Codimension 3 bifurcation from orbit-flip homoclinic orbit of weak type

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Codimension 3 bifurcation from orbit-flip homoclinic orbit of weak type

Qiuying Lu

^B¹

, Guifeng Deng

²

and Hua Luo

¹

1Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China

2School of Mathematics and Information Science, Shanghai Lixin University of Commerce, Shanghai, 201620, China

Received 27 March 2015, appeared 26 October 2015 Communicated by Michal Feˇckan

Abstract. This article is devoted to the research of a new codimension 3 homoclinic orbit bifurcation, which is the orbit-flip of weak type. Such kind of homoclinic orbit is a degenerate case of the orbit-flip homoclinic orbit. We show the existence of 1- homoclinic orbit, 1-periodic orbit, 2ⁿ-homoclinic orbit and 2ⁿ-periodic orbit for arbitrary integern. Our strategy is based on the local moving coordinates method.

Keywords: homoclinic, orbit-flip, weak type, local moving coordinates method.

2010 Mathematics Subject Classification: 34C23, 37G20.

1 Introduction

In the past decades, multi-round homoclinic bifurcations have developed a lot due to their great applications in spatial dynamics, where they correspond to traveling or standing multi- pulses. Besides, cascades of homoclinic doubling bifurcation can be observed for parameter depending vector fields [8], which are similar to the phenomenon of the period doubling bifurcation for diffeomorphism of maps [6]. Since the codimension-one homoclinic bifurcation with real eigenvalues cannot give birth to multi-round homoclinic orbit, see [21], complicated dynamics of codimension 2 cases need to be considered. In [23], Yanagida studied 3 different kinds of codimension 2 cases, which included inclination-flip bifurcation, resonant bifurcation and the orbit-flip bifurcation. Since then, many research works have been devoted to this subject, see [4,5,7,9,10,12,13] for example.

Except for the above codimension 2 mechanism for the occurrence of homoclinic doubling bifurcation, another strategy is to consider the problem in a more degenerate situation, which is codimension 3. [11] presented the existence of infinitely many homoclinic doubling bifurcation from the inclination-flip homoclinic orbit of weak type, where the bifurcated homoclinic orbit ΓN of arbitrary order N were inclination-flip homoclinic orbit. Despite orbit-flip and inclination-flip homoclinic orbits are quite different from their definitions, both of them in- volve the orientation change of their stable manifolds. So lots of similar bifurcation results

BCorresponding author. Email: qiuyinglu@163.com

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have been discovered. For example, there exists non-empty interior region, where a suspended horseshoe is discovered. There are parameter curves for the bifurcation ofN-homoclinic orbit, see [7] and [16] for the inclination flip case and [20] for the orbit-flip case. Moreover, the strange attractors are presented both in the unfolding of inclination-flip case and orbit-flip case, see [17] and [18]. A natural question would then be asking whether similar homoclinic doubling bifurcation can occur infinitely many times from the orbit-flip homoclinic orbit of weak type. To answer this question, we consider a smooth system

˙

z = f(z) +g(z,µ), (1.1)

and its unperturbed system

z˙= f(z), (1.2)

where z ∈ _R³, µ ∈ _R³, 0 < |µ| 1, f(0) = 0, g(z, 0) = 0 and z = 0 is a hyperbolic equilibrium. More precisely, Spec df(0) is real. Without loss of generality, we suppose Spec df(0) = {−α,−β, 1}, where α > β > 0 due to time scaling. We denote the local stable manifold byW_loc^s and the local unstable manifold by W_loc^u . Since α > β, one has a local strong stable manifoldW_loc^ss corresponding to the eigenvalue−α. The local strong stable man- ifold, which is invariant under the flow, belongs to the local stable manifold. We can extend these manifolds by the flow and their extensions are denoted byW^s, W^u, W^ss.

From now on, we always denote the homoclinic orbit of (1.2) byΓ={r(t), t∈ R}. Definition 1.1. Γis called an orbit-flip homoclinic orbit ifΓ∩W^ss6={O}.

Before giving the definition of an orbit-flip homoclinic orbit of weak type, we firstly introduce the so called “weak vector”.

LetΣbeC¹cross-section transverse toW^ss. It turns out thatW^ssplitsΣinto two connected components, say Σ⁺ and Σ⁻. Then the Poincaré return map Φ is only defined on a single component ofΣ\W^s, we suppose this component isΣ⁺. Let C={C(t), t∈ (−1, 1)}be aC¹ curve inΣtransverse to the stable manifold such thatC(₀) =W^ss∩_Σ= p_Σ. Since the Poincaré return map is only defined onΣ⁺, we putΦ(C) = _Φ{C(t), t ∈ (0, 1)}. SetC_Φ(t) = _Φ(C(t)) and define

u_Σ=_lim

t→0 d dtC_Φ(t)

d

dtC_Φ(t)^,

then this vector is called the “weak vector” associated with the sectionΣ.

Definition 1.2. We say that an orbit-flip homoclinic orbit is of weak type if for any cross- sectionΣ,u_Σ ∈ Tp_Σ(W^s∩_Σ)where p_Σ =_Σ∩W^u, andTp_Σ(W^s∩_Σ)is the tangent space of the intersection betweenΣand the stable manifold at point p_Σ.

Let M₁,M₃are Melnikov vectors defined in Section 2. Then, our main theorem is stated as follows.

Theorem 1.3. Assume system(1.2) admitsΓan orbit-flip homoclinic orbit of weak type. The eigenvalues of D f(0) avoid a finite number of resonances and satisfy 1 > β > ¹₂, β+1 > α > β, rank(M₁,M₃) = 2, then there exist a 1-homoclinic bifurcation surface H¹, a 2-fold periodic orbit bifurcation surface SN¹, a period-doubling bifurcation surface P²ⁿ of 2ⁿ⁻¹ periodic orbit and a 2ⁿ- homoclinic bifurcation surface H²ⁿ for ∀ n ∈ N, which share the same normal vector M1 at µ = 0, such that system(1.1)has

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a 1-homoclinic orbit if and only ifµ∈ H¹and|µ| 1;

a 2-fold periodic orbit if and only ifµ∈SN¹;

a2ⁿ⁻¹-periodic orbit changing its stability and a2ⁿ-periodic orbit arising at the same time if and only ifµ∈ P²ⁿ;

a2ⁿ-homoclinic orbit if and only ifµ∈ H²ⁿ.

Furthermore, there exist a bifurcation surface ∆1 (which is a branch of H¹) with codimension 1 and normal vector M₁ such that system (1.1) has a 1-homoclinic orbit as well as a 1-periodic orbit for µ∈_∆₁and|µ| 1.

The paper is organized as follows. In Section 2, the local moving coordinates are intro- duced and the Poincaré return map on a given cross-section is deduced. Section 3 is devoted to proving Theorem1.3.

2 Preliminaries and bifurcation equation

In the following, we assume that the parameter depending vector field (1.1) is locally C² linearizable. This condition is not essential but will simplify computations and notations a lot. Such linearization is possible if the eigenvalues −_α,−β and 1 avoid a finite number of resonances, see [1,2,3,19,22] for more details and discussion. As a consequence, there exist a neighborhoodUof 0 inR³and a neighborhoodV of 0 inR³, such that for allv∈Uand all µ∈V, (1.1) has the following normal form:

˙

x= x, y˙ =−β(µ)y, v˙ =−α(µ)v. (2.1) Lemma 2.1. Suppose β+1 > α> β.LetΣbe a cross section transverse to the orbit-flip homoclinic orbitΓ,then the weak vectorµ_Σexists, which is exactly _∂v^∂.

See [18] for details of the proof.

Now we consider the linear variational system of (1.2) and its adjoint system

˙

z= D f(r(t))z, (2.2)

z˙= −(D f(r(t)))^∗z. (2.3)

Denote r(t) = (r^x(t)_,r^y(t)_,r^v(t)) _{and take} T > 0 large enough such that r(−T) = (_{δ, 0, 0})_, r(T) = (0, 0,δ), whereδis small enough so that{(x,y,v): |x|,|y|,|v|<2δ} ⊂U.

Lemma 2.2. Assume system(1.2)admits an orbit-flip homoclinic orbitΓof weak type andα> β>0, then there exists a fundamental solution matrix Z(t) = (z¹(t),z²(t),z³(t))for system(2.2)with

z¹(t)∈ (T_r_i₍_t₎W^u)^c∩(T_r_i₍_t₎W^ss)^c,

z²(t) =−r˙(t)/|r˙^v(T)| ∈ T_r₍_t₎W^u∩T_r₍_t₎W^ss, z³(t)∈ T_r₍_t₎W^s,

satisfying

Z(−T) =





0 ω₂₁ ω₃₁

1 0 ω32

0 0 ω₃₃



, Z(T) =





ω₁₁ 0 0 ω₁₂ 0 1 ω₁₃ 1 0



, whereω₂₁ <_0, ω₁₁6=_0, ω₃₃6=_0,|ω₃₁| |ω₃₃|_, |ω₃₂| |ω₃₃|_.

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Proof. Firstly, due to the fact that TW_loc^ss = span{(0, 0, 1)^∗} and TW_loc^u = span{(1, 0, 0)^∗}, we can choose z²(T) = (0, 0, 1)^∗ as an initial value with the definition z²(t) = −r˙(t)/|r˙^v(T)| in mind, which impliesz²(−T) = (ω₂₁, 0, 0)^∗, whereω₂₁<0.

Then, forz³(t)∈ T_r₍_t₎W^swith z³(T) = (0, 1, 0)^∗, we havez³(−T) = (ω₃₁,ω₃₂,ω₃₃)^∗. Since system (1.2) admits no inclination-flip, then from the strong-inclination property, it is deduced thatω₃₃ 6=0 and|ω₃₁| |ω₃₃|,|ω₃₂| |ω₃₃|.

Finally, note thatT_r₍₋_T)W^uis thex-axis andT_r(T)W^ssis thev-axis, one can check thatz¹(_t)∈

(T_r₍_t₎W^u)^c∩(T_r₍_t₎W^ss)^c is well defined satisfying z¹(−T) = (0, 1, 0)^∗. Since detZ(−T) 6= 0, then we obtain detZ(T)6=0 thanks to the Liouville formula, which impliesω₁₁ 6=0.

Remark 2.3. If the homoclinic orbit of orbit-flip is weak type, then ω₃₂ =0, see Figure2.1.

Figure 2.1: Orbit-flip homoclinic orbit of weak type.

As well known from the matrix theory, system (2.3) has a fundamental solution matrix Φ(t) = (Z⁻¹(t))^∗. We denote Φ(t) = (φ₁(t)_,φ₂(t)_,φ₃(t)). And for every point z = (x(t),y(t),v(t))nearΓ, introduce the local moving coordinates N= (n₁(t), 0,n₃(t)). Set

z=S(t) =r(t) +Z(t)N^∗ =r(t) +z¹(t)n₁(t) +z³(t)n₃(t). (2.4) With this notation, we can choose the cross sections

S₀={z =S(T)_:|x |_,|y|_,|v|<_2δ} ⊂U, S₁={z =S(−T):|x|,|y|,|v|<2δ} ⊂U.

One can refer [13, 14, 15, 24, 25] for more details and discussions about the local moving coordinates method.

Under the transformation (2.4), system (1.1) has the following form

˙

n_j = (φ_j(t))^∗g_µ(r(t)_{, 0})µ+h.o.t., j=_{1, 3,} _(2.5)

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which isC²and produces the mapP₁: S₁→S₀. Integrating both sides from−TtoT, we have n_j(T) =n_j(−T) +M_jµ+h.o.t., j=1, 3, (2.6) where N(T) = (n₁(T)_{, 0,}n₃(T))_,N(−T) = (n₁(−T)_{, 0,}n₃(−T))_{, and}

M_j =

Z _T

−T

(φ_j(t))^∗g_µ(r(t), 0)dt, j=1, 3 are Melnikov vectors.

Lemma 2.4.

M_j =

Z _T

−Tφ^∗_j(t)fµ(r(t), 0)dt =

Z +_∞

−_∞ φ^∗_j(t)fµ(r(t), 0)dt, j=1, 3. (2.7) Proof. To prove (2.7), it is sufficient to verify thatφ^∗_j(t)fµ(r(t), 0) = 0 fort > T, and t < −T, j = 1, 3. As r(T) = (0, 0,δ), then r(t) = (0, 0,r^v(t)) for t > T with r^v(t) = O(δe⁻^α⁽^t⁻^T⁾). Similarly, we have r(t) = (r^x(t), 0, 0) with r^x(t) = O(δe^t⁺^T) for t < −T, which is due to r(−T) = (δ, 0, 0). According to the flow of linearized system near 0, we have

fµ(r(t), 0) = (0, 0,O(δ)) fort >T, fµ(r(t), 0) = (O(δ), 0, 0) fort <−T.

Denote by φ^∗_j(t) = (φ¹_j(t),φ²_j(t),φ³_j(t)). Since Φ^∗(t)Z(t) = I, we have φ^∗_j(t)z²(t) = 0, j= 1, 3. Then z²(T) = (0, 0, 1)^∗ implies that φ³_j(T) = 0, j = 1, 3. Thereafter, we have φ³_j(t) = 0 for t > T, j = 1, 3. Likewise, we can also obtain φ¹_j(−T) =0, j= 1, 3, due to the fact that z²(−T) = (ω₂₁, 0, 0)^∗. Consequently,φ¹_j(t) = 0, fort <−T, j= 1, 3. Thus, conclusion (2.7) is verified. The proof is completed.

DefineP₀: S₀ →S₁, q₀ →q₁ induced by the flow of (2.1) in the neighborhoodUof z=0.

Set the flying time from q₀ to q₁ as τand the Silnikov times = e⁻^τ (see Figure2.2). Then we have

P₀: q₀(x₀,y₀,v₀)→q₁(x₁,y₁,v₁), x₀=sx₁, y₁=s^βy₀, v₁ =s^αv₀, andx₁=δ, v0 =δ;

n⁰₁= (ω₁₁)⁻¹x0, n⁰₃ =y0−ω₁₂(ω₁₁)⁻¹x0, n¹₁=y₁−ω₃₂(ω₃₃)⁻¹v₁, n¹₃ = (ω₃₃)⁻¹v₁.

From the above, we give the following Poincaré maps:

F₁= P₁◦P₀: S₀ −→S₀,

¯

n⁰₁= y₀s^β−ω₃₂(ω₃₃)⁻¹δs^α+M₁µ+h.o.t.,

¯

n⁰₃= (ω₃₃)⁻¹δs^α+M₃µ+h.o.t.

Now, the successor function is given byG(s,y0) = (G₁,G3) = (F₁(q0)−q0)as follows:

G₁ =−(ω₁₁)⁻¹δs+y0s^β−ω32(ω33)⁻¹δs^α+M₁µ+h.o.t., G₃ =−y₀+ω₁₂(ω₁₁)⁻¹δs+ (ω₃₃)⁻¹δs^α+M₃µ+h.o.t.

By solvingy₀from G₃ =0 and substituting it intoG₁=0, we obtain the bifurcation equation

−(ω₁₁)⁻¹δs−ω₃₂(ω₃₃)⁻¹δs^α+M₁µ+M₃µs^β+h.o.t.=0. (2.8)

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Figure 2.2: Poincaré return map.

3 Bifurcation results

In this section, we consider the codimension 3 bifurcation results of the orbit-flip homoclinic bifurcation of weak type, i.e. ω₃₂=0. Then bifurcation equation is:

−(ω₁₁)⁻¹δs+M₁µ+M₃µs^β+h.o.t.=0. (3.1) Proposition 3.1. Suppose1>β> ¹₂,then the following statements hold.

(1) System (1.1) has a unique periodic orbit for µ ∈ D₋⁺∪D₊⁺ if ω₁₁ > 0, andµ ∈ D⁻₋∪D₊⁻ if ω₁₁<_0.Here

D₋⁺={µ: M₁µ>_0, M₃µ<₀}_, D₊⁺={µ: M₁µ>_0, M₃µ>₀}_, D₊⁻={µ: M₁µ<0, M3µ>0}, D₋⁻={µ: M₁µ<0, M3µ<0}. (2) There exists a bifurcation surface∆1:

M₁µ+h.o.t.=_0, forω₁₁M₃µ>₀

with normal vector M₁ atµ = ₀such that system (1.1) has an 1-homoclinic orbit as well as a 1-periodic orbit forµ∈_∆₁and|µ|1.

(3) There exists a unique bifurcation surface H¹:

M₁µ+h.o.t.=0,

with normal vector M₁ at µ = 0 which coincides with ∆1 in the region defined by {µ : ω₁₁M₃µ>₀}such that system(1.1)has a unique 1-homoclinic orbit forµ∈ H¹and|µ|_1.

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(4) There exists a 2-fold periodic orbit bifurcation surface SN¹:

(ω₁₁)⁻¹δ(δ⁻¹βω₁₁M₃µ)¹⁻¹^β = M₃µ(δ⁻¹βω₁₁M₃µ)¹⁻^β^β+M₁µ+h.o.t.

with normal vector M₁atµ=0such that system(1.1)has a unique 2-fold periodic orbit.

Proof. (1) Denote by

F(s,µ) =−(ω₁₁)⁻¹δs+M₃µs^β+M₁µ+h.o.t.

Lets^β =_t, _L(_t,µ) =_M₃_µt+_M₁µ+_h.o.t., _N(_t,µ) = (ω₁₁)⁻¹_δt¹^β +h.o.t. then forµ∈ D⁺₋, ω₁₁>0,

L(0,µ) =M₁µ+h.o.t.>0, L⁰(0,µ) = M₃µ+h.o.t.<0, N⁰(t,µ) = (βω₁₁)⁻¹δt¹⁻^β^β +h.o.t.>0.

So the line W = L(t,µ) and the curve W = N(t,µ) intersect at a unique sufficiently small positive point ¯t < (δ⁻¹ω₁₁M₁µ)^β and F has a unique sufficiently small positive zero ¯s = (t^¯)^1/β.

Ifµ∈D₊⁺, ω₁₁ >0, then

L(0,µ) =M₁µ+h.o.t.>0, L⁰(t,µ) =M₃µ+h.o.t.>0, N⁰(t,µ) = (βω₁₁)⁻¹δt¹⁻^β^β +h.o.t.>_0,

N⁰⁰(t,µ) = (1−β)(β²ω₁₁)⁻¹δt¹

−2β

β +h.o.t.>0.

Take ¯t= [δ⁻¹ω₁₁(2M₃µ+M₁µ)]^β, then

N(t,^¯ µ)−L(t,^¯ µ) =2M₃µ+M₁µ−M₃µt¯−M₁µ>M₃µ>0.

Therefore, based on the fact that N(·,µ)is a monotone increasing convex function, we see that the lineW =L(t,µ)and the curve W = N(t,µ)intersect uniquely att^∗ ∈ (_{0, ¯}t)_, that is,Fhas a unique sufficiently small positive zero point ¯s∈(0,δ⁻¹ω₁₁(2M₃µ+M₁µ)). (2) Letµ∈_∆₁ ,{µ: F(0,µ) =M₁µ+h.o.t.=0, ω₁₁M₃µ>0}, we have

F(s,µ) =s^β[−(ω₁₁)⁻¹δs¹⁻^β+M₃µ+h.o.t.]_.

Consequently, there are two zero pointss₁ =0, s2 = (ω₁₁δ⁻¹M3µ)¹⁻¹^β +h.o.t.

(3) For µ∈ {M₁µ+h.o.t. = 0}, Eq. (3.1) admits s = 0 as its solution. In this case, system (1.1) has a bifurcated homoclinic orbit. And from the above proof, one can easily check that H¹coincides with∆1in the region defined by{µ:ω₁₁M₃µ>0}.

(4) The 2-fold zero point ¯t should satisfy

L(t,µ) = N(t,µ), L⁰(t,µ) = N⁰(t,µ). (3.2) The second equation turns out to be

(αω₁₁)⁻¹δt¹⁻^β^β +h.o.t.= M₃µ+h.o.t. (3.3)

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which forces ¯t = (δ⁻¹βω₁₁M3µ)¹⁻^β^β +h.o.t. forω₁₁M3µ> 0 due to (3.3). Then from the first equation of (3.2), we get the corresponding 2-fold periodic orbit bifurcation surface SN¹:

(ω₁₁)⁻¹δ(δ⁻¹βω₁₁M3µ)¹⁻¹^β = M3µ(δ⁻¹βω₁₁M3µ)¹⁻^β^β +M₁µ+h.o.t.

with normal vectorM₁ atµ=0.

Now, we turn to study on the bifurcation of 2-homoclinic orbit and the period-doubling bifurcation. The second successor function can be given byG(s₁,s₂,y₀,y₂) = (G¹₁,G¹₃,G₁²,G₃²) = (F₁(q0)−q2,F₁(q2)−q0)as follows:

G₁¹=−(ω₁₁)⁻¹δs₂+y₀s₁^β+M₁µ+h.o.t.,

G₃¹=−y₂+ω₁₂(ω₁₁)⁻¹δs₂+ (ω₃₃)⁻¹δs₁^α+M₃µ+h.o.t.

G₁²=−(ω₁₁)⁻¹δs₁+y₂s₂^β+M₁µ+h.o.t.,

G₃²=−y₀+ω₁₂(ω₁₁)⁻¹δs₁+ (ω₃₃)⁻¹δs₂^α+M₃µ+h.o.t.

By solving y₀ and y₂ from G₃¹ = 0, G²₃ = 0, and then by substituting them into G₁¹ = 0, G²₁ =0, we obtain the bifurcation equation

−(ω₁₁)⁻¹δs₂+ (ω₃₃)⁻¹δs₁^βs^α₂+M₃µs₁^β+M₁µ+h.o.t.= 0. (3.4)

−(ω₁₁)⁻¹δs₁+ (ω₃₃)⁻¹δs₁^αs^β₂+M₃µs₂^β+M₁µ+h.o.t.= 0. (3.5) Proposition 3.2. There exists a unique bifurcation surface H²:

M₃µ(ω₁₁δ⁻¹M₁µ)^β+M₁µ+h.o.t.=_0,

which is well defined in the region{µ : ω₁₁M₁µ > 0, ω₁₁M₃µ < 0} such that system(1.1) has a unique 2-homoclinic orbit forµ∈ H².

Proof. Suppose system (1.1) has a unique 2-homoclinic orbit, then (3.4) admits s₁ >0, s2 = 0 or s₁ = 0, s₂ > 0 as its solution. Due to the symmetry of bifurcation equations, we can supposes₁ >0, s₂ =0. Therefore,

s₁ =ω₁₁δ⁻¹M₁µ+h.o.t.>0.

So, we can get the 2-homoclinic bifurcation surfaceH²:

M3µ(ω₁₁δ⁻¹M₁µ)^β+M₁µ+h.o.t.=0,

which is well defined in the region{µ:ω₁₁M₁µ>_0, ω₁₁M₃µ<₀}such that system (1.1) has a unique 2-homoclinic orbit forµ∈H².

Corollary 3.3. The 1-homoclinic bifurcation surface H¹and 2-homoclinic bifurcation surface H²have the same normal vector M₁ atµ = 0. Then, there is a tongue area bounded by H¹ and H², in which there must be another bifurcation surface P² where a period-doubling bifurcation arises.

Define

P₀^j: q_2j−2(x_2j−2,y_2j−2,v_2j−2)→q_2j−1(x_2j−1,y_2j−1,v_2j−1), x_2j−2= s_jx_2j−1, y_2j−1 =s^β_jy_2j−2, v_2j−1 =s^α_jv_2j−2,

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andx_2j₋₁=δ, v_2j₋₂ =δ, j=1, 2, . . . .

n^2j₁⁻²= (ω₁₁)⁻¹δs_j, n^2j₃⁻² =y_2j−2−ω₁₂(ω₁₁)⁻¹δs_j, n^2j₁⁻¹=y_2j−1−ω32(ω33)⁻¹v_2j−1, n^2j₃⁻¹ = (ω33)⁻¹v_2j−1 = (ω33)⁻¹δs^α_j. From the above, we give the n-th Poincaré return maps:

F₁^j = P₁◦P₀^j: S₀→S₀, q_2j−2 7−→q¯_2j−2, n¯^2j₁⁻²= y_2j−2s^β_j −ω₃₂(ω₃₃)⁻¹δs^α_j +M₁µ+h.o.t.,

¯

n^2j₃⁻²= (ω₃₃)⁻¹δs^α_j +M₃µ+h.o.t.

Consequently, the associated n-th successor function is given by Gⁿ(s₁, . . . ,sn,v0, . . . ,v2n−2) = (G₁¹,G₃¹,G₁²,G₃²,G₁³,G₃³,G⁴₁,G⁴₃)

= (F₁¹(q₀)−q₂,F₁²(q₂)−q₄, . . . ,F₁ⁿ(q_2n−2)−q₀). Taken=4 for example.

G¹₁ =−(ω₁₁)⁻¹_δs₂+_y₀_s^β₁+_M₁µ+_h.o.t.,

G¹₃ =−y₂+ω₁₂(ω₁₁)⁻¹δs₂+ (ω₃₃)⁻¹δs^α₁+M₃µ+h.o.t.

G²₁ =−(ω₁₁)⁻¹δs₃+y₂s^β₂+M₁µ+h.o.t.,

G²₃ =−y₄+ω₁₂(ω₁₁)⁻¹δs₃+ (ω₃₃)⁻¹δs^α₂+M₃µ+h.o.t.

G³₁ =−(ω₁₁)⁻¹δs₄+y₄s^β₃+M₁µ+h.o.t.,

G³₃ =−y6+ω₁₂(ω₁₁)⁻¹δs₄+ (ω33)⁻¹δs^α₃+M3µ+h.o.t.

G⁴₁ =−(ω₁₁)⁻¹δs₁+y₆s^β₄+M₁µ+h.o.t.,

G⁴₃ =−y₀+ω₁₂(ω₁₁)⁻¹δs₁+ (ω₃₃)⁻¹δs^α₄+M₃µ+h.o.t.

By solving (y₀,y₂,y₄,y₆) from (G₃¹,G₃²,G₃³,G₃⁴) = 0 and substituting it into (G¹₁,G₁²,G₁³,G₁⁴), then we get the bifurcation equation:

−(ω₁₁)⁻¹δs₂+ (ω₃₃)⁻¹δs₁^βs^α₄+s^β₁M₃µ+M₁µ+h.o.t.=0, (3.6)

−(ω₁₁)⁻¹δs₃+ (ω₃₃)⁻¹δs₂^βs^α₁+s^β₂M₃µ+M₁µ+h.o.t.=0, (3.7)

−(ω₁₁)⁻¹δs₄+ (ω₃₃)⁻¹δs₃^βs^α₂+s^β₃M₃µ+M₁µ+h.o.t.=0, (3.8)

−(ω₁₁)⁻¹δs₁+ (ω₃₃)⁻¹δs₄^βs^α₃+s^β₄M₃µ+M₁µ+h.o.t.=0. (3.9) Thereafter, similarly as in the analysis in Proposition3.2for 2-homoclinic bifurcation result, we can get the 2²-homoclinic bifurcation surface H⁴. By Repeating the above procedure, we can also get the 2ⁿ-homoclinic bifurcation surface H²ⁿ and the period-doubling bifurcation surface P²ⁿ for arbitraryn∈N. Up to now, it is sufficient to claim that Theorem1.3holds.

The bifurcation diagram (see Figure 3.1) is given to better illustrate our main results. In the diagram, Orepresents that there is no periodic orbits, while P (resp. P^k) represents that there exists a 1-periodic (resp. k-periodic) orbit in the corresponding region.

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Figure 3.1: Bifurcation diagram.

4 Conclusion

This paper is devoted to proving the existence of higher order homoclinic orbits and periodic orbits from the orbit-flip homoclinic orbit of weak type. Such homoclinic orbit is a degenerate version of the so called orbit-flip homoclinic orbit, and it is a new case of codimension 3. The homoclinic orbit of higher order, also named as the multi-round homoclinic orbit, corresponds to the traveling or standing multi-pulse in the spatial dynamics. The method we employ is the local moving coordinates method. The phenomenon of homoclinic doubling bifurcation like we showed in this paper, is just like the cascades of periodic doubling bifurcation found by Feigenbaum and Coullet–Tresser. It is a change of a homoclinic orbit into twice round homoclinic orbit in the neighborhood of the primary homoclinic orbit. More precisely, H¹ is the 1-homoclinic bifurcation surface and H² is the 2-homoclinic bifurcation surface as we found, which have the same normal vectorM₁ atµ= 0. So, there is a tongue area bounded by H¹ and H². In the tongue area, there must be another bifurcation surface P² where a period-doubling bifurcation arises. By repeating the similar procedure, we also obtain the 2ⁿ-homoclinic bifurcation surface H²ⁿ and the period-doubling bifurcation surface P²ⁿ for arbitraryn∈_N.

Acknowledgements

This work is supported by National Natural Science Foundation of P. R. China (11101370, 11101283, 11371140, 11211130093), “521” talent program of ZSTU (11430132521304) and China Scholarship Council.

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