# The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

Authors: Xiaodong Li; Weipeng Zhang; Fengjie Geng; Jicai Huang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.


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## Body

## 1. Introduction and Setting of the Problem

In recent years, there is a large literature concerning the bifurcation problems of homoclinic and heteroclinic loops in dynamical systems (see [1–23] and the references therein). However, to the best of the authors’ knowledge, less attention has been devoted to the bifurcation of double homoclinic loops. Han and Bi [24] investigated the existence of homoclinic bifurcation curves and small and large limit cycles bifurcated from a double homoclinic loop under multiple parameter perturbations for general planar systems. Han and Chen [25] gave the number of limit cycles near double homoclinic loops under perturbations in planar Hamiltonian systems. Homburg and Knobloch [26] considered the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times in conservative and reversible systems. Morales et al. [27, 28] presented contracting and expanding Lorenz attractors through resonant double homoclinic loops. Lu [29] obtained codimension 2 bifurcations of twisted double homoclinic loops in higher dimensional systems. Ragazzo [30] investigated the stability of sets that were generalizations of the simple pendulum double homoclinic loop. In our recent work [31, 32], codimension 2 bifurcations of double homoclinic loops and codimension 3 bifurcations of nontwisted double homoclinic loops with resonant eigenvalues were studied. Concerning this topic, a more extensive list of references can be found in the references mentioned earlier. Generally speaking, when studying the problem of single homoclinic or heteroclinic bifurcation connecting hyperbolic equilibrium, the bifurcation is more complicated as Γ is twisted. Moreover, it is known that double homoclinic loops have higher codimension than a single homoclinic loop under the same conditions. Therefore, it will be more challenging and difficult to analyze the twisting bifurcations of double homoclinic loops.In this paper, on the one hand, using the method which was originally established in [22, 23] and then improved in [9, 17, 21], and so forth, we study the bifurcations of double homoclinic loops with resonant eigenvalues under twisted cases. Besides, the method is more applicable and bifurcation equations obtained in this paper are easier to compute. On the other hand, frequently, the too many equivalent terms in the bifurcation equation will make the bifurcation equation more complex so that it is very difficult to analyze the bifurcation equation. Applying the method used by Homburg and Knobloch [26] to analyze the center-stable and center-unstable tangent bundles, we cannot only get a smooth coordinate transformation in the neighborhood of Γ small enough, but also make the bifurcation equation definite under an additional condition. Such strategy in dealing with the problems of bifurcations from homoclinic and heteroclinic loops is rarely used in the existing literatures. Therefore, a main feature in this paper is a combination of geometrical and analytical methods.Motivated by these points, we will consider the followingCr system and its unperturbed system:
(1)z˙=f(z)+g(z,μ),(2)z˙=f(z),
where r is large enough, z∈R4,  μ∈Rl,  l≥3,   0<|μ|≪1,  f(0)=0,  g(0,μ)=g(z,0)=0.We make the following assumptions, which are shown in Figure1.(H1)
The linearizationDf(0) has simple real eigenvalues at the equilibrium 0:-ρ2,-ρ1,λ1,λ2 satisfying
(3)-ρ2<-ρ1<0<λ1<λ2,ρ1=λ1.(H2)
System (2) has double homoclinic loops Γ=Γ1∪Γ2,  Γi={z=ri(t):t∈R,ri(±∞)=0} and dim(Tri(t)Ws∩Tri(t)Wu)=1, i=1,2, where Ws and Wu are the stable and unstable manifolds of 0, respectively.(H3)
Letei±=limt→∓∞(r˙i(t)/|r˙i(t)|), and ei+∈T0Wu, ei-∈T0Ws unit eigenvectors corresponding to λ1 and -ρ1, respectively, and satisfying e1+=-e2+, e1-=-e2-.(H4)
Span{Tri(t)Wu,Tri(t)Ws,ei+}=R4 as t≫1, and Span{Tri(t)Wu,Tri(t)Ws,ei-}=R4 as t≪-1, i=1,2.Figure 1
Double homoclinic loopsΓ=Γ1∪Γ2.As shown in Figure1, under the hypotheses (H1)–(H4), we can see that the double homoclinic loops Γ are of codimension 3.The single homoclinic loop in high-dimensional systems has been investigated by many authors (see [1, 2, 4–8, 13–15, 17, 21, 22] and the references therein). In this paper, we only focus on bifurcations of the large loop; that is, the double loops Γ=Γ1∪Γ2.A nondegenerate homoclinic orbit is called anontwisted homoclinic orbit if the unstable manifold Wu has an even number of half-twists along the homoclinic orbit, and is called a twisted homoclinic orbit if Wu has an odd number of half-twists along the homoclinic orbit; see more details in Deng [4]. A characterization for a twisted homoclinic orbit is that it arises from and tends to the equilibrium point from different sides of the unstable manifold. However, a nontwisted homoclinic orbit does so from the same side of the manifold. For the double homoclinic loop, Γ=Γ1∪Γ2, it is called double twisted if both Γ1 and Γ2 are twisted, single twisted if and only if one of them is twisted, and nontwisted otherwise.The rest of this paper is organized as follows. In Section2, the normal form in a neighborhood of the equilibrium small enough is established and the bifurcation equations are given. In Section 3, by bifurcation analysis, the results of twisting bifurcations and complete bifurcation diagrams are obtained under different cases. We end the paper with a conclusion in Section 4.


## 2. Normal Form and Bifurcation Equations

From the hypothesis(H1), we can see that we confine ourselves to consider the resonance taking place between the two principal eigenvalues λ1 and -ρ1. Moreover, the corresponding eigenvectors are also the tangent directions of the homoclinic orbits. For simplicity, we may choose a new parameter μ=(α,ν) such that ρ1(μ)=λ1(ν)+αλ1(ν), -1≪α≪1.LetU be a neighborhood of 0 small enough. By analyzing the center-stable and center-unstable tangent bundles, Homburg and Knobloch [26] obtained that there is a smooth coordinate transformation, such that in U system (1) takes the following form:
(4)x˙=x(λ1(ν)+o(1))+O(u)(O(y)+O(v)),y˙=y(-(1+α)λ1(ν)+o(1))+O(v)(O(x)+O(u)),u˙=u(λ2(ν)+o(1))+x2H1(x,y,v),v˙=v(-ρ2(ν)+o(1))+y2H2(x,y,u).
For the definiteness of the bifurcation equation, we make an additional assumption as follows:(H5)
H1(x,0,0)=0, H2(0,y,0)=0; that is,
(5)H1(x,y,v)=a1xk1yk2+a2xk3vk4+a3xk5yk6vk7+h.o.t.,H2(x,y,u)=b1yl1xl2+b2yl3ul4+b3xl5yl6ul7+h.o.t.,where ki>λ2/λ1, i=1,3,5, k2ρ1/λ1>2, k4ρ2/λ1>2, (k6ρ1+k7ρ2)/λ1>2. li>ρ2/ρ1, i=1,3,5, l2λ1/ρ1>2, l4λ2/ρ1>2, (l6λ1+l7λ2)/ρ1>2.By the same process as in [32], which is based on the analysis of the Poincaré return map defined on some local transversal section of the double homoclinic loop Γ, we obtain the bifurcation equations as follows:
(6)s2=(w112)-1s11+α-δ-1M11ν+h.o.t.,s1=(w212)-1s21+α+δ-1M21ν+h.o.t.


## 3. Bifurcation Analysis

Denote thatwi12=Δi|wi12|. We say that Γ is nontwisted as Δ1=Δ2=1 and twisted as Δ1=Δ2=-1 or Δ1Δ2=-1. In this paper, we focus on the twisted bifurcations. It is easy to see that |wi12| is the approximate expanding rate of the solution zi1(t) from Ti0 to -Ti1.Case 1 (Δ1=Δ2=-1 (i.e., double twisted)).
For convenience and simplicity, we use the following notations throughout Case1:
(7)R11={ν:M11ν>0,M21ν<0,|ν|≪1},R21={ν:M11ν<0,M21ν>0,|M11ν|=O|M21ν|,|ν|≪1},D1={ν∈R21:(δ-1M11ν)1+α-w212(21/(1+α)-1)z×δ-1M21ν>0forα>0(δ-1M11ν)1+α},D2={ν∈R21:(-δ-1M21ν)1+α+w112(21/(1+α)-1)=.×δ-1M11ν>0forα>0(-δ-1M21ν)1+α}.If (6) has solution s1=s2=0, then we have
(8)Mi1ν+h.o.t.=0,i=1,2.
If Mi1≠0, then there exists a codimension 1 surface Σi with a normal vector Mi1 at ν=0, such that the ith equation of (6) has solution s1=s2=0 as ν∈Σi and |ν|≪1; that is, Γi is persistent. If rank(M11,M21)=2, then Σ12=Σ1∩Σ2 is a codimension 2 surface with a normal plane span{M11,M21} such that (6) has solution s1=s2=0 as ν∈Σ12 and |ν|≪1; equivalently, the large loop Γ=Γ1∪Γ2 is persistent.Suppose that (6) has solution s1=0,s2>0. Then, we have
(9)s2=-δ-1M11ν+h.o.t.,0=(w212)-1s21+α+δ-1M21ν+h.o.t.
Hence, we obtain the large 1-homoclinic orbit bifurcation surface equation
(10)H21:δ-1w212M21ν+(-δ-1M11ν)1+α+h.o.t.=0,
which is well defined at least in the region R21 and has a normal vector M21 as α>0 (resp., M11 as α<0) at ν=0 such that system (1) has a large 1-homoclinic loop Γ21 near Γ as ν∈H21. It also means that H21 is tangent to Σ2 as α>0 (resp., Σ1 as α<0). When ν∈H21, α>0, s1=0, and s2=-δ-1M11ν+h.o.t., (6) indicates that
(11)s2ν=-δ-1M11+h.o.t.,(12)s1ν=(1+α)(w212)-1(-δ-1M11ν)αs2ν+δ-1M21+h.o.t.
Using (11) ×(1+α)(w212)-1(-δ-1M11ν)α+ (12), we have
(13)s1ν∣H21=δ-1M21+O(|M11ν|α).
So, s1=s1(ν) increases along the direction of the gradient M21 for ν near H21.Similarly, by settingsk=e-ρ1τk, we can derive that s1(ν) increases along the direction -M11 for ν near H21 as α<0.If (6) has solution s1>0 and s2=0, then we have
(14)0=(w112)-1s11+α-δ-1M11ν+h.o.t.,s1=δ-1M21ν+h.o.t.
Therefore, we obtain the existence of the other large 1-homoclinic orbit bifurcation surface equation
(15)H11:δ-1w112M11ν-(δ-1M21ν)1+α+h.o.t.=0,
which is well defined at least in the region R21 and has a normal vector M11 as α>0 (resp., M21 as α<0) at ν=0 such that system (1) has a large 1-homoclinic loop Γ11 near Γ as ν∈H11. Thus, H11 is also tangent to Σ1 as α>0 (resp., Σ2 as α<0). For ν∈H11 and α>0, s1=δ-1M21ν+h.o.t., s2=0, it follows from (6) that
(16)s2ν=(1+α)(w112)-1(δ-1M21ν)αs1ν-δ-1M11+h.o.t.,(17)s1ν=δ-1M21+h.o.t.
Computing (17) ×(1+α)(w112)-1(δ-1M21ν)α+ (16), it leads to
(18)s2ν∣H11=-δ-1M11+O(|M21ν|α).
So, s2=s2(ν) increases along the direction -M11 for ν close to H11.By a similar procedure, we can show thats2(ν) increases along the direction M21 as ν is in the neighborhood of H11 and α<0.Summing up the previous analysis, we have the following theorem.Theorem 1.
Suppose that(H1)–(H5) hold; then, the following conclusions are true.(1)
IfMi1≠0, then there exists a unique surface Σi with codimension 1 and normal vectors Mi1 at ν=0, such that system (1) has a homoclinic loop near Γi if and only if ν∈Σi and |ν|≪1. If rank(M11,M21)=2, then Σ12=Σ1∩Σ2 is a codimension 2 surface and 0∈Σ12 such that system (1) has a large loop consisting of two homoclinic orbits near Γ as ν∈Σ12 and |ν|≪1; that is, Γ is persistent.(2)
In the regionR21, there exists a large 1-homoclinic orbit bifurcation surface H21 which is tangent to Σ2 (resp., Σ1) at ν=0 with the normal vector M21 (resp., M11) as α>0 (resp., α<0), and for ν∈H21, system (1) has a unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ becomes a large 1-periodic orbit when ν moves along the direction M21 (resp., -M11) and nearby H21 as α>0 (resp., α<0). Meanwhile, there exists another large 1-homoclinic orbit bifurcation surface H11 which is tangent to Σ1 (resp., Σ2) at ν=0 with the normal M11 (resp., M21) as α>0 (resp., α<0), and for ν∈H11, system (1) has another one unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ changes into a large 1-periodic orbit when ν shifts along the direction -M11 (resp., M21) and near H11 as α>0 (resp., α<0).Lemma 2.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 are valid; then, in addition to the large 1-homoclinic loop Γ21, system (1) has exactly one simple large 1-periodic orbit near Γ for ν∈H21∩D2. Moreover, the large 1-periodic orbit is persistent as ν changes in the neighborhood of H21.Proof.
Ifν∈H21 and |ν|≪1, then we know that system (1) has one large 1-homoclinic loop Γ21. Now, following (6), we have
(19)[(w112)-1s11+α-δ-1M11ν+h.o.t.]1+α=w212s1-δ-1w212M21ν+h.o.t.
Let N1(s1) and L1(s1) be the left- and right-hand sides of (19), respectively. Then, by (10), we get N1(0)=L1(0) as ν∈H21. Moreover,
(20)N1′(s1)=(1+α)2(w112)-1[(w112)-1s11+α-δ-1M11ν+h.o.t.]αs1α,L1′(s1)=w212+h.o.t.
Thus, 0=N1′(s1)∣s1=0>L1′(s1)∣s1=0. Therefore, there is an s~1, 0<s~1≪1, such that for 0<s1<s~1, N1(s1)>L1(s1).
Lets-1=-δ-1M21ν. Then,
(21)N1(s-1)=[(w112)-1(-δ-1M21ν)1+α-δ-1M11ν+h.o.t.]1+α,L1(s-1)=w212(-δ-1M21ν)-δ-1w212M21ν+h.o.t.=2(-δ-1M11ν)1+α+h.o.t.,
which means that N1(s-1)<L1(s-1) as ν∈H21∩D2. Then, it follows from N1′′(s1)<0 that N1(s1)=L1(s1) has a unique solution s1* satisfying 0<s~1<s1*≪1, which is shown in Figure 2. That is, system (1) has a unique large 1-periodic loop near Γ for ν∈H21∩D2 and 0<|ν|≪1. The proof is complete.Figure 2
N
1
(
s
1
) and L1(s1) for 0≤s1≪1, ν∈H21∩D2.Lemma 3.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 hold. Then, system (1) has a saddle-node bifurcation surface of large 1-periodic orbit in the region R21∩{ν:|M11ν|=o(|M21ν|)} as follows:
(22)SN2:-δ-1w212M21ν=[(w112)-1(w112M11ν(1+α)2M21ν)(1+α)/α-δ-1M11ν]1+α=.-w212(w112M11ν(1+α)2M21ν)1/α+h.o.t.Proof.
It is easy to see thath=N1(s1) is tangent to h=L1(s1) at some point s1, as shown in Figure 3; if
(23)N1(s1)=L1(s1),N1′(s1)=L1′(s1),
then
(24)[(w112)-1s11+α-δ-1M11ν+h.o.t.]1+α=w212s1-δ-1w212M21ν+h.o.t.,(25)(1+α)2(w112)-1s1α[(w112)-1s11+α-δ-1M11ν+h.o.t.]α=w212+h.o.t.
It is not difficult to see that (24) and (25) have a unique small positive solution
(26)s1=(w112M11ν(1+α)2M21ν)1/α+h.o.t.
when ν∈R21 and |M11ν|=o(|M21ν|1+α). Substituting (26) into (24), we get the surface SN2. It is easy to verify that the solution s2 of (6) is positive as s1 is given by (26) and ν∈SN2. Thus, the proof of the lemma is complete.Figure 3
N
1
(
s
1
) and L1(s1) for 0≤s1≪1, ν∈SN2.Remark 4.
Due toN1′′(s1), we see that the line L1(s1) lies above the curve N1(s1) as ν∈SN2. Moreover, L1(0)=-δ-1w212M21ν+h.o.t. decreases as rank(M11,M21)=2 and ν moves along the direction -M21; in this case, system (1) exhibits two large 1-periodic orbits. And as ν leaves SN2 along the opposite direction, there is no large 1-periodic orbit.Corollary 5.
The bifurcation surface{0<α≪1}×SN2 and the region ({0<α≪1}×R21)∩{(α;ν):|M11ν|=O(|M21ν|1+α),αln|M21ν|≪-1} have no intersection point.Proof.
Letw112|M11ν|=-aα|M21ν|1+α, where 0<a<δ-1e2 is a given constant; then, in view of |M21ν|α≪1 and (1+α)1/α→e as α→0, we know the right side of the expression SN2 as follows:
(27)[(w112)-1(w112M11ν(1+α)2M21ν)(1+α)/α-δ-1M11ν]1+α-w212(w112M11ν(1+α)2M21ν)1/α+h.o.t.<[14w112e2a1+α|M21ν|1+α-δ-1w112aα|M21ν|1+α]1+α<.-w212(w112M11ν(1+α)2M21ν)1/α+h.o.t.=(4w112e2)-(1+α)|M21ν|(1+α)2(a1+α-4δ-1e2aα)1+α<.-w212(w112M11ν(1+α)2M21ν)1/α+h.o.t.=aα(1+α)(4w112e2)-(1+α)|M21ν|(1+α)2(a-4δ-1e2)1+α<.-w212ae2M21ν+h.o.t.<δ-1|w212|M21ν,
and δ-1|w212|M21ν is the left side of the expression SN2. Hence, the proof is complete.Corollary 6.
Asw112w212>1, the bifurcation surface {0<α≪1}×SN2 and the region ({0<α≪1}×R21)∩{(α;ν):|M11ν|=O(|M21ν|1+α),-1≪αln|M21ν|<0} have no intersection point.Proof.
Still letw112|M11ν|=-aα|M21ν|1+α, where 0<a<δ-1e2 is a given constant; then, due to |M21ν|α=1+o(|M21ν|)<1, we know that
(28)L≜-δ-1w112w212M21ν+w112w212(w112M11ν(1+α)2M21ν)1/α=w112w212|M21ν|(-δ-1+a(1+α)-2/α),R≜(w112)-α[(w112M11ν(1+α)2M21ν)(1+α)/α-δ-1w112|M11ν|]1+α+h.o.t.=[(1+α)-2(1+α)/αa1+α|M21ν|1+α-δ-1aα|M21ν|1+α]1+α+h.o.t.>aα(1+α)|M21ν|[δ-1+a(1+α)-2/α]1+α+h.o.t.
As w112w212>1, a>0 is a constant, and α>0 is small enough, it is easy to see that R>L. This contradicts the expression of SN2. The proof is complete.By a similar analysis as in Lemmas2 and 3, we can obtain the following results.Lemma 7.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 are valid; then, in addition to the large 1-homoclinic loop Γ11, system (1) has exactly one simple large 1-periodic orbit near Γ for ν∈H11∩D1. Moreover, the large 1-periodic orbit is persistent as ν changes in the neighborhood of H11.Lemma 8.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 hold. Then, system (1) has a saddle-node bifurcation surface of large 1-periodic orbit in the region R11∩{ν:|M11ν|=o(|M21ν|} as follows:
(29)SN1:δ-1w112M11ν=[δ-1M21ν(1+α)2(1+α)2-1]1+α-w112(δ-1w212M21ν(1+α)2-1)1/(1+α)+h.o.t.Remark 9.
LetN2(s2)=[(w212)-1s21+α+δ-1M21ν+h.o.t.]1+α and L2(s2)=w112s2+δ-1w112M11ν+h.o.t. Due to N2′′(s2)>0 and L2(0)=δ-1w112M11ν+h.o.t., we claim that the line L2(s2) lies above the curve N2(s2) as ν∈SN1, and when rank(M11,M21)=2 and ν leaves SN1 along the direction M11, L2(0) decreases; hence, system (1) has two large 1-periodic orbits, and when ν moves along the direction -M11, there is no large 1-periodic orbit.Corollary 10.
The bifurcation surface{0<α≪1}×SN1 and the region ({0<α≪1}×R21)∩{(α;ν):|M11ν|=O(|M21ν|1+α),αln|M21ν|≪-1} have no intersection point.Corollary 11.
Asw112w212>1, the bifurcation surface {0<α≪1}×SN1 and the region ({0<α≪1}×R21)∩{(α;ν):|M11ν|=O(|M21ν|1+α),-1≪αln|M21ν|<0} have no intersection point.In the following we define open regions in the neighborhood of the origin of theν-space, which are shown in Figures 4 and 5.Figure 4
Bifurcation diagram in caseΔ1=Δ2=-1, 0<α≪1, -1≪αln|M21ν|<0, w112w212<1.Figure 5
Bifurcation diagram in caseΔ1=Δ2=-1, 0<α≪1, -1≪αln|M21ν|<0, w112w212>1.(
R
2
1
)
0 is bounded by SN1 and SN2, (R21)1 is bounded by Σ2 and H21, (R21)2 is bounded by H21 and SN2, (R21)1′ is bounded by SN1 and SN2, (R21)2′ is bounded by SN1 and H11, R11 is bounded by Σ1 and Σ2, and (R21)2′′ is bounded by H21 and H11.Now, the previous analysis is summarized in the following three theorems, as shown in Figures4 and 5.Theorem 12.
Suppose that hypotheses(H1)–(H5) hold, 0<α≪1, -1≪αln|M21ν|<0, ν∈R21, and w112w212<1. Then, system (1)(1)
has exactly one simple large 1-periodic orbit nearΓ as ν∈(R21)1;(2)
has a unique double large 1-periodic orbit nearΓ as ν∈SN2;(3)
has exactly two simple large 1-periodic orbits nearΓ as ν∈(R21)2;(4)
has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop nearΓ as ν∈H21∩D2;(5)
does not have any large 1-periodic and large 1-homoclinic loop nearΓ as ν∈(R21)0;(6)
has exactly one simple large 1-periodic orbit nearΓ as ν∈(R21)1′;(7)
has a unique double large 1-periodic orbit nearΓ as ν∈SN1;(8)
has exactly two simple large 1-periodic orbits nearΓ as ν∈(R21)2′;(9)
has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop nearΓ as ν∈H11∩D1.Theorem 13.
Suppose that hypotheses(H1)–(H5) hold, 0<α≪1, -1≪αln|M21ν|<0, ν∈R21, and w112w212>1. Then system (1)(1)
has exactly one simple large 1-periodic orbit nearΓ as ν∈(R21)1;(2)
has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop nearΓ as ν∈H21∩D2;(3)
has exactly two simple large 1-periodic orbits nearΓ as ν∈(R21)2′′;(4)
has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop nearΓ as ν∈H11∩D1;(5)
has exactly one simple large 1-periodic orbit nearΓ as ν∈(R21)1′.Remark 14.
Suppose that hypotheses(H1)–(H5) hold, 0<α≪1, αln|M21ν|≪-1, ν∈R21; then the conclusions of Theorem 13 are still true, and the bifurcation diagram is the same as Figure 5.Denote byD12 the open region with boundaries Σ1 and Σ2, such that D12∩{ν:M11ν>0,M21ν>0,|ν|≪1}≠∅. D21 is the open region with boundaries Σ2 and Σ1, such that D21∩{ν:M11ν<0,M21ν<0,|ν|≪1}≠∅.From the bifurcation equations (6), we have the following theorem.Theorem 15.
Suppose that hypotheses(H1)–(H5) hold, and 0<α≪1. Then, system (1)(1)
does not have any large 1-periodic orbit nearΓ as ν∈D12;(2)
does not have any large 1-periodic orbit nearΓ as ν∈D21;(3)
does not have any large 1-periodic orbit nearΓ as ν∈R11.Remark 16.
For-1≪α<0, by the same analysis, we can obtain the analogous results as those in Theorems 12–15. In fact, in the case we can reverse the time t and change (x,y,u,v) to (y,x,v,u), and then we still get 0<α≪1.At last, we consider the caseα=0. By a similar process to that of Section 3, we obtain the bifurcation equations
(30)-(w112)-1s1+s2+δ-1M11ν+h.o.t.=0,(w212)-1s2-s1+δ-1M21ν+h.o.t.=0.For the same reason as before, we have the following theorem.Theorem 17.
Suppose that(H1)–(H5), α=0, and rank(M11,M21)=2 hold; then,(1)
asν∈R21, system (1) not only has a large 1-homoclinic orbit bifurcation surface
(31)w212M21ν-M11ν+h.o.t.=0,
with a normal vector (w212M21-M11) at ν=0, but also has another large 1-homoclinic orbit bifurcation surface
(32)w112M11ν-M21ν+h.o.t.=0;
with a normal vector (w112M11-M21) at ν=0,(2)
as eitherw112w212<1, M21ν>w112M11ν, and w212M21ν>M11ν or w112w212>1, M21ν<w112M11ν, and w212M21ν<M11ν, system (1) has a unique large 1-periodic orbit near Γ.Case 2 (Δ1=-1, Δ2=1 (i.e., single twisted)).
Similar to Case1, for convenience and simplicity, we use the following notations:
(33)R-11={ν:M11ν<0,M21ν>0,|M21ν|=O(|M11ν|),|ν|≪1},R-21={ν:M11ν<0,M21ν<0,|M11ν|=O(|M21ν|),|ν|≪1},D-12={ν:M11ν>0,M21ν>0,|ν|≪1},D-21={ν:M11ν>0,M21ν<0,|ν|≪1}.
If (6) has a solution s1=s2=0, then we have
(34)Mi1ν+h.o.t.=0,i=1,2.
If Mi1≠0, then there exists a codimension 1 surface Σ-i with a normal vector Mi1 at ν=0, such that the ith equation of (6) has solution s1=s2=0 as ν∈Σ-i and |ν|≪1; that is, Γi is persistent. If rank(M11,M21)=2, then Σ-12=Σ-1∩Σ-2 is a codimension 2 surface with a normal plane span{M11,M21} such that (6) has solution s1=s2=0 as ν∈Σ-12 and |ν|≪1; equivalently, the large loop Γ=Γ1∪Γ2 is persistent.
Suppose that (6) has solution s1=0, s2>0. Then, we have
(35)s2=-δ-1M11ν+h.o.t.,0=(w212)-1s21+α+δ-1M21ν+h.o.t.
Hence, we get the large 1-homoclinic orbit bifurcation surface equation
(36)H-21:δ-1w212M21ν+(-δ-1M11ν)1+α+h.o.t.=0,
which is well defined at least in the region R-21 and has a normal vector M21 as α>0 (resp., M11 as α<0) at ν=0 such that the system (1) has a large 1-homoclinic loop Γ21 near Γ as ν∈H-21. It also means that H-21 is tangent to Σ-2 as α>0 (resp., Σ-1 as α<0). When ν∈H-21, α>0, s1=0, and s2=-δ-1M11ν+h.o.t., (6) implies that
(37)s2ν=-δ-1M11+h.o.t.,(38)s1ν=(1+α)(w212)-1(-δ-1M11ν)αs2ν+δ-1M21+h.o.t.
Using (37) ×(1+α)(w212)-1(-δ-1M11ν)α+ (38), we get
(39)s1ν∣H-21=δ-1M21+O(|M11ν|α).
So, s1=s1(ν) increases along the direction of the gradient M21 for ν near H-21.
Similarly, by settingsk=e-ρ1τk, we can derive that s1(ν) increases along the direction -M11  for  near H-21 as α>0.
If (6) has solution s1>0 and s2=0, then we get
(40)0=(w112)-1s11+α-δ-1M11ν+h.o.t.,s1=δ-1M21ν+h.o.t.
Therefore, we obtain the existence of the other large 1-homoclinic orbit bifurcation surface equation
(41)H-11:δ-1w112M11ν+(δ-1M21ν)1+α+h.o.t.=0,
which is well defined at least in the region R-11 and has a normal vector M11 as α>0 (resp., M21 as α<0) at ν=0 such that the system (1) has a large 1-homoclinic loop Γ11 near Γ as ν∈H-11. Thus, H-11 is also tangent to Σ-1 as α>0 (resp., Σ-2 as α<0). For ν∈H-11 and α>0, s1=δ-1M21ν+h.o.t., s2=0, it follows from (6) that
(42)s2ν=(1+α)(w112)-1(δ-1M21ν)αs1ν-δ-1M11+h.o.t.,(43)s1ν=δ-1M21+h.o.t.
Computing (43) ×(1+α)(w112)-1(δ-1M21ν)α+ (42), it leads to
(44)s2ν∣H-11=-δ-1M11+O(|M21ν|α).
So, s2=s2(ν) increases along the direction -M11 for ν close to H-11.
By a similar procedure, we can show thats2(ν) increases along the direction M21 as ν is in the neighborhood of H-11 and α<0.Summarizing the previous analysis, we have the following theorem.Theorem 18.
Suppose that(H1)–(H5) hold; then, the following conclusions are true.(1)
IfMi1≠0, then there exists a unique surface Σ-i with codimension 1 and normal vectors Mi1 at ν=0, such that system (1) has a homoclinic loop near Γi if and only if ν∈Σ-i and |ν|≪1. If rank(M11,M21)=2, then Σ-12=Σ-1∩Σ-2 is a codimension 2 surface and 0∈Σ-12 such that system (1) has a large loop consisting of two homoclinic orbits near Γ as ν∈Σ-12 and |ν|≪1; that is, Γ is persistent.(2)
In the regionR-21, there exists a unique large 1-homoclinic orbit bifurcation surface H-21 which is tangent to Σ-2 (resp., Σ-1) at ν=0 with the normal vector M21 (resp., M11) as α>0 (resp., α<0), and for ν∈H-21, system (1) has a unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ becomes a large 1-periodic orbit when ν moves along the direction M21 (resp., -M11) and nearby H-21 as α>0 (resp., α<0). In the region R-11, there exists another unique large 1-homoclinic orbit bifurcation surface H-11 which is tangent to Σ-1 (resp., Σ-2) at ν=0 with the normal M11 (resp., M21) as α>0 (resp., α<0), and for ν∈H-11, system (1) has another one unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ changes into a large 1-periodic orbit when ν shifts along the direction -M11 (resp., M21) and nearby H-11 as α>0 (resp., α<0).Lemma 19.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 are valid; system (1) has only a unique large 1-homoclinic orbit Γ21 near Γ for ν∈H-21 and 0<|ν|≪1.Proof.
Ifν∈H-21 and |ν|≪1, from Theorem 18, we know that system (1) has one large 1-homoclinic loop Γ21. Now, following (6), we have
(45)[(w112)-1s11+α-δ-1M11ν+h.o.t.]1+α=w212s1-δ-1w212M21ν+h.o.t.
Let N1(s1) and L1(s1) be the left- and right-hand sides of (45), respectively. Then, by (36), we get N1(0)=L1(0) as ν∈H-21. Moreover,
(46)N1′(s1)=(1+α)2(w112)-1[(w112)-1s11+α-δ-1M11ν(1+α)2(w112)-1=+h.o.t.(w112)-1]αs1α,L1′(s1)=w212+h.o.t.
Thus, N1′(s1)≤0<L1′(s1), and N1′′(s1)<0. Therefore, N1(s1)<L1(s1) for s1≠0, as shown in Figure 6.
Therefore, system (1) has only a unique large 1-homoclinic orbit near Γ for ν∈H-21 and 0<|ν|≪1.Figure 6
N
1
(
s
1
) and L1(s1) for 0≤s1≪1, ν∈H-21.Remark 20.
Due toN1′′(s1)<0, we can see that the line N1(s1) lies under the curve L1(s1) as ν∈H-21. Moreover, L1(0)=-δ-1w212M21ν+h.o.t. decreases as rank(M11,M21)=2 and ν moves along the direction M21; in this case, system (1) has a unique large 1-periodic orbit. And as ν leaves H-21 along the opposite direction, there is no large 1-periodic orbit.By a similar analysis as in Lemma19, we can obtain the following results.Lemma 21.
Suppose that hypotheses(H1)–(H5) and 0<α≪1 are valid; system (1) has only a unique large 1-homoclinic orbit Γ11 near Γ for ν∈H-11 and 0<|ν|≪1.Remark 22.
LetN2(s2)=[(w212)-1s21+α+δ-1M21ν+h.o.t.]1+α and L2(s2)=w112s2+δ-1w112M11ν+h.o.t. Due to N2′′(s2)>0 and L2(0)=δ-1w112M11ν+h.o.t., we claim that the line L2(s2) lies under the curve N2(s2) as ν∈H-11, and when rank(M11,M21)=2 and ν leaves H-11 along the direction -M11, L2(0) increases; hence, system (1) has a unique large 1-periodic orbit, and when ν moves along the direction M11, there is no large 1-periodic orbit.In the following we define open regions in the neighborhood of the origin of theν-space, which are shown in Figure 7.Figure 7
Bifurcation diagram in caseΔ1=-1, Δ2=1, 0<α≪1.(
R
-
2
1
)
0 is bounded by Σ1 and H-21, (R-21)1 is bounded by Σ2 and H-21, (R-11)1 is bounded by H-11 and Σ2, (R-11)0 is bounded by Σ1 and H-11, D-21 is bounded by Σ1 and Σ2, and D-12 is bounded by Σ2 and Σ1.From the previous analysis, we obtain the following three theorems, as shown in Figure7.Theorem 23.
Suppose that hypotheses(H1)–(H5) hold, 0<α≪1, ν∈R-21. Then, system (1)(1)
does not have any large 1-periodic orbit and large 1-homoclinic loop nearΓ as ν∈(R-21)0;(2)
has a unique large 1-homoclinic loop nearΓ as ν∈H-21;(3)
has a unique large 1-periodic orbit nearΓ as ν∈(R-21)1.Theorem 24.
Suppose that hypotheses(H1)–(H5) hold, 0<α≪1, ν∈R-11. Then, system (1)(1)
has no large 1-periodic orbit and large 1-homoclinic loop nearΓ as ν∈(R-11)0;(2)
has a unique large 1-homoclinic loop nearΓ as ν∈H-11;(3)
has a unique large 1-periodic orbit nearΓ as ν∈(R-11)1.From the bifurcation equations, we have the following theorem easily.Theorem 25.
Suppose that hypotheses(H1)–(H5) hold, and 0<α≪1. Then, system (1)(1)
has no large 1-periodic orbit nearΓ as ν∈D-12;(2)
has no large 1-periodic orbit nearΓ as ν∈D-21.Remark 26.
For-1≪α<0, by the same analysis, we can obtain the analogous results as those in Theorems 23–25. In fact, in the case we can reverse the time t and change (x,y,u,v) to (y,x,v,u), and then we still get 0<α≪1.At last, we consider the caseα=0. By a similar process to that of Section 3, we obtain the bifurcation equations
(47)-(w112)-1s1+s2+δ-1M11ν+h.o.t.=0,(w212)-1s2-s1+δ-1M21ν+h.o.t.=0.For the same reason as before, we have the following theorem.Theorem 27.
Suppose that(H1)–(H5), α=0, and rank(M11,M21)=2 hold; then,(1)
asν∈R-21, system (1) has a unique large 1-homoclinic orbit bifurcation surface
(48)w212M21ν-M11ν+h.o.t.=0,
with a normal vector (w212M21-M11) at ν=0;(2)
asν∈R-11, system (1) also has a unique large 1-homoclinic orbit bifurcation surface
(49)w112M11ν-M21ν+h.o.t.=0,
with a normal vector (w112M11-M21) at ν=0;(3)
asM21ν<w112M11ν and w212M21ν>M11ν, system (1) has a unique large 1-periodic orbit near Γ.Remark 28.
AsΓ1 is nontwisted and Γ2 is twisted, we can obtain similar conclusions as shown in Case 2.


## 4. Conclusion

This paper is devoted to investigating the twisting bifurcations of double homoclinic loops with resonant eigenvalues in 4-dimensional systems. We give asymptotic expressions of the bifurcation surfaces and their relative positions, describe the existence regions of large 1-periodic orbits nearΓ in Lemmas 2 and 7, and obtain the sufficient conditions for the existence or nonexistence of saddle-node bifurcation surfaces in Lemmas 3 and 8. More importantly, the complete bifurcation diagrams are given under different cases in Figures 4, 5, and 7. According to our analysis, when Γ is double twisted, in Theorems 12–15, we obtain one bifurcation diagram with saddle-node bifurcation surfaces of large 1-periodic orbit when w112w212<1 and the other bifurcation diagram without saddle-node bifurcation surfaces when w112w212>1. When Γ is single twisted, in Theorems 23–25, we obtain another bifurcation diagram. Compared with the nontwisted cases in Zhang et al. [32], our paper shows completely different results and bifurcation diagrams. It is worthy to be mentioned that the restriction on the dimension is not essential, the method used in this paper can be extended to higher dimensional systems without any difficulty, and the same conclusions can be deduced under the same hypotheses. Furthermore, we mention some problems for future study. (1)  Ws or Wu is inclination flip on one of the double homoclinic loops. (2) Both Ws and Wu are inclination flips on the double homoclinic loops and so on. But the difficulty of these problems will increase with adding codimension of the double homoclinic loops.



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*Source: 152518-2013-05-20.xml*