# Complex Dynamics in Generalized Hénon Map

Authors: Meixiang Cai
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

The complex dynamics of generalized
Hénon map with nonconstant Jacobian determinant are
investigated. The conditions of existence for fold bifurcation,
flip bifurcation, and Hopf bifurcation are derived by using center
manifold theorem and bifurcation theory and checked up by
numerical simulations. Chaos in the sense of Marotto's definition
is proved by analytical and numerical methods. The numerical
simulations show the consistence with the theoretical analysis and
reveal some new complex phenomena which can not be given by
theoretical analysis, such as the invariant cycles which are
irregular closed graphics, the six and forty-one coexisting
invariant cycles, and the two, six, seven, nine, ten, and thirteen
coexisting chaotic attractors, and
some kinds of strange chaotic attractors.


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## Body

## 1. Introduction

The planar map(1)xn+1=1+ayn-bxn2,yn+1=xnwas first introduced by Hénon [1] as a planar diffeomorphism that imitated essential stretching and folding properties of the Poincaré map of the Lorenz system. This original Hénon map (1) had a strange attractor with fractal structure and had constant Jacobian determinant |J|=-b. Since the late 1970s, Hénon map (1) served as an important but artificial example to illustrate many analytical results and numerical techniques of dynamical theory. For example, Feit [2] introduced the characteristic exponents in order to estimate strange attractors numerically. Marotto [3] proved analytically that the map had a transversal homoclinic orbit, which implied the existence of the chaotic behavior for some parameter values. Curry [4] presented a lot of numerical experiments on Hénon map by using the characteristic exponent, frequency spectrum, and a theorem of Smale [5]. Mora and Viana [6] proposed the more general theory of strange attractors. Sonis [7] presented a detailed description of bifurcation phenomena by using the analysis approach. Cao [8] proved that there exists a set with positive Lebesgue measure, which corresponded to a map possessing a strange attractor. Luo and Guo [9] investigated the complete bifurcation and stability of the stable and unstable periodic solutions and the chaotic layers by introducing the positive and negative iterative mappings. Zhang [10] verified the existence of Wada basin boundaries in a switched Hénon map.For the measure preserving Hénon map (|J|=-b=-1), there has been some attention. Such as, Brown [11] proved that the measure preserving Hénon map contained an embedded horseshoe for a>0 by using geometric methods and a contradiction argument. Kirchgraber and Stoffer [12] proved this Hénon map existing a transversal homoclinic point for a set of parameters which was not small by using shadowing techniques. Jensen [13] proved that the unstable manifold of a hyperbolic fixed point was the iterated limit of a very simple set.In this paper, we study the following extension of map (1):(2)xn+1=1+ayn-bxn2+cxnyn+dyn3,yn+1=xn+eyn,c≠0,1,where b, c, and d are constants. Following [14], we call this map the generalized Hénon map (GHM), which has nonconstant Jacobian determinant |J|=-(2be+c)x-3dy2+cey-a.Our motivation to study this particular generalized Hénon map is that it appears in the bifurcation analysis of nontransversal homoclinic orbits and heteroclinic cycles and plays an important role in other homoclinic studies. In [15], if a diffeomorphism in R2 has two saddle fixed points connected by two heteroclinic orbits and one of which is nontransversal, then the GHM appears as a rescaled first return map. In [16], If a diffeomorphism in R3 has a codimension 1 homoclinic tangency to a saddle-focus fixed point, the GHM appears when the eigenvalues satisfy some conditions. Moreover, when the so-called effective dimension of the problem in [17] can change, the GHM also can be expected as a rescaled first-return map in other cases of homoclinic and heteroclinic tangencies.In this paper, the bifurcations and chaos phenomenons in map (2) are investigated. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory [18]; chaotic behavior in the sense of Marotto’s definition [19] is proved. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamics behaviors.This paper is organized as follows. In Section2, the existence and stability of the fixed points are given. In Section 3, the sufficient conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are given. In Section 4, we first rigorously prove the existence of chaos in the sense of Marotto’s definition and then give an example to check up the analytic results. The numerical simulations are presented in Section 5 to verify the theoretical analysis and display the new and interesting dynamics. Finally, we give a conclusion in Section 6.


## 2. Existence and Stability of Fixed Points

The fixed points of map (2) satisfy the following equations:(3)1+ay-bx2+cxy+dy3=x,x+ey=y.By a simple analysis about (3), it is easy to obtain the following proposition.Proposition 1.
(i)
IfΔ>0, map (2) has a unique fixed point at Z1(x10,y10),
(ii)
ifΔ=0, map (2) has two fixed points at Z1(x10,y10) and Z2(x20,y20),
(iii)
ifΔ<0, map (2) has three fixed points at Z1(x10,y10), Z2(x20,y20), and Z3(x30,y30),
whereΔ=4[3d(a+e-1)-(1-e)2(c-b+be)2]3+[2(1-e)3(c-b + be)3-9d(1-e)(c-b+be)(a+e-1)+27d2]2.The Jacobian matrixJ of map (2) evaluated at fixed point Z(x,y) is given by (4)J=-2bx+cya+cx+3dy21e.The characteristic equation of the Jacobian matrix J can be written as (5)λ2+p(x,y)λ+q(x,y)=0,where p(x,y)=2bx-cy-e and q(x,y)=-(2be+c)x-3dy2+cey-a.A simple calculation shows the stability of the fixed points as the following proposition.Proposition 2.
The fixed pointZ(x,y) of map (2) is stable if one of the following conditions holds:(i)
-1<-(2be+c)x-3dy2+cey-a<1;(ii)
-(2be+c)x-3dy2+cey-a<1 and (2b-2be+c)x-3dy2+c(e-1)y+1-e-a>0 and -(2b+2be+c)x-3dy2+c(e+1)y+1+e-a>0.


## 3. Bifurcations

In analysis of the fold bifurcation, flip bifurcation, and Hopf bifurcation of map (2), b is used as the bifurcation parameter. By using center manifold theorems and bifurcation theory, the conditions of existences for the bifurcations are given as follows.Theorem 3.
Map (2) undergoes a fold bifurcation at Z(x(b0),y(b0)) if the following conditions are satisfied at the same time:(6)b0≠cy0+e2x0,b0≠cy0+e-22x0,b0=cx0+3dy02+c1-ey0+a+e-121-ex0,α=1A2b01-e2-c1-e-3dy0≠0.Moreover, if α<0 (resp., α>0), the two fixed points bifurcate from Z for b>b0 (resp., b<b0) and coalesce as the fixed point Z at b=b0 and disappear for b<b0 (resp., b>b0) (see Appendix A for the proof).Theorem 4.
Map (2) undergoes a flip bifurcation at Z(x(b*),y(b*)) if the following conditions are satisfied at the same time:(7)b*≠cy0+e+12x0,b*≠cy0+e+32x0,b*≠-cx0-3dy02+cey0-a2ex0,b*=-cx0-3dy02+c(1+e)y0-a+e+121+ex0,α1≠0,α2≠0.Moreover, if α2>0 (resp., α2<0), the period-2 points that bifurcate from this fixed point are stable (resp., unstable) (see Appendix B for the proof and expressions of α1 and α2).Theorem 5.
Map (2) undergoes a Hopf bifurcation at Z(x(b¯),y(b¯)) if the following conditions are satisfied at the same time:(8)Δ′=-x023dy02+cx0+a>0,-x0e-cy0-2Δ′2x02<b<-x0e-cy0+2Δ′2x02,b¯≠cy0+e2ax0,b¯≠cy0+e+12ax0,b¯=-3dy02+cx0-ey0+a+12ex0,l≠0.Moreover, if l<0 (resp., l>0) and k=(dλb/db)b=b¯>0; then, an attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for b>b¯ (resp., b<b¯) (see Appendix C for the proof and expression of l).


## 4. Existence of Marotto’s Chaos

In this section, we rigorously prove that map (2) possesses chaotic behavior in the sense of Marotto’s definition in [19].Suppose thatZ0(x0,y0) is the fixed point of map (2). We first give the conditions such that the fixed point Z0 is a snap-back repeller. The eigenvalues associated with the fixed point Z0 are given by (9)λ1,2=-p(y0)±p2(y0)-4qy02,where p(y0)=[2b(1-e)-c]y0-e and q(y0)=-3dy02-[2be(1-e)+c(1-2e)]y0-a.According to Definition  1 in [19], we begin to find a set UZ0 in which the norm of conjugate complex eigenvalues exceeds 1 for all Z∈UZ0. Let H(λ)=λ2+p(y)λ+q(y)=0, which represents the characteristic equation for any point (x,y) in R2. If Δ=p2(y)-4q(y)<0 and q(y)>1, then the characteristic equation have a pair of conjugate complex eigenvalues satisfying |λ1,2|>1.Let(10)φ1y=Δ=2b-2be-c2+12dy2=Δ=+22be-2be2-3ce+2cy+4a+e2,φ2y=qy-1=-3dy2-2be1-e+c1-2ey-a-1;then,(11)Δ1′=16c4-be-4bee-124a+e2-4c2+12d+4bce-1+4b2e-12c4-be-4bee-12·4a+e2,Δ2′=2be1-e+c1-2e2-12da+1.IfΔ1′>0, the roots of φ1(y)=0 are y1,2=(-2be+2be2+3ce-2c±2Δ1′)/((2b-2be-c)2+12d), and if Δ2′>0, the roots of φ2(y)=0 are y3,4=(2be(1-e)+c(1-2e)±Δ2′)/6d.From the above discussion, we obtain the following results.Lemma 6.
Supposing thatD1={y∣p2(y)-4q(y)<0}, D2={y∣q(y)>1} and D=D1∩D2, we have the following. (i)
If(2b-2be-c)2+12d>0, d>0, Δ1′>0, Δ2′>0, and y1≤y3<y2 or y1<y4≤y2 or y3≤y1<y4 or y3<y2≤y4, then D=(y1,y2)∩(y3,y4)≠∅.(ii)
If(2b-2be-c)2+12d>0, d<0, Δ1′>0, Δ2′>0, and y1<y3 or y2>y4, then D=(y1,y2)∩((-∞,y3)∪(y4,+∞))≠∅.(iii)
If(2b-2be-c)2+12d>0, d<0, Δ1′>0, Δ2′<0, then D=(y1,y2)≠∅.(iv)
If(2b-2be-c)2+12d<0, d<0, Δ1′>0, Δ2′>0, then D=((-∞,y1)∪(y2,+∞))∩((-∞,y3)∪(y4,+∞))≠∅.(v)
If(2b-2be-c)2+12d<0, d<0, Δ1′>0, Δ2′<0, then D=(-∞,y1)∪(y2,+∞)≠∅.(vi)
If(2b-2be-c)2+12d<0, d<0, Δ1′<0, Δ2′>0, then D=(-∞,y3)∪(y4,+∞)≠∅.(vii)
If(2b-2be-c)2+12d<0, d<0, Δ1′<0, Δ2′<0, then y is arbitrary.Lemma 7.
If one of the conditions in Lemma6 is satisfied and the y-coordinate of fixed point Z0(x0,y0) satisfies y0∈Dy0⊂D, then Z0(x0,y0) is an expanding fixed point of map (2) in UZ0={(x,y)∣y∈Dy0,x∈R}.Due to Definition 2 of snap-back repeller in [19], we need to find one point Z∈UZ0 such that Z≠Z0, FM(Z)=Z0, and |DFM(Z)|≠0 for some positive integer M.In fact, we have(12)1-ay-bx2+cxy+dy3=x1,x+ey=y1,(13)1-ay1-bx12+cx1y1+dy13=x0,x1+ey1=y0.Now, a map F2 has been constructed to map the point Z(x,y) to the fixed point Z0(x0,y0) after two iteration if there are solutions different from Z0 for (12) and (13). By calculation, the solutions different from Z0 for (13) are(14)y1±=-dy0+be2+ce+b2e4+2bce3+c2e2-4ad1/2±-3d2y02+2bde2+cde-2cd-4bdey0+b2e4+2bce3+c2e2-4ad1/22d-1,x1±=y0-ey1±.Substituting (14) into (12), we have(15)dy3-be2+cey2+a+2bey1+cy1y-by12+ey1-y0+1=0,where y1=y1±.Next, we expect to find a real root of (15) satisfying y*∈Dy0. Let y=u+y0, then (15) becomes(16)du3+3dy0-be2-ceu2+3dy02-2be2y0-2cey0+a+cy1+2bey1u+dy03-be2+cey02+a-1y0+1+ey1+cy0y1+2bey0y1-by12=0.Let u=v-(3dy0-be2-ce)/3d; then, (16) becomes(17)v3+Pv+Q=0,where P=(1/3d2)(3ad-c2e2-2bce3-b2e4+3cdy1+6bdey1), Q=(1/27d3){-2e3(c+be)3+9de(c+be)[a+(c+2be)y1]-27d2(-1+y0-ey1+by12)}, and Δ=(P/3)3+(Q/2)2.If the real root of (17)  v*≠(3dy0-be2-ce)/3d, then (15) has at least a real root y*≠y0.Obviously, if one of the conditions in Lemma6 is satisfied and v*≠(3dy0-be2-ce)/3d and the solution of (15) satisfies y*∈UZ0, then Z0 is a snap-back repeller in UZ0. Thus, we have the following theorem.Theorem 8.
If one of the conditions in Lemma6 holds and v*≠(3dy0-be2-ce)/3d and the solutions (x1±,y1±) and (x*,y*) of (14) and (15) satisfy y1±,y*≠y0,(x*,y*)∈UZ0, then the point Z0(x0,y0) is a snap-back repeller of map (2), and hence, map (2) is chaotic in the sense of Marotto’s definition, where UZ0 is given in Lemma 7.Next, we give specific values of the parameters for illustrating and verifying the conditions in Theorem8.Example 9.
Fora=-1.5, b=0.3, c=0.8, d=0.1, e=0.1, map (2) has a fixed point Z0(x0,y0)=(0.417165,0.463517), and the eigenvalues of the fixed point Z0 are λ±=0.110257±1.19306i. From Lemma 6, we have D1=(-41.6035,39.3725), D2=(-0.576696,2.89003), and D=(-0.576696,2.89003), which shows that y0=0.463517∈Dy0⊂D. We choose y1=y1+=3.53245∉D; then, the solution of (15) is y*=1.63427≠y0 and y*∈Dy0. There exists a point Z(x*,y*)=(3.36302,1.69427)∈UZ0, which is different from Z0 and satisfies F2(Z)=Z0 and DF2(Z)≠0. So; Z0 is a snap-back repeller.
The bifurcation diagram and the corresponding maximum Lyapunov exponents of map (2) in (b,x) plane for a=-1.5, c=0.8, d=0.1, and e=0.1 are shown in Figures 1(a) and 1(b), respectively. The attractor of map (2) for a=-1.5, b=0.3, c=0.8, d=0.1, and e=0.1 is given in Figure 1(c) (where the maximum Lyapunov exponent L=0.168>0), which is a Maratto’s chaotic attractor.Figure 1
(a) Bifurcation diagram of map (2) in (b,x) plane for a=-1.5, c=0.8, d=0.1, and e=0.1. (b) Maximum Lyapunov exponents corresponding to (a). (c) Phase portrait of the chaotic attractor for a=-1.5, b=0.3, c=0.8, d=0.1, and e=0.1.
(a)
(b)
(c)


## 5. Numerical Simulations

In this section, we use the bifurcation diagrams, Lyapunov exponents, and phase portraits to illustrate the above analytic results and show the new and more complex dynamical behaviors in the map (2).


### 5.1. Numerical Simulations for Stability and Codimension One Bifurcations of Fixed Points

In this sbusection, we give the numerical simulations for verifying Propositions1 and 2 and Theorems 3–5. Four cases are considered as the follows.Case 1.
The bifurcation diagram of fixed points of map (2) in (b,x) plane for a=0.7, c=0.3, d=-0.15, and e=0.08 is given in Figure 2(a). We show that map (2) has one, two, and three fixed points for b<0.764782, b=0.764782, and b>0.764782, respectively. The fold bifurcation (labeled “SN”) occurs at the fixed point (-2.80455,-2.15734) for bSN=0.764782, and two unstable fixed points bifurcate from this point for b>bSN, which check the correctness of Theorem 3 (here the coefficient α=-0.137389<0). The Hopf bifurcations (labeled “H1” and “H2”) occur at fixed points (0.811635,0.624334) and (0.530768,0.408283) for bH1=-0.378162 and bH2=1.0153, respectively. For example, an attracting invariant cycle bifurcates from the fixed point (0.530768,0.408283) for b>bH2, where the coefficients in Theorem 5 are lH2=-2.02282<0 and kH2=0.15923>0, which check the correctness of Theorem 5. The stable invariant cycles for b=-0.38 and b=1.02 are shown in Figures 2(b) and 2(c), respectively.Figure 2
(a) Bifurcation diagram of the fixed points of map (2) in (b,x) plane for a=-0.1, c=-0.5, d=-0.5, and e=-0.3; (b) and (c): invariant cycles corresponding to Figure 2(a) at b=-0.38 and b=1.02, respectively.
(a)
(b)
(c)Case 2.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=0.7, c=0.3, d=-0.15, and e=0.08 is given in Figure 3(a). We show that there is only one fixed point for 0≤b≤0.6, and the Hopf bifurcation (labeled “H”) occurs at fixed point (2.07188,2.25205) for bH=0.0445132, and flip bifurcation (labeled “P-D”) emerges at the fixed point (1.30361,1.41697) for bP-D=0.479978, which check the correctness of Theorems 5 and 4, respectively. In fact, the fixed point is stable for bH<b<bP-D and loses its stability at flip bifurcation value b=bP-D, an unstable period-2 points bifurcate from the fixed point as b increasing (here the coefficients in Theorem 5 are α1=16.8788≠0 and α2=-0.179751<0).Figure 3
Bifurcation diagrams of the fixed points of map (2) in (b,x) plane for (a) a=0.7, c=0.3, d=-0.15, and e=0.08; (b) a=0.5, c=-0.1, d=-0.04, and e=0.7; (c) a=-0.5, c=0.8, d=0.1, and e=0.1.
(a)
(b)
(c)Case 3.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=0.5, c=-0.1, d=-0.04, and e=0.7 is given in Figure 3(b). We show that the flip bifurcation (labeled “P-D") occurs at fixed point (0.843666,2.81222) for bP-D=0.611937, and fold bifurcation (labeled “SN") emerges from fixed point (-0.970253,-3.23418) for bSN=1.47923.Case 4.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=-0.5, c=0.8, d=0.1, and e=0.1 is given in Figure 3(c). We show that the fold bifurcation (labeled “SN”) emerges from fixed point (1.15118,1.27909) for bSN=0.450125, and flip bifurcation (labeled “P-D”) occurs at fixed point (0.587212,0.652457) for bP-D=1.22047.


### 5.2. Further Numerical Simulations for Map (2)

In this subsection, we study the complex dynamics changes as the parameters varying with the initial value(0,0). The bifurcation parameters are considered in the following five cases.(a)
0≤a≤1.3, and b=0.02, c=0.3, d=-0.15, e=0.08.(b)
0≤b≤1, and a=0.5, c=-0.1, d=-0.04, e=0.7.(c)
0≤c≤1, and a=0.2, b=1, d=-1.5, e=0.23.(d)
-1≤d≤0.4, and a=-0.3, b=0.9, c=-0.8, e=0.5.(e)
0.05≤e≤1.05, and a=-0.4, b=1.1, c=-0.6, d=-0.55.For Case (a). The bifurcation diagram of map (2) in (a,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 4(a) and 4(b), respectively. Figures 4(c)–4(e) are the local amplifications of (a). We observe that there is a stable fixed point for 0≤a<0.624; the fixed point loses its stability as a increases; Hopf bifurcation occurs at a~0.624, and the stable invariant cycle at a=0.625 is shown in Figure 5(a). The region a∈(0.624,0.848) is the invariant cycle region with complex periodic windows, such as period-16, 27 orbits and the invariant cycles at a=0.8 is shown in Figure 5(b). The region a∈(0.848,0.888) is the chaotic region with complex periodic windows, such as period-38 orbit (see Figure 5(c)) and period-11 orbits, and the symmetry breaking of period orbits occurs for the region a∈(0.8535,0.8557), and the chaotic attractors at a=0.862 and a=0.877 are shown in the Figures 5(d) and 5(e), respectively. For a~0.888, the chaotic behavior suddenly converges to period-5 orbits, and the period-5 orbits undergo Hopf bifurcations at a~0.8906 separately, and the six coexisting invariant cycles at a=0.892 are shown in Figure 5(f). For the region a∈(0.8925,1.225), we can see the period-doubling bifurcation to chaos and the chaotic behavior suddenly disappears at a~1.225, and the six coexisting chaotic attractors at a=0.895, the seven coexisting chaotic attractor at a=0.987, and the chaotic attractor at a=1.2 are shown in the Figures 5(g)–5(i), respectively.Figure 4
(a) Bifurcation diagram of map (2) in (a,x) plane for b=0.02, c=0.3, d=-0.15, and e=0.08. (b) Maximum Lyapunov exponents corresponding to (a). (c)–(e). Local amplification of (a) for (c) 0.8≤a≤0.86, (d) 0.86≤a≤0.96, and (e) 0.96≤a≤1.25.
(a)
(b)
(c)
(d)
(e)Figure 5
Phase portraits corresponding to Figure4: (a) a=0.625, (b) a=0.8, (c) a=0.85, (d) a=0.862, (e) a=0.877, (f) a=0.892, (g) a=0.895, (h) a=0.987, and (i) a=1.2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)For Case (b). The bifurcation diagram of map (2) in (b,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 6(a) and 6(b), respectively. In Figure 6, we observe period-doubling bifurcation to chaos and chaotic region with period-6 windows. The two coexisting chaotic attractors at b=0.05 and the chaotic attractor at b=1 are shown in Figures 7(a) and 7(c), respectively, and the period-8 orbits at b=0.85 is shown in Figure 7(b).Figure 6
(a) Bifurcation diagram of map (2) in (b,x) plane for a=0.5, c=-0.1, d=-0.04, and e=0.7. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)Figure 7
Phase portraits corresponding to Figure6: (a) b=0.85, (b) b=0.88, and (c) b=1.
(a)
(b)
(c)For Case (c). The bifurcation diagram of map (2) in (c,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 8(a) and 8(b), respectively, which show the chaotic region with complex periodic windows. The chaotic attractors at c=0.01, c=0.2, and c=0.45 are shown in Figures 9(a)–9(c), respectively.Figure 8
(a) Bifurcation diagram of map (2) in (c,x) plane for a=0.2, b=1, d=-1.5, and e=0.23. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)Figure 9
Phase portraits corresponding to Figure8: (a) c=0.01, (b) c=0.2, and (c) c=0.45.
(a)
(b)
(c)For Case (d). The bifurcation diagram of map (2) in (d,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 10(a) and 10(b), respectively, and the local amplification of (a) for 0.2≤d≤0.4 is shown in Figure 10(c). we observe that period-doubling bifurcation to chaos and inverse period-doubling bifurcation to chaos, and the Hopf bifurcation occurs at d~-0.75. The invariant cycle at d=-0.9, the two coexisting chaotic attractors at d=0.28 and the chaotic attractor at d=0.4 are shown in Figures 11(a) and 11(c), respectively.Figure 10
(a) Bifurcation diagram of map (2) in (d,x) plane for a=-0.3, b=0.9, c=-0.8, and e=0.5. (b) Maximum Lyapunov exponents corresponding to (a). (c) Local amplification of (a).
(a)
(b)
(c)Figure 11
Phase portraits corresponding to Figure10: (a) d=-0.9, (b) d=0.28, and (c) d=0.4.
(a)
(b)
(c)For Case (e). The bifurcation diagram of map (2) in (e,x) plane and the maximum Lyapunov exponents corresponding to (a) are shown in Figures 12(a) and 12(b), respectively. Figures 12(c)–12(h) are the local amplifications of (a). In Figure 12, we observe that the jumping behavior of period-3 orbits to period-1 orbit at e~0.25, Hopf bifurcation occurs at e~0.788, and the invariant cycle at e=0.8234 is shown in Figure 13(a), and the invariant cycle suddenly converges to period-4 orbits at e~0.8235. The region e∈(0.886,0.9558) is the invariant cycle region with complex periodic windows, such as period-21, 17 orbits. The invariant cycle at a=0.934 and the forty-one coexisting invariant cycles at a=0.954 are shown in Figures 13(b) and 13(c), respectively. For e~0.9558, the periodic orbits suddenly become chaotic behaviors. Moreover, the chaotic behaviors suddenly change into a special chaotic behaviors which has 13 small size chaotic attractors as e increases. The chaotic attractor at e=0.957 and the thirteen coexisting chaotic attractors at e=0.959 are shown in Figures 13(d) and 13(e), respectively. For e~0.9611, the chaotic behaviors suddenly converge period-13 orbits, which quickly undergo Hopf bifurcation and change into period-13 orbits again. For the region e∈(0.9633,1.05), the period-doubling bifurcation to chaos occurs several times and the chaotic behaviors with complex periodic windows, and the nine coexisting chaotic attractors at e=0.9725, the chaotic attractor at e=0.9901, the ten coexisting chaotic attractors at e=1.016, and the chaotic attractor at e=1.05 are shown in Figures 13(f)–13(i), respectively.Figure 12
(a) Bifurcation diagram of map (2) in (e,x) plane for a=-0.4, b=1.1, c=-0.6, and d=-0.55. (b) Maximum Lyapunov exponents corresponding to (a). (c)–(h) Local amplification of (a) for (c) 0.9≤e≤0.95, (d) 0.95≤e≤0.956, (e) 0.956≤e≤0.96, (f) 0.96≤e≤0.99, (g) 0.99≤e≤1.01, and (h) 1.01≤e≤1.05.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 13
Phase portraits corresponding to Figure12: (a) e=0.8234, (b) e=0.934, (c) e=0.954, (d) e=0.957, (e) e=0.959, (f) e=0.9725, (g) e=0.9901, (h) e=1.016, and (i) e=1.05.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 5.1. Numerical Simulations for Stability and Codimension One Bifurcations of Fixed Points

In this sbusection, we give the numerical simulations for verifying Propositions1 and 2 and Theorems 3–5. Four cases are considered as the follows.Case 1.
The bifurcation diagram of fixed points of map (2) in (b,x) plane for a=0.7, c=0.3, d=-0.15, and e=0.08 is given in Figure 2(a). We show that map (2) has one, two, and three fixed points for b<0.764782, b=0.764782, and b>0.764782, respectively. The fold bifurcation (labeled “SN”) occurs at the fixed point (-2.80455,-2.15734) for bSN=0.764782, and two unstable fixed points bifurcate from this point for b>bSN, which check the correctness of Theorem 3 (here the coefficient α=-0.137389<0). The Hopf bifurcations (labeled “H1” and “H2”) occur at fixed points (0.811635,0.624334) and (0.530768,0.408283) for bH1=-0.378162 and bH2=1.0153, respectively. For example, an attracting invariant cycle bifurcates from the fixed point (0.530768,0.408283) for b>bH2, where the coefficients in Theorem 5 are lH2=-2.02282<0 and kH2=0.15923>0, which check the correctness of Theorem 5. The stable invariant cycles for b=-0.38 and b=1.02 are shown in Figures 2(b) and 2(c), respectively.Figure 2
(a) Bifurcation diagram of the fixed points of map (2) in (b,x) plane for a=-0.1, c=-0.5, d=-0.5, and e=-0.3; (b) and (c): invariant cycles corresponding to Figure 2(a) at b=-0.38 and b=1.02, respectively.
(a)
(b)
(c)Case 2.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=0.7, c=0.3, d=-0.15, and e=0.08 is given in Figure 3(a). We show that there is only one fixed point for 0≤b≤0.6, and the Hopf bifurcation (labeled “H”) occurs at fixed point (2.07188,2.25205) for bH=0.0445132, and flip bifurcation (labeled “P-D”) emerges at the fixed point (1.30361,1.41697) for bP-D=0.479978, which check the correctness of Theorems 5 and 4, respectively. In fact, the fixed point is stable for bH<b<bP-D and loses its stability at flip bifurcation value b=bP-D, an unstable period-2 points bifurcate from the fixed point as b increasing (here the coefficients in Theorem 5 are α1=16.8788≠0 and α2=-0.179751<0).Figure 3
Bifurcation diagrams of the fixed points of map (2) in (b,x) plane for (a) a=0.7, c=0.3, d=-0.15, and e=0.08; (b) a=0.5, c=-0.1, d=-0.04, and e=0.7; (c) a=-0.5, c=0.8, d=0.1, and e=0.1.
(a)
(b)
(c)Case 3.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=0.5, c=-0.1, d=-0.04, and e=0.7 is given in Figure 3(b). We show that the flip bifurcation (labeled “P-D") occurs at fixed point (0.843666,2.81222) for bP-D=0.611937, and fold bifurcation (labeled “SN") emerges from fixed point (-0.970253,-3.23418) for bSN=1.47923.Case 4.
The bifurcation diagram of fixed point of map (2) in (b,x) plane for a=-0.5, c=0.8, d=0.1, and e=0.1 is given in Figure 3(c). We show that the fold bifurcation (labeled “SN”) emerges from fixed point (1.15118,1.27909) for bSN=0.450125, and flip bifurcation (labeled “P-D”) occurs at fixed point (0.587212,0.652457) for bP-D=1.22047.


## 5.2. Further Numerical Simulations for Map (2)

In this subsection, we study the complex dynamics changes as the parameters varying with the initial value(0,0). The bifurcation parameters are considered in the following five cases.(a)
0≤a≤1.3, and b=0.02, c=0.3, d=-0.15, e=0.08.(b)
0≤b≤1, and a=0.5, c=-0.1, d=-0.04, e=0.7.(c)
0≤c≤1, and a=0.2, b=1, d=-1.5, e=0.23.(d)
-1≤d≤0.4, and a=-0.3, b=0.9, c=-0.8, e=0.5.(e)
0.05≤e≤1.05, and a=-0.4, b=1.1, c=-0.6, d=-0.55.For Case (a). The bifurcation diagram of map (2) in (a,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 4(a) and 4(b), respectively. Figures 4(c)–4(e) are the local amplifications of (a). We observe that there is a stable fixed point for 0≤a<0.624; the fixed point loses its stability as a increases; Hopf bifurcation occurs at a~0.624, and the stable invariant cycle at a=0.625 is shown in Figure 5(a). The region a∈(0.624,0.848) is the invariant cycle region with complex periodic windows, such as period-16, 27 orbits and the invariant cycles at a=0.8 is shown in Figure 5(b). The region a∈(0.848,0.888) is the chaotic region with complex periodic windows, such as period-38 orbit (see Figure 5(c)) and period-11 orbits, and the symmetry breaking of period orbits occurs for the region a∈(0.8535,0.8557), and the chaotic attractors at a=0.862 and a=0.877 are shown in the Figures 5(d) and 5(e), respectively. For a~0.888, the chaotic behavior suddenly converges to period-5 orbits, and the period-5 orbits undergo Hopf bifurcations at a~0.8906 separately, and the six coexisting invariant cycles at a=0.892 are shown in Figure 5(f). For the region a∈(0.8925,1.225), we can see the period-doubling bifurcation to chaos and the chaotic behavior suddenly disappears at a~1.225, and the six coexisting chaotic attractors at a=0.895, the seven coexisting chaotic attractor at a=0.987, and the chaotic attractor at a=1.2 are shown in the Figures 5(g)–5(i), respectively.Figure 4
(a) Bifurcation diagram of map (2) in (a,x) plane for b=0.02, c=0.3, d=-0.15, and e=0.08. (b) Maximum Lyapunov exponents corresponding to (a). (c)–(e). Local amplification of (a) for (c) 0.8≤a≤0.86, (d) 0.86≤a≤0.96, and (e) 0.96≤a≤1.25.
(a)
(b)
(c)
(d)
(e)Figure 5
Phase portraits corresponding to Figure4: (a) a=0.625, (b) a=0.8, (c) a=0.85, (d) a=0.862, (e) a=0.877, (f) a=0.892, (g) a=0.895, (h) a=0.987, and (i) a=1.2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)For Case (b). The bifurcation diagram of map (2) in (b,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 6(a) and 6(b), respectively. In Figure 6, we observe period-doubling bifurcation to chaos and chaotic region with period-6 windows. The two coexisting chaotic attractors at b=0.05 and the chaotic attractor at b=1 are shown in Figures 7(a) and 7(c), respectively, and the period-8 orbits at b=0.85 is shown in Figure 7(b).Figure 6
(a) Bifurcation diagram of map (2) in (b,x) plane for a=0.5, c=-0.1, d=-0.04, and e=0.7. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)Figure 7
Phase portraits corresponding to Figure6: (a) b=0.85, (b) b=0.88, and (c) b=1.
(a)
(b)
(c)For Case (c). The bifurcation diagram of map (2) in (c,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 8(a) and 8(b), respectively, which show the chaotic region with complex periodic windows. The chaotic attractors at c=0.01, c=0.2, and c=0.45 are shown in Figures 9(a)–9(c), respectively.Figure 8
(a) Bifurcation diagram of map (2) in (c,x) plane for a=0.2, b=1, d=-1.5, and e=0.23. (b) Maximum Lyapunov exponents corresponding to (a).
(a)
(b)Figure 9
Phase portraits corresponding to Figure8: (a) c=0.01, (b) c=0.2, and (c) c=0.45.
(a)
(b)
(c)For Case (d). The bifurcation diagram of map (2) in (d,x) plane and the corresponding maximum Lyapunov exponents are shown in Figures 10(a) and 10(b), respectively, and the local amplification of (a) for 0.2≤d≤0.4 is shown in Figure 10(c). we observe that period-doubling bifurcation to chaos and inverse period-doubling bifurcation to chaos, and the Hopf bifurcation occurs at d~-0.75. The invariant cycle at d=-0.9, the two coexisting chaotic attractors at d=0.28 and the chaotic attractor at d=0.4 are shown in Figures 11(a) and 11(c), respectively.Figure 10
(a) Bifurcation diagram of map (2) in (d,x) plane for a=-0.3, b=0.9, c=-0.8, and e=0.5. (b) Maximum Lyapunov exponents corresponding to (a). (c) Local amplification of (a).
(a)
(b)
(c)Figure 11
Phase portraits corresponding to Figure10: (a) d=-0.9, (b) d=0.28, and (c) d=0.4.
(a)
(b)
(c)For Case (e). The bifurcation diagram of map (2) in (e,x) plane and the maximum Lyapunov exponents corresponding to (a) are shown in Figures 12(a) and 12(b), respectively. Figures 12(c)–12(h) are the local amplifications of (a). In Figure 12, we observe that the jumping behavior of period-3 orbits to period-1 orbit at e~0.25, Hopf bifurcation occurs at e~0.788, and the invariant cycle at e=0.8234 is shown in Figure 13(a), and the invariant cycle suddenly converges to period-4 orbits at e~0.8235. The region e∈(0.886,0.9558) is the invariant cycle region with complex periodic windows, such as period-21, 17 orbits. The invariant cycle at a=0.934 and the forty-one coexisting invariant cycles at a=0.954 are shown in Figures 13(b) and 13(c), respectively. For e~0.9558, the periodic orbits suddenly become chaotic behaviors. Moreover, the chaotic behaviors suddenly change into a special chaotic behaviors which has 13 small size chaotic attractors as e increases. The chaotic attractor at e=0.957 and the thirteen coexisting chaotic attractors at e=0.959 are shown in Figures 13(d) and 13(e), respectively. For e~0.9611, the chaotic behaviors suddenly converge period-13 orbits, which quickly undergo Hopf bifurcation and change into period-13 orbits again. For the region e∈(0.9633,1.05), the period-doubling bifurcation to chaos occurs several times and the chaotic behaviors with complex periodic windows, and the nine coexisting chaotic attractors at e=0.9725, the chaotic attractor at e=0.9901, the ten coexisting chaotic attractors at e=1.016, and the chaotic attractor at e=1.05 are shown in Figures 13(f)–13(i), respectively.Figure 12
(a) Bifurcation diagram of map (2) in (e,x) plane for a=-0.4, b=1.1, c=-0.6, and d=-0.55. (b) Maximum Lyapunov exponents corresponding to (a). (c)–(h) Local amplification of (a) for (c) 0.9≤e≤0.95, (d) 0.95≤e≤0.956, (e) 0.956≤e≤0.96, (f) 0.96≤e≤0.99, (g) 0.99≤e≤1.01, and (h) 1.01≤e≤1.05.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 13
Phase portraits corresponding to Figure12: (a) e=0.8234, (b) e=0.934, (c) e=0.954, (d) e=0.957, (e) e=0.959, (f) e=0.9725, (g) e=0.9901, (h) e=1.016, and (i) e=1.05.
(a)
(b)
(c)
(d)
(e)
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(g)
(h)
(i)


## 6. Conclusion

In this paper, the generalized Hénon map with nonconstant Jacobian determinant is investigated. By using the analytic and numerical methods, the existence conditions of fold bifurcation, flip bifurcation, Hopf bifurcation, and Marotto’s chaos are obtained. Moreover, we discover some interesting dynamical behaviors, such as the invariant cycles which are irregular closed graphics, and the six and forty-one coexisting invariant cycles, and the two, six, seven, nine, ten, and thirteen coexisting chaotic attractors, and some kinds of strange chaotic attractors. The numerical simulations show the consistence with the theoretical analysis and exhibit some specific properties which can not be obtained by theoretical analysis.



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*Source: 270604-2015-03-10.xml*