 b = d = h = 6.7 and k ( ]. a.4 b. Lapunov exponents k 4 6 k Fig. 7. Lapunov exponents of sstem (.) ith a = 4 b = d = h = 6.7 and k ( 7] Fix a = 4 b = d = h = 6.7 and var k When k ( 7] varies the corresponding Lapunov exponent spectrum of sstem (.) are shon in Fig. 7. he bifurcation diagram ith respect to k ( 7] is given in Fig. 8. It can be observed that the bifurcation diagram ell coincides ith the spectrum of Lapunov exponents. Fig. 7 shos that sstem (.) is hperchaotic for a ver ide range of k and the sstem can also evolve into chaotic orbits and periodic orbits. From Figs. 7 and 8 the dnamical behaviors of sstem (.) can be clearl observed. When k (. 4.4) the first and the second largest Lapunov exponents are both positive hich implies that sstem (.) is hperchaotic. When k (4.4 6.) it is onl one Lapunov exponent that is bigger than ero hich means that sstem (.) is chaotic. When k (6. 6.6) the first largest Lapunov exponents are almost equal ero.

8 8 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 ISI k Fig. 8. Bifurcation diagram of sstem (.) ith a = 4 b = d = h = 6.7 and k ( 7]. a b 4 Lapunov exponents.. 6 h 7 h Fig. 9. Lapunov exponents of sstem (.) ith a = 4 b = d = k = and h [ 7]. ISI h Fig.. Bifurcation diagram of sstem (.) ith a = 4 b = d = k = and h [ 7]. When k (6.6 7] the first three largest Lapunov exponents are almost equal ero. hus hen k (6. 7] sstem (.) is either periodic or quasi-periodic Fix a = 4 b = d = k = and var h When h [ 7] varies the corresponding Lapunov exponent spectrum of sstem (.) are shon in Fig. 9. he bifurcation diagram ith respect to h [ 7] is given in Fig.. It can be observed that the bifurcation diagram ell coincides ith the spectrum of Lapunov exponents. Fig. 9 shos that sstem (.) is hperchaotic for a ver ide range of h and the sstem can also evolve into chaotic orbits and periodic orbits. From Figs. 9 and the dnamical behaviors of sstem (.) can be clearl observed. When h [ 4) (6.4 7] the first largest Lapunov exponents are ero hich implies

9 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 9 that sstem (.) is periodic. When h (4.4) (7.7 8.) (.9 3) ( ) it is onl one Lapunov exponent that is bigger than ero hich means that sstem (.) is chaotic. When h (.4 7.7) (8.9) (3 3.8) the first and the second largest Lapunov exponents are both bigger than ero hich implies that sstem (.) is hperchaotic. 4. 4D Hopf bifurcation his section emplos the higher-dimensional Hopf bifurcation theor and applies smbolic computations to perform the analsis of parametric k variations ith respect to dnamical bifurcations. First consider the existence of Hopf bifurcation. heorem 4. (Existence of Hopf Bifurcation). Suppose that a > b > k > and k > h ab > hold. hen as k varies and passes through the critical value k = (h ab)(a + b)/b sstem (.) undergoes a Hopf bifurcation at the equilibrium O( ). Proof. Suppose that Eq. (3.) has a pure imaginar root λ = iω (ω R + i = ). Substituting it into Eq. (3.) ields ak (a + b)ω + iω(ab h + k ω ) =. It follos that ω = ab h + k = Solving the above equations gives ω = a(h ab)/b ω = ak/(a + b). k = k = (h ab)(a + b)/b under the condition k > h ab >. Substituting k = k into Eq. (3.) one obtains λ = iω λ = iω λ 3 = (a + b) λ 4 = d here ω = a(h ab)/b. hus hen k > h ab > and k = k the first condition for Hopf bifurcation [8] is satisfied. From Eq. (3.) and k > h ab > it follos that R(λ (k )) λ=iω = b 3 a(h ab) + b (a + b). herefore the second condition for a Hopf bifurcation [8] is also met. Consequentl Hopf bifurcation exists. Remark 4.. When k h ab sstem (.) has no Hopf bifurcation at the equilibrium O( ). In the folloing the stabilit and expression of the Hopf bifurcation of sstem (.) is investigated b using the normal form theor [9] some rigorous mathematical analsis and smbolic computations. heorem 4.. Let a > b > k > h ab > and L h (a b d) + b (a d). hen periodic solutions of sstem (.) from Hopf bifurcation at O( ) have the folloing properties: (i) if L < holds bifurcating periodic solutions exist for sufficient small < k k < k (h ab)(a + b)/b. Moreover periodic solutions of sstem (.) from Hopf bifurcation at O( ) is non-degenerate subcritical and unstable; (ii) if L > holds bifurcating periodic solutions exist for sufficient small < k k < (h ab)(a + b)/b k. Moreover periodic solutions of sstem (.) from Hopf bifurcation at O( ) is non-degenerate supercritical and stable; (iii) the period and characteristic exponent of the bifurcating periodic solution are: = π ω ( + τ ε + O(ε 4 )) β = β ε + O(ε 4 ) here ω = a(h ab)/b and [ a b [ah 4 + bh (a(4b + d) a bd) + ab 3 (d a)] τ = 4(b + ab + ah ) 4ah + bd 4a b ] + (h + b )[b (a d) h (d + b a)][a b(b ) + b 4 + ab 3 + ah ] ((a + b) b )(4ah + bd 4a b) β = ab (h ab)[h (a d b) + b (a d)] (b 3 + ab + ah )[bd + 4a(h ab)] ε = k k µ + O[(k k ) ] µ = a(h ab)[a(h ab) + b (a + b) ][h (a d b) + b (a d)] b(b 3 + ab + ah )[bd + 4a(h ab)]

10 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 (iv) the expression of the periodic solution of sstem (.) from Hopf bifurcation is ( ) πt ab sin ( ) πt ab(h ab) cos ( ) x πt ah sin = ε + a bh K ε + O(ε 3 ) d ( ) πt (a + b)h a(h ab)/b cos here K = ab h bd +8a(h ab) [ ad a(h ab) d cos( 4πt ) (a + d) a(h ab)/b sin( ]. 4πt ) Proof. Let k > h ab > k = k = (h ab)(a + b)/b and t = a(h ab)/b. B straightforard computations ith the help of Mathematica. one can obtain bt abi h ahi b v = v 3 = and v 4 = h(a + b)t h ab hich satisf Av = it v Av 3 = dv 3 Av 4 = (a + b)v 4. For sstem (.) define and hus here P = (Rv Iv v 3 v 4 ) = (x ) = P(x ). ẋ = t + F (x ) ẏ = t x + F (x ) ż = d + F 3 (x ) ẇ = (a + b) + F 4 (x ) bt ab h ah b h(a + b)t h ab F (x ) = (h ab)[a(b + h ) t b x ] ht (b 3 + ab + ah ) F (x ) = [h(b3 + ab + ah ) + ab (h b) + abt (b + h )x ] ah(b 3 + ab + ah ) F 3 (x ) = (h + bt x ab )(b + ah ) F 4 (x ) = (a + b)[a(h + b ) b t x ] b 3 + ab + ah. Furthermore [ F g = 4 x [ F g = 4 x [ F g = 4 x [ 3 F G = 8 x 3 + F F F ( F + i + )] F = x ( F F + i x + F x + i + 3 F + 3 F + 3 F + x x i 3 F + )] F = x x ( F F )] F = x x ( 3 F x 3 (4.) + 3 F 3 F 3 F x x 3 )] =. (4.)

11 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 Meanhile one has h = ( F 3 4 x h = ( F 3 4 x h = ( F 4 4 x + ) F 3 = a bh h = ( F 4 4 x ) F 3 F 3 F 4 B solving the folloing equations: here i = x abh(a + t i) ) F 4 i =. x D = h and (D it I) = h D = one obtains = ( ) ( ) ( ) d h h h (a + b) = h h = h a bh d = abh(a + t i) (d + t i). + ) F 4 = Furthermore G = [( F + ) ( F F + i )] F x x = [ b 3 t + ab (h ab) ah(b 3 + ab + ah + i t ab(h + b ) abt (h + b ) ah(b 3 + ab + ah ) G = [( F + ) ( F F + i )] F = x x G = [( F ) ( F F + i + )] F x x = [ b 3 t ab (h ab) ah(b 3 + ab + ah + i t ab(h + b ) + abt (h + b ) ah(b 3 + ab + ah ) G = [( F ) ( F F + i + )] F = x x g = G + (G k ωk + Gk ωk ) k= [ (h ab)(h (a d b) + 4ab b d) = ab (d + 4t )(b3 + ab + ah ) + i t (ah 4 + bh (a(4b + d) bd a ) + ab 3 (d a)) h(d + 4t )(b3 + ab + ah ) Based on the above calculation and analsis one can compute the folloing quantities: C () = i (g g g 3 ) ω g + g = g µ = R C () α () = a(h ab)[a(h ab) + b (a + b) ][h (a d b) + b (a d)] b(b 3 + ab + ah )[bd + 4a(h ab)] τ = I C () + µ ω () ω [ a b [ah 4 + bh (a(4b + d) a bd) + ab 3 (d a)] = 4(b + ab + ah ) 4ah + bd 4a b ] + (h + b )[b (a d) h (d + b a)][a b(b ) + b 4 + ab 3 + ah ] ((a + b) b )(4ah + bd 4a b) ]. ] ]

12 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 here β = R C () = ab (h ab)[h (a d b) + b (a d)] (b 3 + ab + ah )[bd + 4a(h ab)] ω = a(h ab)/b α () = R(λ (k )) = ω () = (h + b ) ab(h ab) (h ab)[a(h ab) + b(a + b) ]. b 3 a(h ab) + b (a + b) Note α () <. From a > b > and k > h ac > one obtains that if h (a b d) + b (a d) < holds it follos that µ > and β > hich impl that the Hopf bifurcation of sstem (.) at O( ) is non-degenerate and subcritical and the bifurcating periodic solution exists for k > k and is unstable; if h (a b d)+b (a d) > holds it follos that µ < and β < hich impl that the Hopf bifurcation of sstem (.) at O( ) is non-degenerate and supercritical and the bifurcating periodic solution exists for k < k and is stable. Furthermore the period and characteristic exponent are: = π ω ( + τ ε + O(ε 4 )) β = β ε + O(ε 4 ) here ε = k k µ + O[(k k ) ]. And the expression of the bifurcating periodic solution is (except for an arbitrar phase angle): X = (x ) = P(x ) = PY here the matrix P is defined as in (4.) and x = R u = I u ( ) = u + R( u ) + O( u 3 ) u = εe itπ + iε [g e 4itπ 3g e 4itπ + 6g ] + O(ε 3 ) = εe itπ + O(ε 3 ). 6ω B tedious computations one can obtain [ ( ) πt ε ab sin ( )] πt ab(h ab) cos ( ) x πt εah sin = + O(ε 3 ) ε a bh d + ε K ( ) πt ε(a + b)h a(h ab)/b cos here K is defined as in heorem 4.. Based on the above discussion the results of heorem 4. are indeed established. In order to verif the above theoretical analsis let a = 4 b = d = and h = 6.7. and k = 3. According to heorem 4. one has k = 66.. hen from heorem 4. one can get L = 4.8 > hich impl that the Hopf bifurcation of sstem (.) at O( ) is non-degenerate and supercritical a bifurcating periodic solution exists for k < k = 66. and the bifurcating periodic solution is stable. In fact Hopf bifurcation occurs hen k < k = 66. as shon in Fig. (a) (b).. Conclusions In this paper a 4D hperchaotic sstem has been constructed b linearl adding a ne variable to the 3D Rabinovich sstem. Some complex dnamical behaviors such as boundedness chaos and hperchaotic of the 4D autonomous sstem are investigated and analed. he corresponding bounded hperchaotic and chaotic attractor is first numericall verified through investigating phase trajectories Lapunov exponents bifurcation path and Poincaré projections. he geometric structure of the attractor is also investigated. A theoretical and numerical stud indicates that chaos and hperchaos are produced ith the help of a Liénard-like oscillator motion around a hpersaddle stationar point at the origin. Meanhile b using the normal form theor to complete mathematical characteriations for 4D Hopf bifurcation are rigorousl derived and studied. In particular the ultimate bound for the ne hperchaotic sstem and the geometric structure of the attractor are investigated in detail. It ill benefit future theoretical analsis and practical applications of ne hperchaotic sstems. It is hoped that the investigation of the paper ill shed some light to more sstematic studies of 4D quadratic sstems.

13 a 3 Y. Liu et al. / Journal of Computational and Applied Mathematics 34 () 3 3 b. x 3. Fig.. Phase portraits of sstem (.) ith a = 4 b = d = h = 6.7 and k = 66. Acknoledgements he authors ish to thank the revieers for their constructive and pertinent suggestions for improving the presentation of the ork. he first and second authors are partiall supported b National Natural Science Foundation of China (No. 8774) the first author is partiall supported b the Doctorate Foundation of South China Universit of echnolog and the Scientific Research Foundation of Guangxi Education Office of China (No. 9). he third author is partiall supported b the Sustentation Fund of the Elitists for Guangxi Universities (Document Number: [8]4) and the Initiation Fund for the High-level alents of Yulin Normal Universit. References [] O.E. Rössler An equation for hperchaos Phs. Lett. A 7 (979) 7. [] R. Barboa Dnamics of a hperchaotic loren sstem Internat. J. Bifur. Chaos 7 (7) [3] X. Wang M. Wang A hperchaos generated from Loren sstem Phsica A 387 (8) [4] K. hamilmaran M. Lakshmanan A. Venkatesan A hperchaos in a modified canonical Chua s circuit Internat. J. Bifur. Chaos 4 (4) 43. [] Q. Yang K. Zhang G. Chen Hperchaotic attractors from a linearl controlled Loren sstem Nonlinear Anal. RWA (9) [6] Y. Li G. Chen W.K.S. ang Controlling a unified chaotic sstem to hperchaotic IEEE rans. Circuits Sst.-II () 4 7. [7] Q. Yang Y. Liu A hperchaotic sstem from a chaotic sstem ith one saddle and to stable node-foci J. Math. Anal. Appl. 36 (9) [8] G. Leonov Bound for attractors and the existence of homoclinic orbit in the Loren sstem J. Appl. Math. Mech. 6 () 9 3. [9] G. Leonov A. Bunin N. Koksch Attractor localiation of the Loren sstem ZAMM 67 (987) [] X. Liao On the global basin of attraction and positivel invariant set for the Loren chaotic sstem and its application in chaos control and snchroniation Sci. China Ser. E Inform. Sci. 34 (4) [] D. Li X. Wu J. Lu Estimating the ultimate bound and positivel invariant set for the hperchaotic Loren Haken sstem Chaos Solitons Fractals 39 (9) [] A.S. Pikovski M.I. Rabinovich V.Y. rakhtengerts Onset of stochasticit in deca confinement of parametric instabilit Sov. Phs. JEP 47 (978) [3] E.N. Loren Deterministic non-periodic flos J. Atmospheric Sci. (963) 3 4. [4] C. Sparro he Loren Equations: Bifurcations Chaos and Strange Attractors Springer-Verlag NY 98. [] V.A. Boichenko G.A. Leonov V. Reitmann Dimension heor for Ordinar Differential Equations eubner. [6] S. Neukirch Integrals of motion and semipermeable surfaces to bound the amplitude of a plasma instabilit Phs. Rev. E 63 () 36. [7] J. Llibre M. Messias P.R. Silva On the global dnamics of the Rabinovich sstem J. Phs. A: Math. heor. 4 (8) 7. (pp). [8] J. Guckenheimer P. Holmes Nonlinear Oscillations Dnamical Sstems and Bifurcation of Vector Field Springer Ne York 983. [9] B. Hassard N. Kaarinoff Y. Wan heor and Application of Hopf Bifurcation Cambridge Universit Press Cambridge 98. [] Y.A. Kunetsov Elements of Applied Bifurcation heor Springer-Verlag Ne York 998.