Nonlinear Analysis 70 (2009) 3227–3235

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Nonlinear Analysis

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Hopf bifurcation of the third-order Hénon system based on anexplicit criterionEnying 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 ofHopf bifurcation is also investigated by using the normal form method and center manifoldtheory for the discrete time system developed by Kuznetsov. Test results containingsimulations and circuit measurement are shown to demonstrate that the criterion is correctand 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, aremore 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.

The purpose of this paper is to study Hopf bifurcation in a third-order Hénon system via an explicit criterion. In Section 2,an explicit criterion, which is formulated using a set of simple equalities or inequalities that consist of the coefficients of thecharacteristic equation derived from the Jacobian matrix is introduced. Sequentially, the explicit criterion of Hopf bifurcationis applied to the analysis of the third-order Hénon map. In Section 3, determination of the direction of Hopf bifurcation andthe 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-orderHénon 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 explicitcriterion is successfully applied for third-order Hénon in Section 2.2.

2.1. An explicit criterion of Hopf bifurcation

The critical condition for a Hopf bifurcation requires that at a critical value of the bifurcation parameter, a pair of complexconjugate 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 nth order discrete-time dynamical system, assume that at the fixed point x0, it is characteristic polynomial ofJacobian matrix A = (aij)n×n can be written as

pτ(λ) = λn+ a1λ

n−1+ · · · + an−1λ+ an (1)

where ai = ai(τ, k), i = 1, . . . , n, τ is the bifurcation parameter, and k is the control parameter or the other to be determined.Consider the sequence of determinants

∆±0 (τ, k) = 1,∆±1 (τ, k), . . . ,∆±n (τ, k) (2)

where

∆±j (τ, k) =

∣∣∣∣∣∣∣∣∣∣

1 a1 a2 · · · aj−10 1 a1 · · · aj−20 0 1 · · · aj−3· · · · · · · · · · · · · · ·

0 0 0 · · · 1

±an−j+1 an−j+2 · · · an−1 anan−j+2 an−j+3 · · · an 0· · · · · · · · · · · · · · ·

an−1 an · · · 0 0an 0 · · · 0 0

∣∣∣∣∣∣∣∣∣∣, j = 1, . . . , n (3)

(C1) Eigenvalue assignment

∆−n−1(τ0, k) = 0,

pτ0(1) > 0,

(−1)npτ0(−1) > 0,

∆+n−1(τ0, k) > 0,

(4)

∆±j (τ0, k) > 0, j = n− 3, n− 5, . . . 1 (or 2) (5)

when n is even (or odd, respectively).(C2) Transversality condition

d∆−n−1(τ0, k)/dτ 6= 0 (6)

(C3) Nonresonance condition cos(2π/m) 6= ϕ or resonance condition cos(2π/m) = ϕ, where m = 3, 4, 5 . . . and

ϕ = 1− 0.5pτ0(1)∆−n−3(τ0, k)/∆+

n−2(τ0, k). (7)

If (C1)–(C3) hold, then Hopf bifurcation occurs atτ0.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 bythird-order difference equation (8)

x1(k+ 1) = µ− x22(k)− bx3(k)

x2(k+ 1) = x1(k)x3(k+ 1) = x2(k)

(8)

where k is iteration index, x1, x2, x3 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 in

Eq. (8) is obtained as Eq. (9).

(x10, x20, x30) =

(−1.1+

√1.21+ 4 · µ2

,−1.1+

√1.21+ 4 · µ2

,−1.1+

√1.21+ 4 · µ2

). (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 (x10, x20, x30) 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) : λ3− 1.1 · λ+ λ ·

√1.21+ 4 · µ+ 0.1 = 0. (11)

The corresponding coefficients of Eq. (11) are

a0 = 1, a1 = 0, a2 =

√1.21+ 4 · µ− 1.1, a3 = 0.1. (12)

According to Eqs. (3) and (4), for n = 3, equalities and inequalities can be expressed as

∆−2 (µ) =

∣∣∣∣[1 a10 1

]−

[a2 a3a3 0

]∣∣∣∣ =∣∣∣∣∣2.1−

√1.21+ 4µ −0.1−0.1 1

∣∣∣∣∣ = 0 (13)

pµ(1) = 1− 1.1+√

1.21+ 4µ+ 0.1 > 0 (14)

(−1)3pµ(−1) = −[(−1+ 1.1−√

1.21+ 4µ+ 0.1)] > 0 (15)

∆+2 (µ) =

∣∣∣∣[1 a10 1

]+

[a2 a3a3 0

]∣∣∣∣ =∣∣∣∣∣−0.1+

√1.21+ 4µ 0.1

0.1 1

∣∣∣∣∣ > 0. (16)

According to Eqs. (13)–(16), the critical value of Hopf bifurcation of a third-order Hénon system is obtained as µ0 =

0.789525, the fixed point of the system is (x10, x20, x30) = (0.495, 0.495, 0.495). The eigenvalue of the Jacobian matrix A isλ1,2 = 0.05± 0.9987i,λ3 = −0.1

Through calculation, the eigenvalue’s module∣∣λ1,2

∣∣ = 1, |λ3| = 0.1 satisfies the first condition of Hopf bifurcation, thusHopf bifurcation occurs at the equilibrium (x10, x20, x30) = (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∗ = (x10, x20, x30) = (0.495, 0.495, 0.495) is not the origin O(0, 0, 0), the X∗need to transform toorigin by Eq. (17) firstly.

(Y, v) = (X − X∗,µ− µ0). (17)

This transforms the Hénon map into equivalent system

Yk+1 = Fv(Yk). (18)

The essential of Yk = Y∗ = 0 and v = 0 in equivalent systems are the fixed point X∗ and the critical value µ0 in the originalsystem respectively. The system described as Eq. (18) and the system by Eq. (8) have the same eigenvalues of Jacobian matrix,therefore a Hopf bifurcation takes place at origin O(0, 0, 0), the Jacobian matrix A of Eq. (18) at origin is

A =

0 −0.99 −0.11 0 00 1 0

. (19)

At v = 0, the eigenvalues λ1,2(0) = λ1,2(µ0) = e±iθ0 , where 0 < θ0 < π.Let q ∈ Cn be a complex eigenvector corresponding to λ1:

Aq = eiθ0q, (20a)

Aq = e−iθ0q. (20b)

Introduce also the adjoint eigenvector p ∈ Cn admitting the properties:

ATp = e−iθ0p, (21a)

ATp = eiθ0p. (21b)

And satisfying the normalization 〈p, q〉 = 1.

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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:

Yn+1 = AYn +12B(Yn,Yn)+

16C(Yn, Yn, Yn)+ O(‖Yn‖

4) (23)

where B(Yn, Yn) and C(Yn, Yn, Yn) are bilinear and trilinear functions, respectively. In coordinates, we have

Bi(x, y) =n∑

j,k=1

∂2Yi(ξ)

∂ξj∂ξk

∣∣∣∣∣ξ=0

xjyk, (24a)

Ci(x, y, z) =n∑

j,k,l=1

∂3Yi(ξ)

∂ξj∂ξk∂ξl

∣∣∣∣∣ξ=0

xjykzl. (24b)

With the aid of Maple, we obtain

B(ξ,η) =

−2ξ2η200

(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−iθ0g21

2

)− Re

((1− 2eiθ0)e−2iθ0

2(1− eiθ0)g20g11

)−

12|g11|

2−

14|g02|

2 (26)

where

g20 = 〈p, B(q, q)〉

g11 = 〈p, B(q, q)〉

g02 = 〈p, B(q, q)〉

(27)

g21 = 〈p, C(q, q, q)〉 + 2〈p, B(q, (E− A)−1B(q, q))〉 + 〈p, B(q, (e2iθE− A)−1B(q, q))〉

+e−iθ0(1− 2eiθ0)

1− eiθ0〈p, B(q, q)〉 · 〈p, B(q, q)〉 −

21− e−iθ0

|〈p, B(q, q)〉|2 −eiθ0

e3iθ0 − 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:x1(k+ 1) =

(−

R1

R4

)· SET −

R1

R2· 0.4 · x2

2(k)−R1

R3· x3(k)

x2(k+ 1) = x1(k)x3(k+ 1) = x2(k)

(29)

where R1 = 10 k�, R2 = 4 k�, R3 = 100 k�, R4 = 10 k�.

E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235 3231

Fig. 1. The Hopf bifurcation attractor of system x1(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 = µ0 + ∆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

3232 E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235

Fig. 3. Pulse signal.

Table 1Simulation parameters setting for PSPICE

Run time 900 msMaximum step 500 µsRelative accuracy of V’s (RELTOL) 0.0001Best accuracy of voltage (VNTOL) 1.0 µVBest accuracy of amps (ABSTOL) 1.0 pABest accuracy of coulombs (CHGTOL) 0.01 pFMinimum conductance for any 1/� (GMIN) 1.0e−12 1/�DC and bias of “blind” iteration (ITL1) 150DC and bias “best guess” (ITL2) 20Transient time point iteration (ITL4) 50Default nominal temperature (TNOM) 27 ◦C

Fig. 4. Phase plots from simulation by PSPICE.

a potentiometer, this part accomplishes the first equation of Eq. (29) x1(k+1) = (− R1R4

) ·SET− R1R2·0.4 ·x2

2(k)−R1R3·x3(k). Since

states x2 and x3 are time-delayed versions of state x1, an analog shift register comprised of sample and hold (S/H) circuits arechosen as the basis of our circuit implementation (Fig. 2). The system-state vector x in Eq. (29) corresponds to analog circuitvoltages at the output of these S/H circuits [9] is shown in Fig. 2.

The function of sampling and holding is used to change a continuous signal into a discrete signal, a pin on the sampleand hold determines whether it is in a sampling mode (S/H pin high) or a holding mode (S/H pin low) [9].

Three pairs of S/H are adopted and each state uses two S/H circuits. We use third pairs of S/H, each state uses two S/Hcircuits; double buffering is used to hold output state voltages constant while these voltages are sampled by the next stageduring shifting [9]. This action not only converts continue a signal into a discrete signal but also completes the functiondescribed in the second and third equations of Eq. (29). The S/H control signals (INPUT1 and INPUT2) are provided by thecircuit of Fig. 2(b). A pulse signal (as shown in Fig. 3) rising edge at the input of the first piece of 7474 initiates a state updatex(k)→ x(k+ 1) [9].

4.2. Simulation circuit implementation

According to the electrocircuit as shown in Fig. 2, it is easy to construct the simulation circuit’s mode in PSPICE. Eachelectronic component can be found in a corresponding library. The key factor for the simulation procedure is correctparameter-setting for parameters, such as relative accuracy of voltage, best accuracy of amps, etc. Each componentparameter should be correctly set, based on practical physical circuits, otherwise simulation results would not be accurateand converging. For this case, the setting is demonstrated in Table 1 by debugging.

E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235 3233

Fig. 5. Comparison between simulation and theoretical of signal x3 .

Fig. 6. Whole physical circuits of Hénon map.

After completion of the simulation, the phase plot of Fig. 4 is obtained via simulation of Eq. (29) by PSPICE and frommeasurements of the proposed electronic implementation. According to Fig. 5, the quasi-periodic solution of the simulatedcircuit and theoretical analysis are in substantial agreement. It is shown that the explicit criterion is simple and correct.

4.3. Physical circuits implementation

Implementation of the diagram of Fig. 2 is carried out using analog ICs and components. The operational amplifiersare implemented using OP07 from ANALOG DEVICES, sample-and-hold blocks are implemented using a NationalSemiconductor’s sample-and-hold IC LF398, Analog multipliers MLT04 of ANALOG DEVICE are used to implement thenonlinear term of Eq. (29), 3296 from BOURNS is set the value of critical that is added a small perpetuation. Two blocksof DM7474, two pieces of DM7408 and a piece of DM7414 from FAIRCHILD Semiconductor and The variable resistors and

3234 E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235

Fig. 7. Phase plot of the physical circuit.

Fig. 8. Hopf bifurcation time series of x1(k) by PSPICE.

Fig. 9. Hopf bifurcation time series of x1(k) of the physical circuit.

capacitors are implemented for deduced signals which control sample and hold. The physical circuit and correspondingphrase plots are illustrated in Figs. 6 and 7.

It is easy to observe that the Hopf bifurcation attractor of the system obtained from analysis via the explicit criterion asshown in Fig. 1 and simulated by PSPICE as shown in Fig. 7 are in substantial agreement. Accordingly, the Hopf bifurcationtime series output simulated by PSPICE as shown in Fig. 8 and captured from oscillograph as shown in Fig. 9 output quasi-periodic solution of state vector x1(k). Thus, the close resemblances between these plots provide qualitative evidence thatthe theoretical analysis, simulation and physical circuit are in correspondence. It is confirmed that the explicit criterion isvalidated and convenient.

E. Li et al. / Nonlinear Analysis 70 (2009) 3227–3235 3235

5. Conclusions

In summary, a criterion of Hopf bifurcation without using eigenvalues is applied to a third-order Hénon map.Both computer simulation and physical circuit means are implemented to construct a third-order Hénon system. Thecorresponding results demonstrate that the criterion is more convenient and efficient for studying Hopf bifurcation thanthe classical Hopf bifurcation criterion in practice.

Acknowledgements

This work is supported by the National 973 Program of China under the grant number 2004CB719402; the OutstandingYouth Foundation of NSFC under the grant number 50625519; Program for Changjiang Scholars and Innovative ResearchTeam in University.

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