dynamics at infinity, degenerate hopf and zero-hopf ...sprott.physics.wisc.edu/pubs/paper458.pdf ·...

July 8, 2016 10:42 WSPC/S0218-1274 1650125

International Journal of Bifurcation and Chaos, Vol. 26, No. 7 (2016) 1650125 (16 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021812741650125X

Dynamics at Infinity, Degenerate Hopf and Zero-HopfBifurcation for Kingni–Jafari System

with Hidden Attractors

Zhouchao Wei∗School of Mathematics and Physics, China University of Geosciences,

Wuhan 430074, P. R. ChinaDivision of Dynamics, Technical University of Lodz, Lodz,

Stefanowskiego 1/15, 90-924, PolandCollege of Mechanical Engineering, Beijing University of Technology,

Beijing 100124, P. R. ChinaMathematical Institute, University of Oxford,

Oxford, OX2 6GG, [email protected]

[email protected]

Irene MorozMathematical Institute, University of Oxford,

Oxford, OX2 6GG, England

Zhen WangDepartment of Applied Sciences,

Xijing University, Xi’an 710123, P. R. China

Julien Clinton SprottDepartment of Physics, University of Wisconsin,

Madison, WI 53706, USA

Tomasz KapitaniakDivision of Dynamics, Technical University of Lodz,

Lodz, Stefanowskiego 1/15, 90-924, Poland

Received February 15, 2016

To understand the complex dynamics of Kingni–Jafari system with hidden attractors, the firstobjective of this paper is to study the global dynamics, and give a complete description ofthe dynamics of Kingni–Jafari system at infinity by using the Poincare compactification of apolynomial vector field in R

3. The second objective of this paper is to prove the existenceof periodic solutions in the Kingni–Jafari system by classical Hopf bifurcation and degenerateHopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution thatbifurcates from a zero-Hopf equilibrium was derived in the Kingni–Jafari system. The theoreticalanalysis and simulations demonstrate the rich dynamics of the system.

Keywords : Kingni–Jafari system; hidden attractor; dynamics at infinity; zero-Hopf bifurcation;Hopf bifurcation.

∗Author for correspondence

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http://dx.doi.org/10.1142/S021812741650125X

July 8, 2016 10:42 WSPC/S0218-1274 1650125

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1. Introduction

In the past five decades, many 3D autonomoushyperbolic type chaotic systems have been exten-sively studied in science, engineering and mathe-matical communities. From the view of the stabilityof equilibria, many chaotic systems have at least oneunstable equilibrium, and some chaotic attractorsare of Shilnikov type [Silva, 1993].

It turns out that any real dynamical system isbasically nonlinear and has multiple state attractorsdepending on the choice of the initial conditions orsystem parameters. By setting these values for phys-ical meaning, the full information on every possiblesolution is no longer available. In the qualitativetheory of differential equations, it is important toknow whether a given differential system is chaoticor not. By the category of chaotic attractors eitherself-excited or hidden [Kuznetsov et al., 2011a;Kuznetsov et al., 2011b; Leonov et al., 2011, 2012;Leonov & Kuznetsov, 2013], we call the chaoticattractors with no equilibria or only stable equilib-ria “hidden attractors”. From a computational pointof view, these hidden attractors cannot be foundeasily by numerical methods, but they are impor-tant and potentially problematic in engineeringapplications because they can lead to unexpectedand potentially disastrous behavior [Kapitaniak &Leonov, 2015].

Recently, there has been increasing attentionon some unusual systems, such as those having noequilibrium [Wei, 2011; Jafari et al., 2013; Wanget al., 2012; Wei et al., 2014], stable equilibria [Yanget al., 2010; Wei & Yang, 2011, 2012; Wang & Chen,2012; Molaie et al., 2013; Wei & Zhang, 2014], aline equilibrium [Jafari & Sprott, 2013] and coex-isting attractors [Sprott et al., 2013; Li & Sprott,2014a, 2014b]. Motivated by the paper by Wang andChen [2012], Kingni, Jafari and co-workers joinedthis exploration and performed a systematic searchto find an additional three-dimensional chaotic sys-tem with only one stable equilibrium [Kingni et al.,2014], which belongs to a newly introduced cat-egory of chaotic systems with hidden attractors.Nevertheless, knowledge about equilibria does nothelp in the localization of hidden attractors [Laoet al., 2014; Leonov et al., 2014; Jafari et al., 2015;Shahzad et al., 2015; Sharma et al., 2015a, 2015b].Therefore, understanding the local and the globalbehaviors of chaotic systems with hidden attractorsis of great importance [Kapitaniak & Leonov, 2015;Wang et al., 2015; Dudkowski et al., 2015].

The present paper is a continuation of workby Kingni and Jafari [Kingni et al., 2014]. A moredetailed theoretical analysis need to be demon-strated for the proposed Kingni–Jafari system withhidden attractors. The rest of this paper is orga-nized as follows. In Sec. 2, the Kingni–Jafari sys-tem is introduced and stable equilibrium point cancoexist with a strange attractor. In Sec. 3, dynam-ical behaviors at infinity are obtained by using thePoincare compactification of polynomial vector fieldin R

3. In Sec. 4, by using the normal form theoryand symbolic computations, we present the out-line of the Hopf bifurcation methods on codimen-sion one and two Hopf bifurcations, in particular,how to calculate the Lyapunov coefficients relatedto the stability of the equilibrium. In Sec. 5, usingaveraging theory, we prove the existence of unstableperiodic orbits bifurcating from zero-Hopf equilib-rium. Finally, in Sec. 6, we make some concludingremarks.

2. The Kingni–Jafari System andHidden Chaos

In this section, we first present the Kingni–Jafarisystem proposed in [Kingni et al., 2014]:

x = −z,

y = −x − z,

z = 3x − ay + x2 − z2 − yz + b,

(1)

where the state variables are (x, y, z) ∈ R3 and theparameters (a, b) ∈ R2. It is easy to see that whena = 0, there is no equilibria. When a = 0, system (1)has only one equilibrium E0(0, b

a , 0). By liberaliza-tion around E0, the Jacobian matrix of system (1)is given by

J(E0) =

0 0 −1−1 0 −1

3 −a − b

a

.

Obviously, the following characteristic equationabout the equilibrium E0 is:

det(λI − J(E0))

= λ3 +b

aλ2 + (3 − a)λ + a = 0. (2)

According to the Routh–Hurwitz criterion, thereal parts of all the roots λ are negative if and

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Dynamics at Infinity, Degenerate Hopf and Zero-Hopf Bifurcation for Kingni–Jafari System

only if

a > 0, b > 0,b

a(3 − a) − a > 0.

Therefore, E0 is asymptotically stable if the follow-ing condition is satisfied

0 < a < 3, b >a2

3 − a. (3)

As can be seen in Fig. 1, system (1) has threetypes of equilibrium E0 in the region (a, b) | 0 <a < 2, 0 < b < 3. Based on the above discussions,we find hidden chaos in the above parameterregions. Therefore, in the following we focus onsome behavior previously unobserved-phenomenonin 3D system: hidden attractors with only one sta-ble equilibrium, which have a basin of attractionthat does not intersect with small neighborhoods ofany equilibrium points. By choosing some parame-ter values a = 1.3, b = 1.01 [Kingni et al., 2014], andthe initial condition (0.7, 2.3,−1.5), the system hasonly one stable equilibrium, whose characteristicvalues are λ1 = −0.7679, λ2,3 = −0.0045 ± 1.3012i.In this case, the system (1) has no homoclinic(heteroclinic) orbits but has a single-scroll chaoticattractor (see Fig. 2). The corresponding Lyapunovexponents [Wolf et al., 1985; Kuznetsov & Leonov,2005, 2007] are L1 = 0.094, L2 = 0, L3 = −1.297.

Fig. 1. The equilibrium point E0 of system (1) is stable nodein the black region, stable focus in white region and saddlefocus in yellow region.

−4−2

02

−2

0

2

4−2

0

2

4

6

z(t)x(t)

y(t)

(a)

−2 −1 0 1 2 3−5

0

5

x

z

−2 −1 0 1 2 3−4

−2

0

2

x

y

(b)

Fig. 2. Hidden chaotic attractor of system (1) with param-eters a = 1.3, b = 1.01 and initial condition (0.7, 2.3,−1.5).

The Kaplan–Yorke dimension [Kuznetsov et al.,2014] is defined by

DL = j +1

|Lj+1|j∑

i=1

Li,

where j is the largest integer satisfying∑j

i=1 Li ≥ 0and

∑j+1i=1 Li < 0. For the hidden chaotic attrac-

tor in Fig. 2, the Kaplan–Yorke dimension is DL =2.072, which means that the Lyapunov dimension ofthe hidden chaotic attractor is fractional. As cross-section plot shown for z = 0 with a = 1.3, b = 1.01in Fig. 3, a black dot is the stable equilibriumand surrounded by light blue. Moreover, the lightblue region contains initial conditions (x, y, 0) thatattract the stable equilibrium. The two black linesare a cross-section of the strange attractor in the

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Fig. 3. For system (1), the strange attractor basin in redand the stable equilibrium basin in light blue for z = 0 witha = 1.3, b = 1.01.

z = 0 plane, and the red region contains initial con-ditions (x, y, 0) that attract the strange attractor.

In addition, the different dynamical regions inthe space of the parameters a and b are as deter-mined in Fig. 4. Initial conditions for each pointin the plot are chosen randomly from a Gaussian

Fig. 4. Regions of various dynamical behaviors as a func-tion of the bifurcation parameters a and b. The region forstable equilibrium are shown in green, the chaotic regionsare shown in red, the limit cycles are shown in light blue,and unbounded solutions are shown in white.

distribution with unit variance centered on (x, y,z) = (0, 0, 0). These regions of coexisting attrac-tors are thus shown with colors intermingled, forexample red and green in the narrow region around(a, b) = (1.3, 1.01) where a strange attractor coex-ists with a stable equilibrium. In particular, thereis a narrow region around (a, b) = (1.3, 1.01) wherechaotic solutions occur in the presence of a weaklystable focus equilibrium. In that region, the strangeattractor is hidden. The region is small, and initialconditions must be chosen carefully. The basin plotshows the coexistence of the two attractors, whichfurther indicates that the system has extremelyrich dynamics. Such systems belong to a newlyintroduced category of chaotic systems with hid-den attractors that are important and potentiallyproblematic in engineering applications.

3. Infinity Dynamics inKingni–Jafari System

As we know, the Kingni–Jafari system (1) can beextended to an analytic system defined on a closedball of radius one, whose interior is diffeomorphic toR3 and its invariant boundary, the two-dimensionalsphere S

2 = (x, y, z) |x2 + y2 + z2 = 1 playsthe role of the infinity. This ball shall be denotedhere called the Poincare ball, since the technique forthis extension is the Poincare compactification fora polynomial vector field in R

n, which is describedin detail in [Cima & Llibre, 1990; Velasco, 1969].

Under the linear transformation y → y + ba ,

only one equilibrium E0 is transformed to the ori-gin O(0, 0, 0), and system (1) becomes

x = −z,

y = −x − z,

z = 3x − ay + x2 − z2 − yz − b

az.

(4)

In order to study the behavior of the trajectories ofthe system (4) near infinity, we will use the theory ofPoincare compactification in R3. Let Poincare ballS3 = r = (r1, r2, r3, r4) ∈ R4 | ‖r‖ = 1 be theunit sphere, S+ = r ∈ S3, r4 > 0 and S− =r ∈ S3, r4 < 0 be the northern and southernhemispheres, denote the tangent hyperplanes at thepoint (±1, 0, 0, 0), (0,±1, 0, 0), (0, 0,±1, 0), (0, 0,0,±1) by the local chart Ui, Vi for i = 1, 2, 3, 4,where Ui = r ∈ S3, ri > 0, Vi = r ∈ S3, ri < 0.We only consider the local chart Ui, Vi for i = 1, 2, 3for obtaining the dynamics at x, y, z infinity.

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In the local charts U1 and V 1

Take the change of variables (x, y, z) = (w−1, uw−1,vw−1), and t = wτ , the system (4) becomes

du

dτ= uvw − w − wv,

dv

dτ= wv2 + 3w − awu + 1

− v2 − uv − b

awv,

dw

dτ= vw2.

(5)

If w = 0, the system (5) reduces to

du

dτ= 0,

dv

dτ= 1 − v2 − uv.

(6)

Solving the system (6) directly, one can obtain thesolution

u= c1,

v =−c1

2+

12

tanh

(t√

c21 + 42

+ arctanh2c2 + c1√

c21 + 4

)

×√

c21 + 4,

Fig. 5. Trajectories in the local phase portraits ofsystem (6).

where c1 ∈ R. We find the general directions of thevector field in Fig. 5. The flow in the local chart V1 isthe same as the flow in the local chart U1 reversingthe time, hence, the phase portrait on the infinitesphere at the negative end point of the x axis isshown in Fig. 5, reversing the time direction.


Next, we study the dynamics of the system (4) atinfinity of the y axis. Taking the transformation(x, y, z) = (uw−1, w−1, vw−1), and t = wτ , the sys-tem (4) becomes

du

dτ= wu2 + uvw − wv,

dv

dτ= uvw + wv2 + 3wu − wa

+ u2 − v2 − v − b

awv,

dw

dτ= w2u + w2v.

(7)


du

dτ= 0,

dv

dτ= u2 − v2 − v.

(8)

Solving the system (8) directly, one can obtain thesolution

u = c1,

v = −12

+12

tanh

(t√

4c21 + 12

+ arctanh2c2 + c1√

4c21 + 1

)√4c2

1 + 1,

where c1 ∈ R. We find the general directions of thevector field in Fig. 6. The flow in the local chart V2

is the same as the flow in the local chart U2, hence,the phase portrait of the system (4) on the infinitesphere at the negative end point of the y axis isshown in Fig. 6, reversing the time direction.


Finally, we consider infinity at the z axis. Let(x, y, z) = (uw−1, vw−1, w−1), and t = wτ , the

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Fig. 6. Trajectories in the local phase portraits of sys-tem (8).

system (4) becomes

du

dτ= −3wu2 + awuv − u3 + u

+ uv − w +b

awu,

dv

dτ= −3uvw + awv2 − u2v

+ v + v2 +b

awv − wu − w,

dw

dτ= −3uw2 + aw2v − wu2

+ w + wv +b

aw2.

(9)


du

dτ= −u3 + u + uv,

dv

dτ= −vu2 + v + v2.

(10)

The system (10) has an unstable node (0, 0) anda singular stable parabola v = u2 − 1. Also, we canobtain the local phase portraits as shown in Fig. 7.The flow in the local chart V3 is the same as theflow in the local chart U3, hence, the phase por-trait of the system (4) on the infinite sphere at thenegative end point of the z axis is shown in Fig. 7,

Fig. 7. Trajectories in the local phase portraits of sys-tem (10).

reversing the time direction. From the above anal-ysis, we have the following proposition. The phaseportrait of the system (4) on the Poincare sphereat infinity is as shown in Fig. 8. There exists twosingular stable parabolas at the upper hemisphereand lower hemisphere and two nodes at the positive(unstable) and negative (stable) of the z axis.

Fig. 8. Phase portrait of the system (4) on the Poincaresphere at infinity. The blue curves and red curves are equi-librium points.

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4. Hopf Bifurcation and Generationof Hidden Chaos in theKingni–Jafari System

Suppose that the characteristic equation of Kingni–Jafari system (4) has a pair of pure imaginaryroots ±iω (ω ∈ R+). It is easy to show that whenb = b0 = a2

3−a , (2) yields

λ1 =a

a − 3< 0, λ2,3 = ±√

3 − ai,

where 0 < a < 3.

Proposition 4.1. Define

T = (a, b) | 0 < a < 3, b = b0 =a2

3 − a,

then the Jacobian matrix of system (4) at O(0, 0, 0)has one negative real eigenvalue a

a−3 and a pair ofpurely imaginary eigenvalues ±√

3 − ai.

Taking b as the Hopf bifurcation parameter, thetransverse condition

Re(λ′(b0))|λ=√

3−ai

= − (−3 + a)3

2a(−27 + 27a − 10a2 + a3)< 0 (11)

is also satisfied. Therefore, we have the followingtheorem.

Theorem 4.2. If (a, b) ∈ T and b varies andpasses through the critical value b0 = a2

3−a , sys-tem (4) undergoes Hopf bifurcation at the equilib-rium O(0, 0, 0).

We first review the projection methoddescribed in Chapters 3 and 5 [Kuznetsov, 2004],but following the analysis of [Mello & Coelho, 2009;Sotomayor et al., 2007a, 2007b], for the calculationof the first Lyapunov coefficient l1, associated withthe stability of a Hopf bifurcation. Consider thedifferential equation

X = f(X,µ), (12)

where X ∈ R3 and µ ∈ R2 are respectively vectorsrepresenting phase variables and control parame-ters. Assume that f is a class of C∞ in R3 × R2.Suppose that (12) has an equilibrium point X = X0

at µ = µ0, and denoting the variable X − X0 alsoby X, write

F (X) = f(X,µ0), (13)

as

F (X) = AX +12B(X,X)

+16C(X,X,X) + O(‖X‖4),

where A = fx(0, µ0) and, for i = 1, 2, 3,

B(X,Y ) =3∑

j,k=1

∂2Fi(ξ)∂ξj∂ξk

∣∣∣∣∣∣ξ=0

XjYk,

C(X,Y,Z) =3∑

j,k,l=1

∂3Fi(ξ)∂ξj∂ξk∂ξl

∣∣∣∣∣∣ξ=0

XjYkZl.

Suppose that A has a pair of complex eigenvalues onthe imaginary axis: λ2,3 = ±iw0 (w0 > 0), and theseeigenvalues are the only eigenvalues with Reλ = 0.Let T c be the generalized eigenspace of A corre-sponding to λ2,3. Let p, q ∈ C3 be vectors such that

Aq = iw0q, AT p = −iw0p, 〈p, q〉 = 1,

where AT is the transpose of the matrix A. Anyvector y ∈ T c can be represented as y = wq + wq,where w = 〈p, y〉 ∈ C. The two-dimensional centermanifold associated to the eigenvalues λ2,3 can beparameterized by w and w, by means of an immer-sion of the form X = H(w, w), where H : C2 → R3

has a Taylor expansion of the form

H(w, w) = wq + wq

+∑

2≤j+k≤3

1j!k!

hjkwjwk + O(|w|4),

with hjk ∈ C3 and hjk = hkj. Substituting thisexpression into (13) we obtain the following differ-ential equation

Hww′ + Hww′ = F (H(w, w)),

where F is given by (13). The complex vectors hij

are obtained by solving the system of linear equa-tions defined by the coefficients of (13). Taking intoaccount the coefficients of F , system (13), on thechart w for a central manifold, can be written asfollows

w = iw0w +12G21w|w|2 + O(|w|4),

with G21 ∈ C. The first Lyapunov coefficient can bewritten as

l1 =12Re G21,

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where

G21 = 〈p,C(q, q, q) + B(q, h20) + 2B(q, h11)〉.Defining H32 as

H32 = 6B(h11, h21) + B(h20, h30) + 3B(h21, h20)

+ 3B(q, h22) + 2B(q, h31) + 6C(q, h11, h11)

+ 3C(q, h20, h20) + 3C(q, q, h21)

+ 6C(q, q, h21) + 6C(q, h20, h11)

+ C(q, q, h30) + D(q, q, q, h20)

+ 6D(q, q, q, h11) + 3D(q, q, q, h20)

+ E(q, q, q, q, q) − 6G21h21 − 3G21h21

and G32 = 〈p,H32〉, the second Lyapunov coefficientl2 is given by

l2 =112

ReG32. (14)

A Hopf bifurcation point (X0, µ0) is an equilib-rium point of (12) where the Jacobian matrix A hasonly a pair of purely imaginary eigenvalues ±iw0

(w0 > 0), and the other eigenvalue with nonzeroreal part. At a Hopf point a two-dimensional centermanifold is well defined, it is invariant under theflow generated by (12) and can be continued witharbitrary high class of differentiability to nearbyparameter values.

A Hopf point is called transversal if theparameter-dependent complex eigenvalues cross theimaginary axis with nonzero derivative. In a neigh-borhood of a transversal Hopf point with l1 = 0 thedynamic behavior of the system (12), reduced to the

family of parameter-dependent continuations of thecenter manifold, is orbitally topologically equivalentto the following complex normal form

w′ = (η + iw)w + l1w|w|2,where w ∈ C, η, w and l1 are real functions hav-ing derivatives of arbitrary high order, which arecontinuations of 0, w0 and the first Lyapunov coef-ficient at the Hopf point [Sotomayor et al., 2007a].As l1 < 0 (l1 > 0) one family of stable (unstable)periodic orbits can be found on this family of mani-folds, shrinking to an equilibrium point at the Hopfpoint.

A Hopf point of codimension two is a Hopfpoint where l1 vanishes. It is called transversal ifη = 0 and l1 = 0 have transversal intersections,where η = η(µ) is the real part of the critical eigen-values. In a neighborhood of a transversal Hopfpoint of codimension two with l2 = 0 the dynamicbehavior of the system (12) reduced to the familyof parameter-dependent continuations of the centermanifold, is orbitally topologically equivalent to

w′ = (η + iw)w + τw|w|2 + l2w|w|4,where η and τ are unfolding parameters.

In the rest of this section we employ the three-dimensional Hopf bifurcation theory and applysymbolic computations to perform the analysis ofparametric variations with respect to dynamicalbifurcations. Because the system has only one equi-librium, we consider the bifurcation of the sys-tem (4) at O(0, 0, 0) (system (1) at E0).

Theorem 4.3. Consider the two-parameter fam-ily of differential equations (4). The first Lyapunovcoefficient associated with the equilibrium O is givenby

l1 =(a − 3)(−648 + 1350a − 1185a2 + 551a3 − 131a4 + 12a5)a(2916 − 5832a + 4995a2 − 2295a3 + 586a4 − 77a5 + 4a6)

.

If l1 > 0 then the Hopf point at O is unstable (weakrepelling focus) and for each b > b0 = a2

3−a , but closeto b0, there exists an unstable limit cycle near theasymptotically stable equilibrium O; if l1 < 0 thenthe Hopf point at O is asymptotically stable (weakattractor focus) and for each b > b0 = a2

3−a , butclose to b0, there exists a stable limit cycle near theunstable equilibrium O.

Proof Under this condition 0< a< 3, the transver-sality condition (11) is also satisfied. Accordingly,

Hopf bifurcation at O occurs. The stability of Odepends on the value of the first Lyapunov coeffi-cient l1, which shows the stability of the equilibriumpoint and the periodic orbits which appear. Usingthe notation of the previous section, the multilinearsymmetric functions can be written as

B(x, y) = (0, 0, 2x1y1 − 2x3y3 − x2y3 − x3y2),

C(x, y, z) = (0, 0, 0, 0).

Furthermore, we can also obtain

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p =

(−a

√3 − a + 3(a − 3)i

2[(3 − a)3/2 + ia],− i(3 − a)a

2[(3 − a)3/2 + ia],

(3 − a)3/2

2[(3 − a)3/2 + ia]

),

q =(

i√3 − a

,−1 +

√3 − ai

3 − a, 1)

,

h11 =(

0,−2(1 − a)(3 − a)a

, 0)

,

h20 =(− 4(i +

√3 − a)

3[(3 − a)3/2 − ia],

10 − 4a + 6√

3 − ai

−6(a − 3)2 + 3a√

3 − ai,−8(

√3 − a + (a − 3)i)

3[(3 − a)3/2 − ia]

).

Then we can compute the following value

G21 =−108 + 153a − 57a2 + 6a3 + a(57 − 47a + 8a2)

√3 − ai

3a(−54 + 54a − 17a2 + 2a3) + 9(3 − a)3/2a2i

and

l1 =(a − 3)(−648 + 1350a − 1185a2 + 551a3 − 131a4 + 12a5)

2a(2916 − 5832a + 4995a2 − 2295a3 + 586a4 − 77a5 + 4a6).

Therefore, Theorem 4.3 is proved.

In order to investigate the relationship betweenthe Hopf bifurcation and hidden chaos, we justifythe above theoretical analysis of the first Lyapunovcoefficient for the Hopf bifurcation of system (1)with a = 1.3, according to Theorem 4.3, we haveb0

.= 0.9941 and l1.= 0.0682 > 0. Choosing ini-

tial condition (0.1, 0.87,−0.09), an unstable peri-odic solution could be found near the stable equilib-rium point E0 for b = 0.996 > b0. This is indeed thecase, as shown in Fig. 9 (blue orbits). In addition,when parameters a and b are kept unchanged, itshows that the initial condition changes slightly andbecomes (0.15, 0.87,−0.09), hidden chaotic attrac-tor occurs from the unstable limit cycle that arose inthe Hopf bifurcation (see Fig. 9 (red orbits)). There-fore, this system has a coexisting point, a periodiccycle and a hidden chaotic attractor, demonstrat-ing that this new system is truly complicated andinteresting.

In the following theorem we study the sign ofthe second Lyapunov coefficient l2 when the firstcoefficient l1 vanishes.

Theorem 4.4. For system (1) there are two param-eter sets

D1 =

(a, b) | a = a1, b =a2

3 − a

,

D2 =

(a, b) | a = a2, b =a2

3 − a

,

where a1.= 1.5359 and a2

.= 2.2768 are the onlytwo roots in the region 0 < a < 3 for l1 = 0. Thesecond Lyapunov coefficients at E0 for parametervalues (a, b) ∈ D1 or D2 are given by

l2|D1

.= −0.3423,

l2|D2

.= −0.2132.

Therefore, for the set of parameter values in D1 orD2, system (1) has a transversal Hopf point of codi-mension two at E0 which is stable since l2 < 0.

Proof. The second Lyapunov coefficient can beobtained for the parameters on the set D1. Fora = a1, one has

G21 = −0.37936i

and

h21 = (−0.12824 + 0.01838i,−0.35756 − 0.08761i,

−0.29128 + 0.15517i),

h30 = (0.65523 + 0.28148i, 0.57769 + 0.46199i,

1.02181 − 2.37852i),

h31 = (0.18321 − 0.09332i, 0.22177 − 0.01762i,

4.46976 + 0.87345i),

h22 = (0, 0.20030, 0),

G32 = −4.10790 − 2.97868i.

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By the above theorem and calculation, one has

l1 =12ReG21 = 0,

l2|D =112

Re G32 = −0.3423 < 0.

Similarly, we can get

l1 =12ReG21 = 0,

l2|D =112

Re G32 = −0.2132 < 0.

Therefore, we can arrive at the above Theorem 4.4.

5. Zero-Hopf Bifurcation andPeriodic Orbits in Kingni–JafariSystem

We first present a result from the averaging theory[Sanders et al., 2007; Llibre & Zhang, 2009; Llibre &Valls, 2011; Llibre & Chavela, 2014; Llibre et al.,2015].

Theorem 5.1. We consider the two following ini-tial value problems

x = εf(t, x) + ε2g(t, x, ε), x(0) = x0 (15)

and

y = εf0(y), y(0) = y0, (16)

where x, y, x0 ∈ Ω is an open subset of Rn, t ∈[0,∞), ε ∈ (0, ε0), f and g are periodic of T inthe variable t, and f0(y) is the averaged function off(t, x) with respect to t, i.e.

f0(y) =1T

∫ T

0f(t, y)dt. (17)

Suppose:

(i) f, its Jacobian ∂f∂x , its Hessian ∂2f

∂x2 , g, its Jaco-bian ∂g

∂x are defined, continuous and bounded bya constant independent of ε ∈ [0,∞) × Ω andε ∈ (0, ε0);

(ii) T is a constant independent of ε;(iii) y(t) belongs to the interval of time [0, 1/ε].

Then the following statements hold [8, 10 ]:

(a) For the time scale 1/ε, we have that x(t)−y(t) = O(ε), as ε → 0.

(b) If p is a singular point of the averagedsystem (16) such that the determinant ofthe Jacobian matrix is not zero, then there

exists a limit cycle φ(t, ε) of period T forsystem (15) which is close to p and suchthat φ(t, ε) → p as ε → 0.

(c) The stability or instability of the limit cycleφ(t, ε) is given by the stability or instabil-ity of the singular point p of the averagedsystem (16). In fact, the singular point phas the stability behavior of the Poincaremap associated to the limit cycle φ(t, ε).

It is obvious that the averaging method gives aquantitative relation between the solutions of somenonautonomous periodic differential system and thesolutions of its averaged differential system, which isan autonomous one. It provides a first order approx-imation in ε for the limit cycles of a periodic differ-ential system.

We recall that an equilibrium point isa zero-Hopf equilibrium of a three-dimensionalautonomous differential system, if it has a zero realeigenvalue and a pair of purely imaginary eigenval-ues. Therefore, this system (1) has no zero-Hopfbifurcations from Eq. (2). In order to exhibit dif-ferent topological type of dynamics in the smallneighborhood of this isolated equilibrium as thetwo parameters vary in a small neighborhood ofthe origin, in the next result, we characterize theequilibrium point (0, 0, 0) is zero-Hopf equilibriumwhen Kingni–Jafari system (1) has the parameterb = ka2:

x = −z,

y = −x − z,

z = 3x − ay + x2 − z2 − yz − kaz.

(18)

According to the condition (3), the equilibriumpoint (0, 0, 0) is stable when k > 1

3−a . Therefore,we limit k > 1

3−a .

Proposition 5.2. The differential system (18) has aunique zero-Hopf equilibrium localized at the originof the coordinates when a = 0. Moreover, the eigen-values at the origin for this one-parameter familyare 0, and ±√

3i.

We shall study the system with a zero-Hopfbifurcation producing some unstable periodic orbitfor a > 0 sufficiently small that bifurcates from azero-Hopf equilibrium point localized at the originof coordinates when a = 0. It is worth noting thatthe unstable periodic orbit has been numericallyextracted from chaotic attractors of continuous

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dynamical systems. Using unstable periodic solu-tions, we attempt to explain typical behavior andthe statistical properties of the chaotic solutions.Therefore, we have the following result.

Theorem 5.3. When k > 13−a , the system (18)

has a zero-Hopf bifurcation at the equilibrium point(0, 0, 0) when a = 0, and produces one small peri-odic solution for a > 0 sufficiently small :

x =a√

2(3k − 1) cos θ√3

,

y =a√

2(3k − 1) cos θ√3

− a√

2(3k − 1) sin θ

3+

a(1 − 3k)3

,

z = a√

2(3k − 1) sin θ.

Moreover, the periodic solution is unstable andhas two invariant manifolds, one stable and oneunstable, each of them formed by two cylinders.

Proof. In order to find the change of variables whilewriting our differential system (18) in normal formfor applying the averaging theory, we first writethe linear part at the origin of the differential sys-tem (18) when a = 0 into its real Jordan normalform, i.e. into the form

0

√3 0

−√3 0 0

0 0 0

.

For that we consider the linear change

x =u√3, y =

u√3− v

3+ w, z = v.

Using the new variables (u, v,w), the differentialsystem (18) can be written as

u = −√

3v,

v =(√

3 − 1√3a

)u +

(13a − ak

)v − aw +

u2

3− 2v2

3− uv√

3− vw,

w = − a

3√

3u +

(a

9− ak

3

)v − aw

3+

u2

9− uv

3√

3− 2v2

9− vw

3.

(19)

Now we write the differential system (19) in cylindrical coordinates (u, v) = (r cos θ, r sin θ)

r = −16

sin θ(r2 + 6aw + 2√

3ar cos θ − 3r2 cos 2θ − 2ar sin θ + 6akr sin θ + 6rw sin θ +√

3r2 sin 2θ),

θ =√

3 − 16r cos θ − aw cos θ

r− a cos2θ√

3+

12r cos θ cos 2θ +

(13a − ak − w

)cos θ sin θ − r cos θ sin 2θ

2√

3,

w =118

(−r2 − 6aw − 2√

3ar cos θ + 3r2 cos 2θ + 2ar sin θ − 6akr sin θ − 6rw sin θ −√

3r2 sin 2θ).

(20)

If the rescaling (r, w) = (εR, εW ) with a = ε > 0 being a sufficiently small parameter, then thesystem (20) becomes

dR

dθ= εF11(θ,R,W ) + O(ε2),

dW

dθ= εF21(θ,R,W ) + O(ε2),

(21)

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−3−2

−10

1

−2−1

01

23

−2

0

2

4

6

E0

z(t)

*

x(t)

y(t)

Hidden chaosPeriodic orbit

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−1

0

1

2

3

4

5

6

*

E0

x(t)

y(t)


(a) (b)

−1 0 1 2 3 4 5−4

−3

−2

−1

0

1

2

*

E0

y(t)

x(t)

Hidden chaosPeridoc orbit

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−4

−3

−2

−1

0

1

2

*

E0

x(t)

z(t)


(c) (d)

Fig. 9. Coexistence of hidden chaotic attractor and periodic orbit of system (1) with parameters a = 1.3, b = 0.996 > b0.=

0.9941; the hidden chaotic attractor (red) from initial condition (0.15, 0.87,−0.09) and the periodic attractor (blue) from initialcondition (0.1, 0.87,−0.09).

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−1−0.5

00.5

1

x 10−3

−5−3

−11

35

x 10−4

−1.5

−1

−0.5

0

x 10−3

z(t)x(t)

y(t)

−5 −3 −1 1 3 5

x 10−4

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4x 10

−3

x(t)

y(t)

(a) (b)

−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4

x 10−3

−8

−6

−4

−2

0

2

4

6

8x 10

−4

y(t)

z(t)

−5 −3 −1 1 3 5

x 10−4

−8

−6

−4

−2

0

2

4

6

8x 10

−4

x(t)

z(t)

(c) (d)

Fig. 10. Phase diagram (red parts) and theoretical approximate orbit (black parts) of system (18) with a = 0.0001, k = 10and initial condition (4.3970a,−5.2697a, 0).

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where

F11(θ,R,W ) = −sin θ

6√

3(R2 + 6W + 2

√3R cos θ

− 3R2 cos 2θ − 2R sin θ + 6kR sin θ

+ 6RW sin θ +√

3R2 sin 2θ),

F21(θ,R,W ) = − 118√

3(R2 + 6W + 2

√3R cos θ

− 3R2 cos 2θ − 2R sin θ + 6kR sin θ

+ 6RW sin θ +√

3R2 sin 2θ).

Using the notation of the averaging theory wehave t = θ, T = 2π, x = (R,W ). It is immediate tocheck that system (21) satisfies all the assumptionsof the averaging theory for periodic orbits.

Now we denote g(R,W ) = (g11(R,W ), g21(R,W )) and compute the integrals (21), i.e. the aver-aged functions

g11(R,W ) =12π

∫ 2π

0F11(θ,R,W )dθ

= −R(−1 + 3k + 3W )6√

3,

g21(R,W ) =12π

∫ 2π

0F21(θ,R,W )dθ

= −R2 + 6w18√

3.

Therefore, the g11(θ,R,W ) = g21(θ,R,W ) = 0 hasa unique positive real solution (R∗,W∗) when k > 1

3 ,where

R∗ =√

2(3k − 1), W∗ =13(1 − 3k).

In addition, we know the Jacobian at the point(R∗,W∗) is

1 − 3k27

< 0

and the eigenvalues at the solution (R∗,W∗) are

−√3 ± 3

√−1 + 4k18

.

This shows that the periodic orbit is unstable hav-ing a stable manifold and an unstable manifold bothformed by two cylinders. Therefore, it guaranteesfor ε > 0 sufficiently small the existence of a peri-odic solution (R,W ) → (R∗,W∗) when ε → 0. So

system (18) for ε = 0 sufficiently small has one peri-odic orbit χε → w = W∗∩u2 + v2 = R2∗. Goingback to the differential system (20), we get thatsuch a system for a = ε > 0 sufficiently small hasone periodic solution of period approximately 2π ofthe form

(r, w) = (εR∗ + O(a), εW∗ + O(a)).

Moreover, the differential system (19) of two peri-odic solutions of period close to 2π is of the form

u(t) = a√

2(3k − 1) cos θ,

v(t) = a√

2(3k − 1) sin θ,

w(t) =a(1 − 3k)

3,

for a > 0 sufficiently small. Finally, we get for thedifferential system (18) the periodic solution

x =a√

2(3k − 1) cos θ√3

,

y =a√

2(3k − 1)(√

3 cos θ − sin θ)3

+ a(1 − 3k),

z =√

2(3k − 1) sin θ,

(22)

a > 0 being sufficiently small. It is a periodic solu-tion starting at the zero-Hopf equilibrium point.This completes the proof of Theorem 5.3.

For a = 0.0001, k = 10 and initial condi-tion (4.3970a,−5.2697a, 0), we obtain the numericalphase diagram (the red parts in Fig. 10). In addi-tion, according to the solution (22), we also presentthe theoretical orbit using averaging method (theblack areas in Fig. 10). It is easy to find the correct-ness and effectiveness of the analytical results. Insome cases, the zero-Hopf bifurcation could implya local birth of “chaos”, see for instance the arti-cles [Scheurle & Marsden, 1984; Champneys & Kirk,2004; Baldoma & Seara, 2006].

6. Conclusion

It is apparent that the Kingni–Jafari system belongsto the newly introduced class of chaotic systemswith hidden attractors. There are still abundant andcomplex dynamical behaviors and the topologicalstructure of the new system should be completelyand thoroughly investigated and exploited. In this

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paper, some complex dynamics of the Kingni–Jafarisystem have been studied theoretically. A completedescription of the dynamics of this system at infin-ity is given using the Poincare compactification ofpolynomial vector in R

3. The generation mechanismof the hidden chaos for this special system is inves-tigated. It exhibits the result of hidden chaos froma Hopf bifurcation point. In order to determine theLyapunov stability when the first Lyapunov coeffi-cient vanishes, we make an extension of the analy-sis to degenerate cases. In addition, in the certainparameter region that the equilibrium is stable, ananalytic proof of the existence of zero-Hopf bifurca-tion is exhibited by the averaging theory. An unsta-ble periodic orbit from the zero-Hopf bifurcationis obtained. This approach may be useful to clar-ify the complicated phenomena that hidden chaoticattractors coexist with stable equilibrium.

Indeed, the global dynamical behaviors and thegeometrical structure of this system have not beenpresented completely. It is expected that a moredetailed theoretical analysis and simulation investi-gation will be provided in a forthcoming study.

Acknowledgments

We would like to express our gratitude to the ref-erees for their valuable comments and suggestionsthat led to a truly significant improvement of themanuscript. This work was supported by the PolishNational Science Centre, MAESTRO Programme–Project (No. 2013/08/A/ST8/00/780), the Natu-ral Science Foundation of China (No. 11401543),the China Scholarship Council (CSC) (No.201506415023), Beijing Postdoctoral ResearchFoundation (No. 2015ZZ17), the China Postdoc-toral Science Foundation funded project (Nos.2014M560028 and 2015T80029), the Fundamen-tal Research Funds for the Central Universi-ties, China University of Geosciences (Wuhan)(No. CUGL150419), the Government of ChaoyangDistrict Postdoctoral Research Foundation (No.2015ZZ-7) and the Funding Project for AcademicHuman Resources Development in Institutions ofHigher Learning under the Jurisdiction of BeijingMunicipality (PHRIHLB).

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dynamics at infinity, degenerate hopf and zero-hopf ...sprott.physics.wisc.edu/pubs/paper458.pdf ·...

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