Hopf bifurcation of the third-order Hénon system based on an explicit criterion

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  Hopf bifurcation of the third-order Hénon system based on an explicit criterion
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  Nonlinear Analysis 70 (2009) 3227–3235 Contents lists available at ScienceDirect Nonlinear Analysis  journal homepage: www.elsevier.com/locate/na Hopf bifurcation of the third-order Hénon system based on anexplicit criterion Enying Li, Guangyao Li, Guilin Wen, Hu Wang ∗ The State Key Laboratory of Advanced Technology for Vehicle Design and Manufacture, College of Mechanical and Automotive Engineering, Hunan University,Hunan Changshai 410082, China a r t i c l e i n f o  Article history: Received 7 November 2007Accepted 21 April 2008 Keywords: Hopf bifurcationExplicit criterionStabilityHénon mapCircuit implementationSimulationLimit cycle a b s t r a c t In this paper, Hopf bifurcation of the third-order Hénon system is studied via a simpleexplicit criterion, which is derived from the Schur–Cohn Criterion. Moreover stability of Hopfbifurcationisalsoinvestigatedbyusingthenormalformmethodandcentermanifoldtheory for the discrete time system developed by Kuznetsov. Test results containingsimulationsandcircuitmeasurementareshowntodemonstratethatthecriterioniscorrectand feasible. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Qualitative change in the nature of the solution occurs if a parameter passes through a critical point means bifurcation.The type of bifurcation that connects equilibrium with periodic solutions is called Hopf bifurcation. For determining thecritical points for a given system, eigenvalues of the Jacobian are required for calculation in the classical criterion. For low-dimensional problems, this kind of process is easy to perform. On the contrary, such a procedure is difficult to obtainthe analytic solution and might easily result in errors in high order systems. In most cases, it is necessary to find theconditions in terms of system parameters due to a consideration of stability changes with respect to the parameters.Therefore, in a continuous system, the criterion of Hopf bifurcation was deduced by Poter [1] based on the Hurwitzcriterion. Other singularities such as double  k -Hopf, Hopf-zero was studied by Yu [2]. The well-know Schur–Cohn stabilitycriterion [3,4], which is stated in terms of the coefficients of characteristic equations instead of those of eigenvalues, are more convenient and efficient for detecting the existence of Hopf bifurcation in high order and multi-parameters systemswas also demonstrated [5,6] in discrete dynamic systems. ThepurposeofthispaperistostudyHopfbifurcationinathird-orderHénonsystemviaanexplicitcriterion.InSection2,anexplicitcriterion,whichisformulatedusingasetofsimpleequalitiesorinequalitiesthatconsistofthecoefficientsofthecharacteristicequationderivedfromtheJacobianmatrixisintroduced.Sequentially,theexplicitcriterionofHopfbifurcationis applied to the analysis of the third-order Hénon map. In Section 3, determination of the direction of Hopf bifurcation and the stability of quasi-periodic solutions are obtained by using the normal form method and the center manifold theory forthe discrete time system developed by Kuznetsov [7]. Finally, in order to verify the result of the criterion, the third-order Hénon system is represented by an electrocircuit via an electronic simulation system and practical physical circuits. Thecorresponding phenomena produced by simulation and practical circuits are matched with analysis by explicit criterion. ∗  Corresponding author. Tel.: +86 0731 8821445; fax: +86 0731 8821445. E-mail address:  wanghuenying@hotmail.com (H. Wang).0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.04.038  3228  E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235 2. Application of explicit criterion of Hopf bifurcation in third-order Hénon map In this section, basic theories of the explicit criterion of Hopf bifurcation are outlined in 2.1. Consequently, the explicit criterion is successfully applied for third-order Hénon in Section 2.2. 2.1. An explicit criterion of Hopf bifurcation ThecriticalconditionforaHopfbifurcationrequiresthatatacriticalvalueofthebifurcationparameter,apairofcomplexconjugate eigenvalues lie on the unit circle and the other eigenvalue lie inside the unit circle. But for high-dimensional andmulti-parameter systems, the Jacobian matrix may involve certain singularities which result from numeric inaccuraciesin eigenvalue computations. The explicit criterion even if the Jacobian matrix involves some unknown parameters, therelationship between unknown parameters and the critical bifurcation constraint condition is explicitly expressed.For an  n th order discrete-time dynamical system, assume that at the fixed point  x 0 , it is characteristic polynomial of  Jacobian matrix  A  =  ( a ij ) n × n  can be written as  p τ  ( λ )  =  λ n +  a 1 λ n − 1 + ··· +  a n − 1 λ  +  a n  (1)where a i  =  a i ( τ  , k ), i  =  1 , . . . , n , τ   isthebifurcationparameter,and k isthecontrolparameterortheothertobedetermined.Consider the sequence of determinants ∆ ± 0  ( τ  , k )  =  1 , ∆ ± 1  ( τ  , k ), . . . , ∆ ± n  ( τ  , k )  (2)where ∆ ±  j  ( τ  , k )  =  1  a 1  a 2  ···  a  j − 1 0 1  a 1  ···  a  j − 2 0 0 1  ···  a  j − 3 ··· ··· ··· ··· ··· 0 0 0  ···  1  ±  a n −  j + 1  a n −  j + 2  ···  a n − 1  a n a n −  j + 2  a n −  j + 3  ···  a n  0 ··· ··· ··· ··· ··· a n − 1  a n  ···  0 0 a n  0  ···  0 0  ,  j  =  1 , . . . , n  (3)(C1) Eigenvalue assignment ∆ − n − 1 ( τ  0 , k )  =  0 ,  p τ  0 ( 1 )  >  0 ,( − 1 ) n  p τ  0 ( − 1 )  >  0 , ∆ + n − 1 ( τ  0 , k )  >  0 , (4) ∆ ±  j  ( τ  0 , k )  >  0 ,  j  =  n  −  3 , n  −  5 , . . . 1  ( or 2 )  (5)when  n  is even (or odd, respectively).(C2) Transversality conditiond ∆ − n − 1 ( τ  0 , k ) / d τ     =  0 (6)(C3) Nonresonance condition cos ( 2 π/ m )    =  ϕ  or resonance condition cos ( 2 π/ m )  =  ϕ , where  m  =  3 , 4 , 5 . . .  and ϕ  =  1  −  0 . 5  p τ  0 ( 1 ) ∆ − n − 3 ( τ  0 , k ) / ∆ + n − 2 ( τ  0 , k ).  (7)If (C1)–(C3) hold, then Hopf bifurcation occurs at τ  0 .The details of deduction can be found in Ref. [5]. 2.2. Applications of third-order Hénon map The discrete-time system considered here is the Hopf bifurcation generalized Hénon map [8–11] which is described by third-order difference equation (8)   x 1 ( k  +  1 )  =  µ  −  x 22 ( k )  −  bx 3 ( k )  x 2 ( k  +  1 )  =  x 1 ( k )  x 3 ( k  +  1 )  =  x 2 ( k ) (8)where  k  is iteration index,  x 1 ,  x 2 ,  x 3  and  µ , b  ∈  R ,  µ  is a bifurcation parameter,  b  =  0 . 1 in this case.The objective in this study is to determine the value of bifurcation parameter. The fixed point of the system described inEq. (8) is obtained as Eq. (9). (  x 10 ,  x 20 ,  x 30 )  =  − 1 . 1  +√  1 . 21  +  4  ·  µ 2  ,  − 1 . 1  +√  1 . 21  +  4  ·  µ 2  ,  − 1 . 1  +√  1 . 21  +  4  ·  µ 2  .  (9)  E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235  3229 The corresponding Jacobian matrix  A  of the Hénon map at a fixed point  (  x 10 ,  x 20 ,  x 30 )  can be written as Eq. (10)  A  =  0 1 . 1  −   1 . 21  +  4  ·  µ  − 0 . 11 0 00 1 0  .  (10)The characteristic polynomial of Jacobian matrix  A  can be obtained as P  ( λ )  :  λ 3 −  1 . 1  ·  λ  +  λ  ·   1 . 21  +  4  ·  µ  +  0 . 1  =  0 .  (11)The corresponding coefficients of Eq. (11) are a 0  =  1 ,  a 1  =  0 ,  a 2  =   1 . 21  +  4  ·  µ  −  1 . 1 ,  a 3  =  0 . 1 .  (12)According to Eqs. (3) and (4), for  n  =  3, equalities and inequalities can be expressed as ∆ − 2  ( µ )  =  1  a 1 0 1  −  a 2  a 3 a 3  0  =  2 . 1  −   1 . 21  +  4 µ  − 0 . 1 − 0 . 1 1  =  0 (13)  p µ ( 1 )  =  1  −  1 . 1  +   1 . 21  +  4 µ  +  0 . 1  >  0 (14) ( − 1 ) 3  p µ ( − 1 )  = −[ ( − 1  +  1 . 1  −   1 . 21  +  4 µ  +  0 . 1 ) ]  >  0 (15) ∆ + 2  ( µ )  =  1  a 1 0 1  +  a 2  a 3 a 3  0  =  − 0 . 1  +   1 . 21  +  4 µ  0 . 10 . 1 1  >  0 .  (16)According to Eqs. (13)–(16), the critical value of Hopf bifurcation of a third-order Hénon system is obtained as  µ 0  = 0 . 789525, the fixed point of the system is  (  x 10 ,  x 20 ,  x 30 )  =  ( 0 . 495 , 0 . 495 , 0 . 495 ) . The eigenvalue of the Jacobian matrix  A  is λ 1 , 2  =  0 . 05  ±  0 . 9987i , λ 3  = − 0 . 1Through calculation, the eigenvalue’s module  λ 1 , 2   =  1 , | λ 3 | =  0 . 1 satisfies the first condition of Hopf bifurcation, thusHopf bifurcation occurs at the equilibrium  (  x 10 ,  x 20 ,  x 30 )  =  ( 0 . 495 , 0 . 495 , 0 . 495 ) . 3. Direction and stability of the Hopf bifurcations In this section, we shall study the direction, stability and period of the bifurcating periodic solutions in a Hénon mapdescribed in Eq. (8). The method, we use is based on the theories of discrete system by Kuznetsov [7]. Since the equilibrium  X  ∗  =  (  x 10 ,  x 20 ,  x 30 )  =  ( 0 . 495 , 0 . 495 , 0 . 495 )  is not the srcin  O ( 0 , 0 , 0 ) , the  X  ∗ need to transform tosrcin by Eq. (17) firstly. ( Y  , v )  =  (  X   −  X  ∗ , µ  −  µ 0 ).  (17)This transforms the Hénon map into equivalent system Y  k + 1  =  F  v ( Y  k ).  (18)The essential of   Y  k  =  Y  ∗  =  0 and  v  =  0 in equivalent systems are the fixed point  X  ∗  and the critical value  µ 0  in the srcinalsystemrespectively.ThesystemdescribedasEq.(18)andthesystembyEq.(8)havethesameeigenvaluesofJacobianmatrix, therefore a Hopf bifurcation takes place at srcin  O ( 0 , 0 , 0 ) , the Jacobian matrix  A  of Eq. (18) at srcin is  A  =  0  − 0 . 99  − 0 . 11 0 00 1 0  .  (19)At  v  =  0, the eigenvalues  λ 1 , 2 ( 0 )  =  λ 1 , 2 ( µ 0 )  =  e ± i θ  0 , where 0  <  θ  0  <  π .Let  q  ∈  C  n be a complex eigenvector corresponding to  λ 1 :  Aq  =  e i θ  0 q ,  (20a)  Aq  =  e − i θ  0 q .  (20b)Introduce also the adjoint eigenvector  p  ∈  C  n admitting the properties:  A T   p  =  e − i θ  0  p ,  (21a)  A T   p  =  e i θ  0  p .  (21b)And satisfying the normalization    p , q  =  1.  3230  E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235 With the aid of Matlab, the vectors are obtained as: q  =  10 . 05  −  0 . 99875i − 0 . 995  −  0 . 099875i   (22a)  p  =  0 . 4951  −  0 . 024295i0 . 0004902  −  0 . 49569i − 0 . 0004902  −  0 . 049326i  .  (22b)Let the system equation (18) be written as: Y  n + 1  =  AY  n  + 12 B ( Y  n , Y  n )  + 16 C  ( Y  n , Y  n , Y  n )  +  O (  Y  n  4 )  (23)where  B ( Y  n , Y  n )  and  C  ( Y  n , Y  n , Y  n )  are bilinear and trilinear functions, respectively. In coordinates, we have B i (  x ,  y )  = n   j , k = 1 ∂ 2 Y  i ( ξ  ) ∂ξ   j ∂ξ  k  ξ  = 0  x  j  y k ,  (24a) C  i (  x ,  y ,  z  )  = n   j , k , l = 1 ∂ 3 Y  i ( ξ  ) ∂ξ   j ∂ξ  k ∂ξ  l  ξ  = 0  x  j  y k  z  l .  (24b)With the aid of Maple, we obtain B ( ξ  , η )  =  − 2 ξ  2 η 2 00   (25a) C  ( ξ  , η , ζ  )  =  000  .  (25b)The direction of bifurcation of a closed invariant curve can be calculated by Eq. (26) [7] a ( 0 )  =  Re  e − i θ  0  g  21 2  −  Re  ( 1  −  2e i θ  0 ) e − 2i θ  0 2 ( 1  −  e i θ  0 )  g  20  g  11  − 12  |  g  11 | 2 − 14  |  g  02 | 2 (26)where  g  20  =   p , B ( q , q )   g  11  =   p , B ( q , q )   g  02  =   p , B ( q , q )  (27)  g  21  =   p , C  ( q , q , q )  +  2   p , B ( q , ( E   −  A ) − 1 B ( q , q ))  +   p , B ( q , ( e 2i θ  E   −  A ) − 1 B ( q , q )) + e − i θ  0 ( 1  −  2e i θ  0 ) 1  −  e i θ  0   p , B ( q , q )  ·   p , B ( q , q )  − 21  −  e − i θ  0 |  p , B ( q , q ) | 2 − e i θ  0 e 3i θ  0 −  1  |  p , B ( q , q ) | 2 .  (28)From Eqs. (26) to (28),  a ( 0 )  = − 1 . 4472  <  0 can be obtained. According criterion of stability, when a small perturbationis added ∆ µ  =  0 . 003, where it is a sufficiently small positive real number, so the system has a stable limit cycle around theequilibrium (quasi-periodic solution). When  µ  =  0 . 792525, the result of numerical analysis is illustrated in Fig. 1. 4. Validation via third-order Hénon circuit In this section, based on the circuit was proposed by Miller [9] who constructed Hyperchaotic Hénon circuit, the third- order Hénon simulation model and practical circuit are built respectively and are applied verify the explicit criterion. 4.1. Circuit diagram and performance The circuit diagram of  Fig. 2 is proposed to realize a third-order Hénon map, in the figure part (a) represents an analogof the system circuit; the digital signal of the system is derived from part (b).The equations of Hénon circuit can be expressed as:   x 1 ( k  +  1 )  =  − R 1 R 4  ·  SET   − R 1 R 2 ·  0 . 4  ·  x 22 ( k )  − R 1 R 3 ·  x 3 ( k )  x 2 ( k  +  1 )  =  x 1 ( k )  x 3 ( k  +  1 )  =  x 2 ( k ) (29)where  R 1  =  10  k , R 2  =  4  k , R 3  =  100  k , R 4  =  10  k  .  E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235  3231 Fig. 1.  The Hopf bifurcation attractor of system  x 1 ( k ) .(a) Analog circuit of the Hénon map.(b) Digital circuit of the Hénon map. Fig. 2.  Hénon circuit diagram. When the value of SET is  µ  =  µ 0  + ∆ µ ( ∆ µ  =  0 . 003 ) , ∆ µ  is sufficiently small, positive and real, the system disturbedby a small perturbation the Hopf bifurcation (stable quasi-periodic solution) should happen.The block diagram of  Fig. 2(a) which serves as the basis of the hardware of analog design and is readily obtained fromEq. (29). The mathematical operation is fulfilled by two standard operational amplifiers, resistors, an analog multiplier and
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