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Non-resonance 3D homoclinic bifurcation with an inclination flip
Chaos, Solitons and Fractals 42 (2009) 2597–2605
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Non-resonance 3D homoclinic bifurcation with an inclination flip q Qiuying Lu Department of Mathematics, East China Normal University, Shanghai 200241, China Laboratoire Paul Painlevé, UFR de Mathématiques Université de Lille 1, 59655 Villeneuve d’Ascq, France
a r t i c l e
i n f o
Article history: Accepted 30 March 2009
Communicated by Prof. Ji-Huan He
a b s t r a c t Local active coordinates approach is employed to study the bifurcation of a non-resonance three-dimensional smooth system which has a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues a; b; 1 satisfying a > b > 0. A homoclinic orbit is called an inclination-flip homoclinic orbit if the strong inclination property of the stable manifold is violated. In this paper, we show the existence of 1-homoclinic orbit, 1-periodic orbit, 2n -homoclinic orbit and 2n -periodic orbit in the unfolding of an inclination-flip homoclinic orbit. And we figure out the bifurcation diagram based on the existence region of the corresponding bifurcation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction and hypotheses It is well known that the bifurcation of a degenerate dynamical system on R3 may be very complicated. An earlier example is the case where there exists a homoclinic orbit to a hyperbolic equilibrium point. In [13], Shil’nikov studied the codimension 1 homoclinic bifurcation with two complex conjugated eigenvalues. He pointed out that if the eigenvalues a and b satisfied Re a ¼ Reb < 1, then the dynamical behavior in a small neighborhood of the homoclinic orbit was chaotic. However, in [14], he showed that, under the generic hypothesis, the homoclinic bifurcation was relatively simple. Precisely, the vector field X c had neither homoclinic orbit nor periodic orbit in a small neighborhood of the primary homoclinic orbit for c > 0. While, for c < 0, only one hyperbolic periodic orbit bifurcated from the homoclinic orbit. Accordingly, more degenerate cases should be considered for more complicated dynamics. In [17], Yanagida studied the inclination-flip homoclinic orbit together with two other codimension 2 homoclinic bifurcations, which were the cases of the resonant bifurcation and the orbit-flip bifurcation. Since then, many research works have been devoted to this subject, see [1–6,8–12,18–21]. Among these works, Dumortier et al. [3] gave the persistence condition for inclination-flip homoclinic orbits in terms of Melnikov integrals. Whereas, Kisaka [9] studied the homoclinic doubling bifurcation of an inclination-flip homoclinic orbit under the assumption ku < ks < 2ku . In [2], it was presented that a perturbation of an inclination-flip homoclinic orbit would lead to the occurrence of Smale horseshoes. In [6], the author proved the existence of Smale horseshoe under the condition 2ku < minfks ; kuu g by using the invariant foliation to reduce the study of the return map into the analysis of one-dimensional multi-valued map. In [12], the author showed the existence of a strange attractor in the unfolding of an inclination-flip homoclinic orbit by comparing the Poincaré return map with the Hénon family. In this paper, we consider the 1-homoclinic orbit, 1-periodic orbit, 2n -homoclinic orbit and 2n -periodic orbit bifurcated from an inclination-flip homoclinic orbit. We use the local active coordinates method which was originally established in [21] and then was further developed in [19,20].
q
Work supported by National Natural Science Foundation of China (# 10671069). E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.112
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We consider the following smooth system
z_ ¼ f ðzÞ þ gðz; lÞ;
ð1Þ
and its unperturbed system
z_ ¼ f ðzÞ;
ð2Þ
where z 2 R3 ; l 2 R2 ; 0 6 jlj 1; f ð0Þ ¼ 0; gðz; 0Þ ¼ 0. First of all, we assume that: ðH1 Þ System (2) has a hyperbolic equilibrium O and the relevant linearization matrix Df ð0Þ has simple real eigenvalues: a; b; 1 satisfying a > b > 0. Without loss of generality, we can always assume that the hyperbolic fixed point is the origin, i.e., gð0; lÞ ¼ 0. Since the implicit function theorem tells us that the hyperbolic fixed point persists through out the unfolding. Moreover we assume that the eigenvalues of Df ð0Þ avoid a finite number of resonances so that system (1) is uniformly C 2 linearizable according to [15]. Thereafter, up to a C 2 diffeomorphism, there exists U, a small neighborhood of the origin in R3 and V, a neighborhood of the origin in R2 , so that for all z 2 U and all l 2 V, system (1) has the following C 2 normal form:
x_ ¼ x;
y_ ¼ bðlÞy;
v_ ¼ aðlÞv :
ð3Þ
Besides, we make the following assumptions: _ . Then eþ 2 T 0 W u ; e 2 T 0 W s are unit eigenðH2 Þ System (2) has a homoclinic loop C ¼ fz ¼ rðtÞ; t 2 Rg. Let e ¼ limt!1 jrr_ ðtÞ ðtÞj vectors corresponding to 1 and b. ðH3 Þ Denote by e s the unit eigenvector corresponding to a, then
SpanðT rðtÞ W u ; T rðtÞ W s ; es Þ ¼ R3 ;
for t 1
With the above assumptions, the homoclinic orbit C is of codimension 2. Remark 1.1 (a) ðH3 Þ is equivalent to T rðtÞ W s ! eþ e ; when t ! 1. (b) For the existing loop C, ðH2 Þ is generic, which guarantees that C has no orbit flip. While ðH3 Þ is not generic, which indicates that W s occurs an inclination flip when t ! 1 (see Fig. 1).
2. Preliminaries and bifurcation equation Now we consider the linear variational system of (2) and its adjoint system
z_ ¼ Df ðrðtÞÞz; z_ ¼ ðDf ðrðtÞÞÞ z:
ð4Þ ð5Þ
We denote rðtÞ ¼ ðrx ðtÞ; ry ðtÞ; r v ðtÞÞ and take T > 0 large enough so that rðTÞ ¼ ðd; 0; 0Þ; rðTÞ ¼ ð0; d; dv Þ, where jdv j ¼ Oðd2 Þ and d is small enough so that fðx; y; v Þ : jxj; jyj; jv j < 2dg U.
Fig. 1. Inclination flip.
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Lemma 2.1 (cf. [7,16]). There exists a fundamental solution matrix ZðtÞ ¼ ðz1 ðtÞ; z2 ðtÞ; z3 ðtÞÞ for system (4) with
c c z1 ðtÞ 2 T ri ðtÞ W u \ T ri ðtÞ W s ; z2 ðtÞ ¼ r_ ðtÞ=jr_ y ðTÞj 2 T rðtÞ W u \ T rðtÞ W s ; z3 ðtÞ 2 T rðtÞ W ss ; satisfying
0
0 x21
B zðTÞ ¼ @ 0
0
1
0
1
0
1
x31 x11 0 0 B C x32 C 1 0 A; A; zðTÞ ¼ @ x12 x33 x13 x23 1
where jx23 j 1; x21 < 0; x11 – 0; x32 – 0. From the matrix theory, system (5) has a fundamental matrix solution UðtÞ ¼ ðZ 1 ðtÞÞ . We denote UðtÞ ¼ ð/1 ; /2 ; /3 Þ. We introduce the local active coordinates near the orbit C as (z1 ðtÞ; z2 ðtÞ; z3 ðtÞ) with the components N ¼ ðn1 ; 0; n3 Þ, and set
z ¼ SðtÞ ¼ rðtÞ þ zðtÞN ¼ rðtÞ þ z1 ðtÞn1 þ z3 ðtÞn3 :
ð6Þ
With this notation, we can choose the cross sections
S0 ¼ fz ¼ SðTÞ : jxj; jyj; jv j < 2dg U; S1 ¼ fz ¼ SðTÞ : jxj; jyj; jv j < 2dg U: Under the transformation (6), system (1) has the following form
n_ j ¼ ð/j ðtÞÞ g l ðrðtÞ; 0Þl þ h:o:t:;
j ¼ 1; 3;
ð7Þ
2
which is C and produces the map P 1 : S1 ! S0 . Integrating both sides from T to T, we have
nj ðTÞ ¼ nj ðTÞ þ M j l þ h:o:t:;
j ¼ 1; 3;
ð8Þ RT
where NðTÞ ¼ ðn1 ðTÞ; 0; n3 ðTÞÞ; NðTÞ ¼ ðn1 ðTÞ; 0; n3 ðTÞÞ, and Mj ¼ T ð/j ðtÞÞ g l ðrðtÞ; 0Þ dt; j ¼ 1; 3 are Melnikov vectors. RT R þ1 Lemma 2.2 (cf. [7,16]). Let notations be as above. Then M 1 ¼ T ð/1 ðtÞÞ g l ðrðtÞ; 0Þ dt ¼ 1 ð/1 ðtÞÞ g l ðrðtÞ; 0Þ dt. We define P 0 : S0 ! S1 ; q0 ! q1 induced by the flow of (3) in the neighborhood U of z ¼ 0. And we set the flying time from q0 to q1 as s and the Shil’nikov time s ¼ es (see Fig. 2). Then we have
P0 : q0 ðx0 ; y0 ; v 0 Þ ! q1 ðx1 ; y1 ; v 1 Þ; x0 ¼ sx1 ;
y1 ¼ sb y0 ;
v 1 ¼ sa v 0 ;
and x1 d; y0 d;
n01 ¼ ðx11 Þ1 x0 ;
n03 ¼ v 0 dv ðx11 Þ1 x13 x0 ; 1
n11 ¼ v 1 ðx32 Þ x33 y1 ; n13 ¼ ðx32 Þ1 y1 : From the above, we give the following Poincaré maps:
F 1 ¼ P1 P 0 : S0 ! S0 ; 01 ¼ v 0 sa ðx32 Þ1 x33 dsb þ M1 l þ h:o:t:; n 03 ¼ ðx32 Þ1 dsb þ M 3 l þ h:o:t: n
Fig. 2. Poincaré return map.
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Now, the successor function is given by Gðs; v 0 Þ ¼ ðG1 ; G3 Þ ¼ ðF 1 ðq0 Þ q0 Þ as follows:
G1 ¼ ðx11 Þ1 ds þ v 0 sa ðx32 Þ1 x33 dsb þ M1 l þ h:o:t:; G3 ¼ v 0 þ dv þ ðx11 Þ1 x13 ds þ ðx32 Þ1 dsb þ M 3 l þ h:o:t: By solving
v 0 from G3 ¼ 0 and substituting it into G1 ¼ 0, we obtain the bifurcation equation
ðx11 Þ1 ds þ dv sa þ ðx32 Þ1 dsaþb þ M 3 lsa ðx32 Þ1 x33 dsb þ M 1 l þ h:o:t: ¼ 0:
ð9Þ
3. Bifurcation analysis and bifurcation diagram Definition 1. We shall say that the strong foliation W ss rðtÞ occurs a strong inclination flip as t ! 1, if the stable manifold W srðtÞ turns to be an inclination flip and the strong stable component x33 ¼ 0 as t ¼ T; T 1. We will distinguish the following cases: Case (1): 1 > a > b > 0. The bifurcation Eq. (9) is reduced to the following
if x33 ¼ 0;
for dv –0; for dv ¼ 0;
if x33 – 0;
dv sa þ M 1 l þ h:o:t: ¼ 0; ðx11 Þ1 ds þ ðx32 Þ1 dsaþb þ M 3 lsa þ M 1 l þ h:o:t: ¼ 0;
ðx32 Þ1 x33 dsb þ M1 l þ h:o:t: ¼ 0:
ð10Þ ð11Þ ð12Þ
Case (2): a > 1 > b > 0. We obtain the following bifurcation equations
if x33 ¼ 0; if x33 – 0;
ðx11 Þ1 ds þ M 1 l þ h:o:t: ¼ 0; 1
ðx32 Þ x33 ds þ M1 l þ h:o:t: ¼ 0: b
ð13Þ ð14Þ
Case (3): a > b > 1. We have the following bifurcation equation
ðx11 Þ1 ds þ M 1 l þ h:o:t: ¼ 0:
ð15Þ
Remark 3.1. Notice that, the local weak stable manifold is not unique. In fact, one can fill up a wedge area in W s \ U with these manifolds (curves). Obviously, dv ¼ 0 means that in the coordinate system corresponding to the normal form (3), the local weak stable manifold C \ U is exactly a segment of the y-axis. And we name this kind of homoclinic orbit a homoclinic orbit of ‘weak’ type. First, we assume x33 –0, that is to say, W ss rðtÞ does not occur a strong inclination flip. According to (12), (14) and (15), we have the following results. Theorem 3.1. If M 1 – 0; x33 – 0, then system (1) has at most one periodic orbit in the small neighborhood of C. And it does exist if and only if l 2 fx11 M 1 l > 0g; 0 < jlj 1, for a > b > 1; l 2 fx32 x33 M 1 l > 0g; 0 < jlj 1, for 1 > a > b > 0 and a > 1 > b > 0. Theorem 3.2. If M1 – 0; x33 – 0, then there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists as l 2 H1 . Second, we consider the case of strong inclination flip, that is, x33 ¼ 0. Owing to (10), (13) and (15), we can state the following result. Theorem 3.3. Assume x33 ¼ 0, then (1) if 1 > a > b > 0 and dv –0, system (1) has a unique 1-periodic orbit if and only if l 2 fdv M 1 l < 0g; 0 < jlj 1, and there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists for l 2 H1 . (2) if a > 1 > b > 0 or a > b > 1, system (1) has a unique 1-periodic orbit if and only if l 2 fx11 M 1 l > 0g; 0 < jlj 1, and there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists for l 2 H1 . Then, it remains the case concerning the bifurcation equation (11) which we deal in the sequel. We state the results for this case in the following. Theorem 3.4. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then there exists a 1-homoclinic bifurcation n surface H1 , a twofold periodic orbit bifurcation surface SN 1 , a period-doubling bifurcation surface P2 of 2n1 periodic orbit and a n 2n 2 -homoclinic bifurcation surface H for 8 n 2 N, which share the same normal vector M 1 at l ¼ 0, so that system (1) has
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a a a a
1-homoclinic orbit if and only if l 2 H1 and jlj 1; twofold periodic orbit if and only if l 2 SN 1 ; 2n1 -periodic orbit changing its stability and a 2n -periodic orbit arising at the same time if and only if n 2n -homoclinic orbit if and only if l 2 H2 .
n
l 2 P2 ;
Furthermore there exists a bifurcation surface D1 (which is a branch of H1 ) with codimension 1 and normal vector M 1 so that system (1) has a 1-homoclinic orbit as well as a 1-periodic orbit for l 2 D1 and jlj 1. The rest of the paper is devoted to the proof of the above theorem which follows from several propositions. We denote the left side of (11) by Fðs; lÞ ¼ Lðs; lÞ Nðs; lÞ, where
Lðs; lÞ ¼ M 3 lsa þ M1 l þ h:o:t:;
Lð0; lÞ ¼ Fð0; lÞ;
Nðs; lÞ ¼ ðx11 Þ1 ds ðx32 Þ1 dsaþb þ h:o:t:; Nð0; lÞ ¼ 0; and
Dþþ ¼ fl : M 1 l > 0; M 3 l > 0g; Dþ ¼ fl : M1 l > 0; M3 l < 0g; Dþ ¼ fl : M 1 l < 0; M 3 l > 0g; D ¼ fl : M1 l < 0; M3 l < 0g: It is evident that the four areas are not empty when rank ðM 1 ; M 3 Þ ¼ 2. Proposition 3.1. Suppose 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then Fðs; lÞ has a unique sufficiently small positive zero point s so that system (1) has a unique periodic orbit. More precisely, þ (1) when a þ b > 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if x11 > 0, and l 2 D [ Dþ if x11 < 0. þ (2) when 0 < a þ b < 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if x32 < 0, and l 2 D [ Dþ if x32 > 0. 1 þ > ðx11 Þ1 , and (3) when a þ b ¼ 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if ðx32 Þ l 2 D [ Dþ if ðx32 Þ1 < ðx11 Þ1 .
Proof. When a þ b > 1,
Fðs; lÞ ¼ ðx11 Þ1 ds þ M 3 lsa þ M1 l þ h:o:t: We set sa ¼ t. Then in case x11 > 0, we have the following possibilities: If l 2 Dþ , then
L0 ðt; lÞ ¼ M 3 l þ h:o:t: < 0;
Lð0; lÞ ¼ M1 l þ h:o:t: > 0; 0
1
N ðt; lÞ ¼ ðax11 Þ dt
1a
a
þ h:o:t: > 0:
So the line W ¼ Lðt; lÞ and the curve W ¼ Nðt; lÞ intersect at a unique sufficiently small positive point t < ðd1 x11 M 1 lÞa . Hence F has a unique sufficiently small positive zero s ¼ ðtÞ1=a . If l 2 Dþ þ , then
Lð0; lÞ ¼ M 1 l þ h:o:t: > 0; N0 ðt; lÞ ¼ ðax11 Þ1 dt
1a
a
L0 ðt; lÞ ¼ M 3 l þ h:o:t: > 0;
þ h:o:t: > 0;
N ðt; lÞ ¼ ð1 aÞða x11 Þ1 dt 00
2
12a
a
þ h:o:t: > 0:
We take t ¼ ½d1 x11 ð2M3 l þ M 1 lÞ a . Then
Nðt; lÞ Lðt; lÞ ¼ 2M 3 l þ M 1 l M 3 lt M 1 l > M 3 l > 0: Therefore, based on the fact that Nð ; lÞ is a monotone increasing convex function, we see that the line W ¼ Lðt; lÞ and the curve W ¼ Nðt; lÞ intersect uniquely at t 2 ð0; tÞ, that is to say, F has a unique sufficiently small positive zero point s 2 ð0; d1 x11 ð2M3 l þ M 1 lÞÞ. The proof for the rest cases can be given similarly. h Proposition 3.2. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then (1) for a þ b > 1, there exists a bifurcation surface D1 with codimension 1 and normal vector M 1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit for l 2 D1 and jlj 1; (2) for 0 < a þ b < 1, there exists a bifurcation surface D2 with codimension 1 and normal vector M 1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit as l 2 D2 and jlj 1;
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(3) for a þ b ¼ 1, there exists a bifurcation surface D3 with codimension 1 and normal vector M1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit as l 2 D3 and jlj 1. Proof. When a þ b > 1; l 2 D1 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x11 M 3 l > 0g, we have
h i Fðs; lÞ ¼ sa ðx11 Þ1 ds1a þ M 3 l þ h:o:t: : 1
Consequently, there are two zero points s1 ¼ 0; s2 ¼ ðx11 d1 M 3 lÞ1a þ h:o:t. When 0 < a þ b < 1; l 2 D2 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x32 M 3 l < 0g, one has
h i Fðs; lÞ ¼ sa ðx32 Þ1 dsb þ M 3 l þ h:o:t: ¼ 0 1
which admits s1 ¼ 0; s2 ¼ ðx32 d1 M 3 lÞb þ h:o:t. as its solutions. When a þ b ¼ 1; l 2 D3 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x11 x32 ðx32 x11 ÞM3 l > 0g, we obtain
h i Fðs; lÞ ¼ sa ððx32 Þ1 ðx11 Þ1 Þds1a þ M3 l þ h:o:t: :
1
Thereafter, it has two zero points s1 ¼ 0; s2 ¼ ðx11 x32 ðx32 x11 Þ1 d1 M 3 lÞ1a þ h:o:t. h Proposition 3.3. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM 1 ; M 3 Þ ¼ 2, then Fðs; lÞ has a unique twofold positive zero point s so that system (1) has a unique twofold periodic orbit. Precisely speaking, 1
(1) when a þ b > 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ðd1 ax11 M 3 lÞ1a þ h:o:t. for x11 M 3 l > 0 and the twofold periodic bifurcation surface is SN1 :
1 a ðx11 Þ1 d d1 ax11 M 3 l 1a ¼ M 3 l d1 ax11 M3 l 1a þ M 1 l þ h:o:t: with normal vector M 1 at l ¼ 0; 1 (2) when 0 < a þ b < 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ½ðdða þ bÞÞ1 ax32 M 3 l b þ h:o:t. for x32 M 3 l < 0 1 and the twofold periodic orbit bifurcation surface SN :
h i1þab h iab ðx32 Þ1 d ðdða þ bÞÞ1 ax32 M 3 l ¼ M 3 l ðdða þ bÞÞ1 ax32 M3 l þ M 1 l þ h:o:t: has normal vector M 1 at l ¼ 0; 1 (3) when a þ b ¼ 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ½ðdðx32 x11 ÞÞ1 ax11 x32 M 3 l 1a for 1 1 1 ððx11 Þ ðx32 Þ ÞM3 l > 0, and the corresponding twofold periodic orbit bifurcation surface SN :
h i11 a ðx11 Þ1 ðx32 Þ1 d ðdðx32 x11 ÞÞ1 ax11 x32 M3 l h i1a a ¼ M 3 l ðdðx32 x11 ÞÞ1 ax11 x32 M 3 l þ M1 l þ h:o:t: has normal vector M 1 at
l ¼ 0.
Proof. The twofold zero point t should satisfy
Lðt; lÞ ¼ Nðt; lÞ;
L0 ðt; lÞ ¼ N0 ðt; lÞ:
ð16Þ
The second equation turns out to be
ðax11 Þ1 dt
1a
a
b
ða þ bÞðax32 Þ1 dta þ h:o:t: ¼ M3 l: 1
ð17Þ
a
When a þ b > 1, we have t ¼ ðd ax11 M 3 lÞ1a þ h:o:t. for x11 M 3 l > 0 due to (17). Then from the first equation of (16), we get the corresponding twofold periodic orbit bifurcation surface SN 1 :
1 a ðx11 Þ1 d d1 ax11 M 3 l 1a ¼ M 3 l d1 ax11 M3 l 1a þ M 1 l þ h:o:t: with normal vector M1 at l ¼ 0. The other two cases can be proved similarly.
h
Now we study the bifurcation of 2-homoclinic orbit and the period-doubling bifurcation for the case of 1 > a > b > 0; x33 ¼ 0; dv ¼ 0. As above, we set t1 and t 2 as the flying time from q0 to q1 and from q2 to q3 , respectively, si ¼ eti ; i ¼ 1; 2. Then the second successor function can be expressed by
G2 ðs1 ; s2 ; v 0 ; v 2 Þ ¼ G11 ; G13 ; G21 ; G23 ¼ ðF 1 ðq0 Þ q2 ; F 1 ðq2 Þ q0 Þ
Q. Lu / Chaos, Solitons and Fractals 42 (2009) 2597–2605
2603
with:
G11 ¼ ðx11 Þ1 ds2 þ v 0 sa1 ðx32 Þ1 x33 dsb1 þ M 1 l þ h:o:t:; G13 ¼ v 2 þ dv þ ðx11 Þ1 x13 ds2 þ ðx32 Þ1 dsb1 þ M 3 l þ h:o:t: G21 ¼ ðx11 Þ1 ds1 þ v 2 sa2 ðx32 Þ1 x33 dsb2 þ M 1 l þ h:o:t:; G23 ¼ v 0 þ dv þ ðx11 Þ1 x13 ds1 þ ðx32 Þ1 dsb2 þ M 3 l þ h:o:t: Solving ðv 0 ; v 2 Þ from ðG13 ; G23 Þ ¼ 0 and substituting it into G11 ¼ 0 and G21 ¼ 0, we obtain the bifurcation equations a ðx11 Þ1 ds2 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsa1 sb2 þ M3 lsa1 þ M 1 l þ h:o:t: ¼ 0; 1
ð18Þ
a ðx11 Þ1 ds1 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsb1 sa2 þ M3 lsa2 þ M 1 l þ h:o:t: ¼ 0: 2
ð19Þ
It is easy to see that system (1) has a 2-homoclinic orbit near C if and only if the above equation has s1 ¼ 0; s2 > 0 as its solution by the symmetry of G2 . If s1 ¼ 0; s2 > 0 is the solution of the bifurcation equation, then s2 ¼ d1 x11 M 1 l þ h:o:t. for x11 M 1 l > 0, and the 2-homoclinic bifurcation surface H2 : x11 M 3 l ¼ dðd1 x11 M 1 lÞ1a þ h:o:t. has codimension 1 with normal vector M 1 at l ¼ 0. Thus we have: Proposition 3.4. Suppose 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then there exists a unique 1-homoclinic bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with codimension 1 and normal vector M 1 at l ¼ 0, which coincides with D1 in the region defined by fl : x11 M 3 l > 0g. For l 2 H1 and jlj 1, system (1) has a unique 1-homoclinic orbit. There exists a unique bifurcation surface H2 : x11 M 3 l ¼ dðd1 x11 M 1 lÞ1a þ h:o:t. which is well defined in the region fl : x11 M 1 l > 0; x11 M 3 l < 0g, so that system (1) has a unique 2-homoclinic orbit for l 2 H2 . From Proposition 3.4, we know that H1 and H2 have the same normal vector M1 at l ¼ 0, and M3 l ¼ OðjM1 lj1a Þ for l 2 H2 . So, there is a tongue area bounded by H1 and H2 . And in the tongue area, there must be another bifurcation surface P 2 where a period-doubling bifurcation arises. Similarly as in Section 2, we define
Pj0 : q2j2 ðx2j2 ; y2j2 ; v 2j2 Þ ! q2j1 ðx2j1 ; y2j1 ; v 2j1 Þ; x2j2 ¼ sj x2j1 ;
v 2j1 ¼ saj v 2j2 ;
y2j1 ¼ sbj y2j2 ;
and x2j1 d; y2j2 d; j ¼ 1; 2; . . ..
n2j2 ¼ ðx11 Þ1 x2j2 ; 1 n2j1 1
n2j2 ¼ v 2j2 dv ðx11 Þ1 x13 x2j2 ; 3
¼ v 2j1 ðx32 Þ x33 y2j1 ; n2j1 ¼ ðx32 Þ1 y2j1 : 3 1
From the above, we can give the nth Poincaré return maps:
F j1 ¼ P1 P j0 : S0 ! S0 ;
2j2 ; q2j2 #q
2j2 n 1
¼ v 2j1 ðx32 Þ x33 y2j1 þ M1 l þ h:o:t:;
2j2 n 3
¼ ðx32 Þ1 y2j1 þ M3 l þ h:o:t:
1
Consequently, the associated n-th successor function is given by
Gn ðs1 ; . . . ; sn ; v 0 ; . . . ; v 2n2 Þ ¼ G11 ; G13 ; G21 ; G23 ; G31 ; G33 ; G41 ; G43 ¼ F 11 ðq0 Þ q2 ; F 21 ðq2 Þ q4 ; . . . ; F n1 ðq2n2 Þ q0 : Now, we study the 4-homoclinic bifurcation surface H4 with the condition 1 > a > b > 0; x33 ¼ 0; dv ¼ 0. By solving ðv 0 ; v 2 ; v 4 ; v 6 Þ from ðG13 ; G23 ; G33 ; G43 Þ ¼ 0 and substituting it into ðG11 ; G21 ; G31 ; G41 Þ, we obtain the bifurcation equations: a ðx11 Þ1 ds2 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsa1 sb4 þ M3 lsa1 þ M 1 l þ h:o:t: ¼ 0; 1 1
1
1þa 13 ds2
ðx11 Þ ds3 þ ðx11 Þ x
b a 32 Þ ds1 s2 1 b a 32 Þ ds2 s3 1 b a 32 Þ ds3 s4
þ ðx
a þ ðx ðx11 Þ1 ds4 þ ðx11 Þ1 x13 ds1þ 3 1
1
1þa 13 ds4
ðx11 Þ ds1 þ ðx11 Þ x
1
þ ðx
a
ð20Þ
þ M3 ls2 þ M 1 l þ h:o:t: ¼ 0;
ð21Þ
þ M3 lsa3 þ M 1 l þ h:o:t: ¼ 0;
ð22Þ
a
þ M3 ls4 þ M 1 l þ h:o:t: ¼ 0:
ð23Þ
So, we just need to consider the above equations that admit s1 ¼ 0; s2 > 0; s3 > 0; s4 > 0 as their solutions. Correspondingly, from (20), we have s2 ¼ x11 d1 M 1 l þ h:o:t. for x11 M 1 l > 0. Then (21) yields
h i a s3 ¼ x11 d1 M3 l x11 d1 M1 l þ M 1 l þ h:o:t: ¼ x11 d1 M 1 l þ h:o:t:
for
l 2 fl : x11 d1 M1 l > 0g \ fl : M3 l ¼ oðjM1 lj1a Þg.
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Fig. 3. Bifurcation diagram in the case of 1 > a > b > 0; a þ b > 1; dv ¼ 0; x11 > 0; x33 ¼ 0.
If a þ b > 1, we obtain s4 ¼ x11 d1 M 1 l þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Then, (23) gives the 4-homoclinic bifurcation surface H4 :
a M3 l x11 d1 M1 l þ M 1 l þ h:o:t: ¼ 0:
ð24Þ
Here, H4 is defined on fl : M 3 l ¼ OðjM1 lj1a Þg. If 0 < a þ b < 1, we get s4 ¼ x11 ðx32 Þ1 ðx11 d1 M 1 lÞaþb þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Consequently, from (23), the 4-homoclinic bifurcation surface H4 should be:
b h aþb ia ðx32 Þ1 d x11 d1 M 1 l x11 ðx32 Þ1 x11 d1 M1 l h aþb ia þ M 3 l x11 ðx32 Þ1 x11 d1 M 1 l þ h:o:t: ¼ 0:
ð25Þ
In this case, H4 is defined on fl : M 3 l ¼ OðjM1 ljb Þg. If a þ b ¼ 1, then s4 ¼ x11 d1 ½1 þ ðx32 Þ1 M 1 l þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Then, we get the 4-homoclinic bifurcation surface H4 :
n h i oa h ia M3 l x11 d1 1 þ ðx32 Þ1 M 1 l þ M 1 l þ ðx32 Þ1 x11 1 þ ðx32 Þ1 M1 l þ h:o:t: ¼ 0:
ð26Þ
Owing to the bifurcation surface equation, H4 is defined on fl : M 3 l ¼ OðjM1 lj1a Þg as well. Summing up, we get the 4-homoclinic bifurcation surface H4 : (24)–(26) on the parameter surface fl : M 3 l ¼ oðjM 1 lj1a Þg. n Repeating the above procedure, we can also get the 2n -homoclinic bifurcation surface H2 and the period-doubling bifurcation 2n surface P for arbitrary n 2 N. To well illustrate our main theorem, we give the bifurcation diagrams under the assumptions 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, where O represents that there is no periodic orbits, while P (respectively, Pk ) represents that there exists a 1-periodic (respectively, k-periodic) orbit in the corresponding region (see Fig. 3).
4. Conclusion By employing the local active coordinates approach, we have established the complete bifurcation results for non-resonance 3D homoclinic orbit with an inclination flip. More precisely, when the homoclinic orbit takes place an inclination flip but not a strong inclination flip or it occurs a strong inclination flip but not of the ‘weak’ type, then the bifurcation result is unique, i.e. either a homoclinic orbit or a periodic orbit persists. Otherwise, the uniqueness of solution cannot be guaranteed, which is quite challenging for the bifurcation discussion. However, with the help of induction, we manage to establish the 1n homoclinic bifurcation surface H1 , twofold periodic orbit bifurcation surface SN 1 , a period-doubling bifurcation surface P2 of n1 n 2n periodic orbit and a 2 -homoclinic bifurcation surface H for 8 n 2 N. Thus, complete bifurcation results are obtained 2 and complete bifurcation diagram is given. Acknowledgements The author is grateful to Professor Deming Zhu for having introduced to her the subject and for his many helpful suggestions. She also wishes to thank Dr. Guoting Chen for a careful reading of the article and the anonymous referees for their comments and suggestions which improved the results.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Chow SN, Deng B, Fiedler B. Homoclinic bifurcation at resonant eigenvalues. J Dyn Syst Diff Eqs 1990;2:177–244. Deng B. Homoclinic twisting bifurcation and cusp horseshoe maps. J Dyn Diff Eqs 1993;5:417–67. Dumortier F, Kokubu H, Oka H. A degenerate singularity generating geometric Lorenz attractors. Ergod Theory Dyn Syst. 1995;15:833–56. Geng FJ, Zhu D, Xu Y. Bifurcations of heterodimensional cycles with two saddle points. Chaos, Solitons & Fractals 2009;39:2063–75. He JH. Limit cycle and bifurcation of nonlinear problems. Chaos, Solitons & Fractals 2005;26:827–33. Homburg AJ, Kokubu H, Krupa M. The cusp horse-shoe and its bifurcation in the unfolding of an inclination-flip homoclinic orbit. Ergod Theory Dyn Syst 1994;14:667–93. Jin Y, Zhu D. Bifurcations of rough heteroclinic loops with three saddle points. Acta Math Sin Eng Ser 2002;18:199–208. Kisaka M, Kokubu H, Oka H. Bifurcations to N-homoclinic orbit and N-periodic orbits in vector fields. J Dyn Diff Eqs 1993;5:305–57. Kisaka M, Kokubu H, Oka H. Supplement to homoclinic doubling bifurcation in vector fields, Dynamical systems (Santiago, 1990), Pitman Res. Notes Math. Ser., 285, Longman Sci. Tech., Harlow; 1993. p. 92–116. Lau YT. Global aspects of homoclinic bifurcations of three-dimensional saddles. Chaos, Solitons & Fractals 1993;3:369–82. Lu Q. Codimension 2 bifurcation of twisted double homoclinic loops. Comput Math Appl 2009;57:1127–41. Naudot V. Strange attractor in the unfolding of an inclination-flip homoclinic orbit. Ergod Theory Dyn. Syst 1996;16:1071–86. Shil’nikov LP. A case of the existence of a denumerable set of periodic motions. Sov Math Dockl 1965;6:163–6. Shil’nikov LP. On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium of state of saddle type. Math USSR Sbornik 1968;6:427–37. Sternberg S. On the local homeomorphisms of Euclidean n-space, Am. J. Math, 80 (1958), 623-631 and 81 (1959), 578-604. Tian Q, Zhu D. Bifurcations of nontwisted heteroclinic loops. Sci Chin Ser A 2000;30:193–202. Yanagida E. Branching of double pulse solutions from single pulse solutions in nerve axon equations. J Diff Eqs 1987;66:243–62. Zhang W, Wang FX, Zu JW. Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation. Chaos, Solitons & Fractals 2005;24:977–98. Zhang T, Zhu D. Codimension 3 homoclinic bifurcation of orbit flip with resonant eigenvalues corresponding to the tangent directions. Int J Bifurcat Chaos 2004;14:4161–75. Zhang WP, Zhu D. Codimension 2 bifurcations of double homoclinic loops. Chaos, Solitons & Fractals 2009;39:295–303. Zhu D. Problems in homoclinic bifurcation with higher dimensions. Acta Math Sin Engl Ser 1998;14:341–52.
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Non-resonance 3D homoclinic bifurcation with an inclination flip q Qiuying Lu Department of Mathematics, East China Normal University, Shanghai 200241, China Laboratoire Paul Painlevé, UFR de Mathématiques Université de Lille 1, 59655 Villeneuve d’Ascq, France
a r t i c l e
i n f o
Article history: Accepted 30 March 2009
Communicated by Prof. Ji-Huan He
a b s t r a c t Local active coordinates approach is employed to study the bifurcation of a non-resonance three-dimensional smooth system which has a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues a; b; 1 satisfying a > b > 0. A homoclinic orbit is called an inclination-flip homoclinic orbit if the strong inclination property of the stable manifold is violated. In this paper, we show the existence of 1-homoclinic orbit, 1-periodic orbit, 2n -homoclinic orbit and 2n -periodic orbit in the unfolding of an inclination-flip homoclinic orbit. And we figure out the bifurcation diagram based on the existence region of the corresponding bifurcation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction and hypotheses It is well known that the bifurcation of a degenerate dynamical system on R3 may be very complicated. An earlier example is the case where there exists a homoclinic orbit to a hyperbolic equilibrium point. In [13], Shil’nikov studied the codimension 1 homoclinic bifurcation with two complex conjugated eigenvalues. He pointed out that if the eigenvalues a and b satisfied Re a ¼ Reb < 1, then the dynamical behavior in a small neighborhood of the homoclinic orbit was chaotic. However, in [14], he showed that, under the generic hypothesis, the homoclinic bifurcation was relatively simple. Precisely, the vector field X c had neither homoclinic orbit nor periodic orbit in a small neighborhood of the primary homoclinic orbit for c > 0. While, for c < 0, only one hyperbolic periodic orbit bifurcated from the homoclinic orbit. Accordingly, more degenerate cases should be considered for more complicated dynamics. In [17], Yanagida studied the inclination-flip homoclinic orbit together with two other codimension 2 homoclinic bifurcations, which were the cases of the resonant bifurcation and the orbit-flip bifurcation. Since then, many research works have been devoted to this subject, see [1–6,8–12,18–21]. Among these works, Dumortier et al. [3] gave the persistence condition for inclination-flip homoclinic orbits in terms of Melnikov integrals. Whereas, Kisaka [9] studied the homoclinic doubling bifurcation of an inclination-flip homoclinic orbit under the assumption ku < ks < 2ku . In [2], it was presented that a perturbation of an inclination-flip homoclinic orbit would lead to the occurrence of Smale horseshoes. In [6], the author proved the existence of Smale horseshoe under the condition 2ku < minfks ; kuu g by using the invariant foliation to reduce the study of the return map into the analysis of one-dimensional multi-valued map. In [12], the author showed the existence of a strange attractor in the unfolding of an inclination-flip homoclinic orbit by comparing the Poincaré return map with the Hénon family. In this paper, we consider the 1-homoclinic orbit, 1-periodic orbit, 2n -homoclinic orbit and 2n -periodic orbit bifurcated from an inclination-flip homoclinic orbit. We use the local active coordinates method which was originally established in [21] and then was further developed in [19,20].
q
Work supported by National Natural Science Foundation of China (# 10671069). E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.112
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We consider the following smooth system
z_ ¼ f ðzÞ þ gðz; lÞ;
ð1Þ
and its unperturbed system
z_ ¼ f ðzÞ;
ð2Þ
where z 2 R3 ; l 2 R2 ; 0 6 jlj 1; f ð0Þ ¼ 0; gðz; 0Þ ¼ 0. First of all, we assume that: ðH1 Þ System (2) has a hyperbolic equilibrium O and the relevant linearization matrix Df ð0Þ has simple real eigenvalues: a; b; 1 satisfying a > b > 0. Without loss of generality, we can always assume that the hyperbolic fixed point is the origin, i.e., gð0; lÞ ¼ 0. Since the implicit function theorem tells us that the hyperbolic fixed point persists through out the unfolding. Moreover we assume that the eigenvalues of Df ð0Þ avoid a finite number of resonances so that system (1) is uniformly C 2 linearizable according to [15]. Thereafter, up to a C 2 diffeomorphism, there exists U, a small neighborhood of the origin in R3 and V, a neighborhood of the origin in R2 , so that for all z 2 U and all l 2 V, system (1) has the following C 2 normal form:
x_ ¼ x;
y_ ¼ bðlÞy;
v_ ¼ aðlÞv :
ð3Þ
Besides, we make the following assumptions: _ . Then eþ 2 T 0 W u ; e 2 T 0 W s are unit eigenðH2 Þ System (2) has a homoclinic loop C ¼ fz ¼ rðtÞ; t 2 Rg. Let e ¼ limt!1 jrr_ ðtÞ ðtÞj vectors corresponding to 1 and b. ðH3 Þ Denote by e s the unit eigenvector corresponding to a, then
SpanðT rðtÞ W u ; T rðtÞ W s ; es Þ ¼ R3 ;
for t 1
With the above assumptions, the homoclinic orbit C is of codimension 2. Remark 1.1 (a) ðH3 Þ is equivalent to T rðtÞ W s ! eþ e ; when t ! 1. (b) For the existing loop C, ðH2 Þ is generic, which guarantees that C has no orbit flip. While ðH3 Þ is not generic, which indicates that W s occurs an inclination flip when t ! 1 (see Fig. 1).
2. Preliminaries and bifurcation equation Now we consider the linear variational system of (2) and its adjoint system
z_ ¼ Df ðrðtÞÞz; z_ ¼ ðDf ðrðtÞÞÞ z:
ð4Þ ð5Þ
We denote rðtÞ ¼ ðrx ðtÞ; ry ðtÞ; r v ðtÞÞ and take T > 0 large enough so that rðTÞ ¼ ðd; 0; 0Þ; rðTÞ ¼ ð0; d; dv Þ, where jdv j ¼ Oðd2 Þ and d is small enough so that fðx; y; v Þ : jxj; jyj; jv j < 2dg U.
Fig. 1. Inclination flip.
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Lemma 2.1 (cf. [7,16]). There exists a fundamental solution matrix ZðtÞ ¼ ðz1 ðtÞ; z2 ðtÞ; z3 ðtÞÞ for system (4) with
c c z1 ðtÞ 2 T ri ðtÞ W u \ T ri ðtÞ W s ; z2 ðtÞ ¼ r_ ðtÞ=jr_ y ðTÞj 2 T rðtÞ W u \ T rðtÞ W s ; z3 ðtÞ 2 T rðtÞ W ss ; satisfying
0
0 x21
B zðTÞ ¼ @ 0
0
1
0
1
0
1
x31 x11 0 0 B C x32 C 1 0 A; A; zðTÞ ¼ @ x12 x33 x13 x23 1
where jx23 j 1; x21 < 0; x11 – 0; x32 – 0. From the matrix theory, system (5) has a fundamental matrix solution UðtÞ ¼ ðZ 1 ðtÞÞ . We denote UðtÞ ¼ ð/1 ; /2 ; /3 Þ. We introduce the local active coordinates near the orbit C as (z1 ðtÞ; z2 ðtÞ; z3 ðtÞ) with the components N ¼ ðn1 ; 0; n3 Þ, and set
z ¼ SðtÞ ¼ rðtÞ þ zðtÞN ¼ rðtÞ þ z1 ðtÞn1 þ z3 ðtÞn3 :
ð6Þ
With this notation, we can choose the cross sections
S0 ¼ fz ¼ SðTÞ : jxj; jyj; jv j < 2dg U; S1 ¼ fz ¼ SðTÞ : jxj; jyj; jv j < 2dg U: Under the transformation (6), system (1) has the following form
n_ j ¼ ð/j ðtÞÞ g l ðrðtÞ; 0Þl þ h:o:t:;
j ¼ 1; 3;
ð7Þ
2
which is C and produces the map P 1 : S1 ! S0 . Integrating both sides from T to T, we have
nj ðTÞ ¼ nj ðTÞ þ M j l þ h:o:t:;
j ¼ 1; 3;
ð8Þ RT
where NðTÞ ¼ ðn1 ðTÞ; 0; n3 ðTÞÞ; NðTÞ ¼ ðn1 ðTÞ; 0; n3 ðTÞÞ, and Mj ¼ T ð/j ðtÞÞ g l ðrðtÞ; 0Þ dt; j ¼ 1; 3 are Melnikov vectors. RT R þ1 Lemma 2.2 (cf. [7,16]). Let notations be as above. Then M 1 ¼ T ð/1 ðtÞÞ g l ðrðtÞ; 0Þ dt ¼ 1 ð/1 ðtÞÞ g l ðrðtÞ; 0Þ dt. We define P 0 : S0 ! S1 ; q0 ! q1 induced by the flow of (3) in the neighborhood U of z ¼ 0. And we set the flying time from q0 to q1 as s and the Shil’nikov time s ¼ es (see Fig. 2). Then we have
P0 : q0 ðx0 ; y0 ; v 0 Þ ! q1 ðx1 ; y1 ; v 1 Þ; x0 ¼ sx1 ;
y1 ¼ sb y0 ;
v 1 ¼ sa v 0 ;
and x1 d; y0 d;
n01 ¼ ðx11 Þ1 x0 ;
n03 ¼ v 0 dv ðx11 Þ1 x13 x0 ; 1
n11 ¼ v 1 ðx32 Þ x33 y1 ; n13 ¼ ðx32 Þ1 y1 : From the above, we give the following Poincaré maps:
F 1 ¼ P1 P 0 : S0 ! S0 ; 01 ¼ v 0 sa ðx32 Þ1 x33 dsb þ M1 l þ h:o:t:; n 03 ¼ ðx32 Þ1 dsb þ M 3 l þ h:o:t: n
Fig. 2. Poincaré return map.
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Now, the successor function is given by Gðs; v 0 Þ ¼ ðG1 ; G3 Þ ¼ ðF 1 ðq0 Þ q0 Þ as follows:
G1 ¼ ðx11 Þ1 ds þ v 0 sa ðx32 Þ1 x33 dsb þ M1 l þ h:o:t:; G3 ¼ v 0 þ dv þ ðx11 Þ1 x13 ds þ ðx32 Þ1 dsb þ M 3 l þ h:o:t: By solving
v 0 from G3 ¼ 0 and substituting it into G1 ¼ 0, we obtain the bifurcation equation
ðx11 Þ1 ds þ dv sa þ ðx32 Þ1 dsaþb þ M 3 lsa ðx32 Þ1 x33 dsb þ M 1 l þ h:o:t: ¼ 0:
ð9Þ
3. Bifurcation analysis and bifurcation diagram Definition 1. We shall say that the strong foliation W ss rðtÞ occurs a strong inclination flip as t ! 1, if the stable manifold W srðtÞ turns to be an inclination flip and the strong stable component x33 ¼ 0 as t ¼ T; T 1. We will distinguish the following cases: Case (1): 1 > a > b > 0. The bifurcation Eq. (9) is reduced to the following
if x33 ¼ 0;
for dv –0; for dv ¼ 0;
if x33 – 0;
dv sa þ M 1 l þ h:o:t: ¼ 0; ðx11 Þ1 ds þ ðx32 Þ1 dsaþb þ M 3 lsa þ M 1 l þ h:o:t: ¼ 0;
ðx32 Þ1 x33 dsb þ M1 l þ h:o:t: ¼ 0:
ð10Þ ð11Þ ð12Þ
Case (2): a > 1 > b > 0. We obtain the following bifurcation equations
if x33 ¼ 0; if x33 – 0;
ðx11 Þ1 ds þ M 1 l þ h:o:t: ¼ 0; 1
ðx32 Þ x33 ds þ M1 l þ h:o:t: ¼ 0: b
ð13Þ ð14Þ
Case (3): a > b > 1. We have the following bifurcation equation
ðx11 Þ1 ds þ M 1 l þ h:o:t: ¼ 0:
ð15Þ
Remark 3.1. Notice that, the local weak stable manifold is not unique. In fact, one can fill up a wedge area in W s \ U with these manifolds (curves). Obviously, dv ¼ 0 means that in the coordinate system corresponding to the normal form (3), the local weak stable manifold C \ U is exactly a segment of the y-axis. And we name this kind of homoclinic orbit a homoclinic orbit of ‘weak’ type. First, we assume x33 –0, that is to say, W ss rðtÞ does not occur a strong inclination flip. According to (12), (14) and (15), we have the following results. Theorem 3.1. If M 1 – 0; x33 – 0, then system (1) has at most one periodic orbit in the small neighborhood of C. And it does exist if and only if l 2 fx11 M 1 l > 0g; 0 < jlj 1, for a > b > 1; l 2 fx32 x33 M 1 l > 0g; 0 < jlj 1, for 1 > a > b > 0 and a > 1 > b > 0. Theorem 3.2. If M1 – 0; x33 – 0, then there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists as l 2 H1 . Second, we consider the case of strong inclination flip, that is, x33 ¼ 0. Owing to (10), (13) and (15), we can state the following result. Theorem 3.3. Assume x33 ¼ 0, then (1) if 1 > a > b > 0 and dv –0, system (1) has a unique 1-periodic orbit if and only if l 2 fdv M 1 l < 0g; 0 < jlj 1, and there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists for l 2 H1 . (2) if a > 1 > b > 0 or a > b > 1, system (1) has a unique 1-periodic orbit if and only if l 2 fx11 M 1 l > 0g; 0 < jlj 1, and there exists a codimension 1 bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with normal vector M 1 at l ¼ 0 so that C persists for l 2 H1 . Then, it remains the case concerning the bifurcation equation (11) which we deal in the sequel. We state the results for this case in the following. Theorem 3.4. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then there exists a 1-homoclinic bifurcation n surface H1 , a twofold periodic orbit bifurcation surface SN 1 , a period-doubling bifurcation surface P2 of 2n1 periodic orbit and a n 2n 2 -homoclinic bifurcation surface H for 8 n 2 N, which share the same normal vector M 1 at l ¼ 0, so that system (1) has
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a a a a
1-homoclinic orbit if and only if l 2 H1 and jlj 1; twofold periodic orbit if and only if l 2 SN 1 ; 2n1 -periodic orbit changing its stability and a 2n -periodic orbit arising at the same time if and only if n 2n -homoclinic orbit if and only if l 2 H2 .
n
l 2 P2 ;
Furthermore there exists a bifurcation surface D1 (which is a branch of H1 ) with codimension 1 and normal vector M 1 so that system (1) has a 1-homoclinic orbit as well as a 1-periodic orbit for l 2 D1 and jlj 1. The rest of the paper is devoted to the proof of the above theorem which follows from several propositions. We denote the left side of (11) by Fðs; lÞ ¼ Lðs; lÞ Nðs; lÞ, where
Lðs; lÞ ¼ M 3 lsa þ M1 l þ h:o:t:;
Lð0; lÞ ¼ Fð0; lÞ;
Nðs; lÞ ¼ ðx11 Þ1 ds ðx32 Þ1 dsaþb þ h:o:t:; Nð0; lÞ ¼ 0; and
Dþþ ¼ fl : M 1 l > 0; M 3 l > 0g; Dþ ¼ fl : M1 l > 0; M3 l < 0g; Dþ ¼ fl : M 1 l < 0; M 3 l > 0g; D ¼ fl : M1 l < 0; M3 l < 0g: It is evident that the four areas are not empty when rank ðM 1 ; M 3 Þ ¼ 2. Proposition 3.1. Suppose 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then Fðs; lÞ has a unique sufficiently small positive zero point s so that system (1) has a unique periodic orbit. More precisely, þ (1) when a þ b > 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if x11 > 0, and l 2 D [ Dþ if x11 < 0. þ (2) when 0 < a þ b < 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if x32 < 0, and l 2 D [ Dþ if x32 > 0. 1 þ > ðx11 Þ1 , and (3) when a þ b ¼ 1; Fðs; lÞ has a unique sufficiently small positive zero point s for l 2 Dþ [ Dþ if ðx32 Þ l 2 D [ Dþ if ðx32 Þ1 < ðx11 Þ1 .
Proof. When a þ b > 1,
Fðs; lÞ ¼ ðx11 Þ1 ds þ M 3 lsa þ M1 l þ h:o:t: We set sa ¼ t. Then in case x11 > 0, we have the following possibilities: If l 2 Dþ , then
L0 ðt; lÞ ¼ M 3 l þ h:o:t: < 0;
Lð0; lÞ ¼ M1 l þ h:o:t: > 0; 0
1
N ðt; lÞ ¼ ðax11 Þ dt
1a
a
þ h:o:t: > 0:
So the line W ¼ Lðt; lÞ and the curve W ¼ Nðt; lÞ intersect at a unique sufficiently small positive point t < ðd1 x11 M 1 lÞa . Hence F has a unique sufficiently small positive zero s ¼ ðtÞ1=a . If l 2 Dþ þ , then
Lð0; lÞ ¼ M 1 l þ h:o:t: > 0; N0 ðt; lÞ ¼ ðax11 Þ1 dt
1a
a
L0 ðt; lÞ ¼ M 3 l þ h:o:t: > 0;
þ h:o:t: > 0;
N ðt; lÞ ¼ ð1 aÞða x11 Þ1 dt 00
2
12a
a
þ h:o:t: > 0:
We take t ¼ ½d1 x11 ð2M3 l þ M 1 lÞ a . Then
Nðt; lÞ Lðt; lÞ ¼ 2M 3 l þ M 1 l M 3 lt M 1 l > M 3 l > 0: Therefore, based on the fact that Nð ; lÞ is a monotone increasing convex function, we see that the line W ¼ Lðt; lÞ and the curve W ¼ Nðt; lÞ intersect uniquely at t 2 ð0; tÞ, that is to say, F has a unique sufficiently small positive zero point s 2 ð0; d1 x11 ð2M3 l þ M 1 lÞÞ. The proof for the rest cases can be given similarly. h Proposition 3.2. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then (1) for a þ b > 1, there exists a bifurcation surface D1 with codimension 1 and normal vector M 1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit for l 2 D1 and jlj 1; (2) for 0 < a þ b < 1, there exists a bifurcation surface D2 with codimension 1 and normal vector M 1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit as l 2 D2 and jlj 1;
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(3) for a þ b ¼ 1, there exists a bifurcation surface D3 with codimension 1 and normal vector M1 at l ¼ 0 so that system (1) has an 1-homoclinic orbit as well as an 1-periodic orbit as l 2 D3 and jlj 1. Proof. When a þ b > 1; l 2 D1 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x11 M 3 l > 0g, we have
h i Fðs; lÞ ¼ sa ðx11 Þ1 ds1a þ M 3 l þ h:o:t: : 1
Consequently, there are two zero points s1 ¼ 0; s2 ¼ ðx11 d1 M 3 lÞ1a þ h:o:t. When 0 < a þ b < 1; l 2 D2 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x32 M 3 l < 0g, one has
h i Fðs; lÞ ¼ sa ðx32 Þ1 dsb þ M 3 l þ h:o:t: ¼ 0 1
which admits s1 ¼ 0; s2 ¼ ðx32 d1 M 3 lÞb þ h:o:t. as its solutions. When a þ b ¼ 1; l 2 D3 :¼ fl : Fð0; lÞ ¼ M1 l þ h:o:t: ¼ 0; x11 x32 ðx32 x11 ÞM3 l > 0g, we obtain
h i Fðs; lÞ ¼ sa ððx32 Þ1 ðx11 Þ1 Þds1a þ M3 l þ h:o:t: :
1
Thereafter, it has two zero points s1 ¼ 0; s2 ¼ ðx11 x32 ðx32 x11 Þ1 d1 M 3 lÞ1a þ h:o:t. h Proposition 3.3. Suppose that 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM 1 ; M 3 Þ ¼ 2, then Fðs; lÞ has a unique twofold positive zero point s so that system (1) has a unique twofold periodic orbit. Precisely speaking, 1
(1) when a þ b > 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ðd1 ax11 M 3 lÞ1a þ h:o:t. for x11 M 3 l > 0 and the twofold periodic bifurcation surface is SN1 :
1 a ðx11 Þ1 d d1 ax11 M 3 l 1a ¼ M 3 l d1 ax11 M3 l 1a þ M 1 l þ h:o:t: with normal vector M 1 at l ¼ 0; 1 (2) when 0 < a þ b < 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ½ðdða þ bÞÞ1 ax32 M 3 l b þ h:o:t. for x32 M 3 l < 0 1 and the twofold periodic orbit bifurcation surface SN :
h i1þab h iab ðx32 Þ1 d ðdða þ bÞÞ1 ax32 M 3 l ¼ M 3 l ðdða þ bÞÞ1 ax32 M3 l þ M 1 l þ h:o:t: has normal vector M 1 at l ¼ 0; 1 (3) when a þ b ¼ 1, Fðs; lÞ has a unique twofold positive zero point s ¼ ½ðdðx32 x11 ÞÞ1 ax11 x32 M 3 l 1a for 1 1 1 ððx11 Þ ðx32 Þ ÞM3 l > 0, and the corresponding twofold periodic orbit bifurcation surface SN :
h i11 a ðx11 Þ1 ðx32 Þ1 d ðdðx32 x11 ÞÞ1 ax11 x32 M3 l h i1a a ¼ M 3 l ðdðx32 x11 ÞÞ1 ax11 x32 M 3 l þ M1 l þ h:o:t: has normal vector M 1 at
l ¼ 0.
Proof. The twofold zero point t should satisfy
Lðt; lÞ ¼ Nðt; lÞ;
L0 ðt; lÞ ¼ N0 ðt; lÞ:
ð16Þ
The second equation turns out to be
ðax11 Þ1 dt
1a
a
b
ða þ bÞðax32 Þ1 dta þ h:o:t: ¼ M3 l: 1
ð17Þ
a
When a þ b > 1, we have t ¼ ðd ax11 M 3 lÞ1a þ h:o:t. for x11 M 3 l > 0 due to (17). Then from the first equation of (16), we get the corresponding twofold periodic orbit bifurcation surface SN 1 :
1 a ðx11 Þ1 d d1 ax11 M 3 l 1a ¼ M 3 l d1 ax11 M3 l 1a þ M 1 l þ h:o:t: with normal vector M1 at l ¼ 0. The other two cases can be proved similarly.
h
Now we study the bifurcation of 2-homoclinic orbit and the period-doubling bifurcation for the case of 1 > a > b > 0; x33 ¼ 0; dv ¼ 0. As above, we set t1 and t 2 as the flying time from q0 to q1 and from q2 to q3 , respectively, si ¼ eti ; i ¼ 1; 2. Then the second successor function can be expressed by
G2 ðs1 ; s2 ; v 0 ; v 2 Þ ¼ G11 ; G13 ; G21 ; G23 ¼ ðF 1 ðq0 Þ q2 ; F 1 ðq2 Þ q0 Þ
Q. Lu / Chaos, Solitons and Fractals 42 (2009) 2597–2605
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with:
G11 ¼ ðx11 Þ1 ds2 þ v 0 sa1 ðx32 Þ1 x33 dsb1 þ M 1 l þ h:o:t:; G13 ¼ v 2 þ dv þ ðx11 Þ1 x13 ds2 þ ðx32 Þ1 dsb1 þ M 3 l þ h:o:t: G21 ¼ ðx11 Þ1 ds1 þ v 2 sa2 ðx32 Þ1 x33 dsb2 þ M 1 l þ h:o:t:; G23 ¼ v 0 þ dv þ ðx11 Þ1 x13 ds1 þ ðx32 Þ1 dsb2 þ M 3 l þ h:o:t: Solving ðv 0 ; v 2 Þ from ðG13 ; G23 Þ ¼ 0 and substituting it into G11 ¼ 0 and G21 ¼ 0, we obtain the bifurcation equations a ðx11 Þ1 ds2 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsa1 sb2 þ M3 lsa1 þ M 1 l þ h:o:t: ¼ 0; 1
ð18Þ
a ðx11 Þ1 ds1 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsb1 sa2 þ M3 lsa2 þ M 1 l þ h:o:t: ¼ 0: 2
ð19Þ
It is easy to see that system (1) has a 2-homoclinic orbit near C if and only if the above equation has s1 ¼ 0; s2 > 0 as its solution by the symmetry of G2 . If s1 ¼ 0; s2 > 0 is the solution of the bifurcation equation, then s2 ¼ d1 x11 M 1 l þ h:o:t. for x11 M 1 l > 0, and the 2-homoclinic bifurcation surface H2 : x11 M 3 l ¼ dðd1 x11 M 1 lÞ1a þ h:o:t. has codimension 1 with normal vector M 1 at l ¼ 0. Thus we have: Proposition 3.4. Suppose 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, then there exists a unique 1-homoclinic bifurcation surface H1 : M 1 l þ h:o:t: ¼ 0 with codimension 1 and normal vector M 1 at l ¼ 0, which coincides with D1 in the region defined by fl : x11 M 3 l > 0g. For l 2 H1 and jlj 1, system (1) has a unique 1-homoclinic orbit. There exists a unique bifurcation surface H2 : x11 M 3 l ¼ dðd1 x11 M 1 lÞ1a þ h:o:t. which is well defined in the region fl : x11 M 1 l > 0; x11 M 3 l < 0g, so that system (1) has a unique 2-homoclinic orbit for l 2 H2 . From Proposition 3.4, we know that H1 and H2 have the same normal vector M1 at l ¼ 0, and M3 l ¼ OðjM1 lj1a Þ for l 2 H2 . So, there is a tongue area bounded by H1 and H2 . And in the tongue area, there must be another bifurcation surface P 2 where a period-doubling bifurcation arises. Similarly as in Section 2, we define
Pj0 : q2j2 ðx2j2 ; y2j2 ; v 2j2 Þ ! q2j1 ðx2j1 ; y2j1 ; v 2j1 Þ; x2j2 ¼ sj x2j1 ;
v 2j1 ¼ saj v 2j2 ;
y2j1 ¼ sbj y2j2 ;
and x2j1 d; y2j2 d; j ¼ 1; 2; . . ..
n2j2 ¼ ðx11 Þ1 x2j2 ; 1 n2j1 1
n2j2 ¼ v 2j2 dv ðx11 Þ1 x13 x2j2 ; 3
¼ v 2j1 ðx32 Þ x33 y2j1 ; n2j1 ¼ ðx32 Þ1 y2j1 : 3 1
From the above, we can give the nth Poincaré return maps:
F j1 ¼ P1 P j0 : S0 ! S0 ;
2j2 ; q2j2 #q
2j2 n 1
¼ v 2j1 ðx32 Þ x33 y2j1 þ M1 l þ h:o:t:;
2j2 n 3
¼ ðx32 Þ1 y2j1 þ M3 l þ h:o:t:
1
Consequently, the associated n-th successor function is given by
Gn ðs1 ; . . . ; sn ; v 0 ; . . . ; v 2n2 Þ ¼ G11 ; G13 ; G21 ; G23 ; G31 ; G33 ; G41 ; G43 ¼ F 11 ðq0 Þ q2 ; F 21 ðq2 Þ q4 ; . . . ; F n1 ðq2n2 Þ q0 : Now, we study the 4-homoclinic bifurcation surface H4 with the condition 1 > a > b > 0; x33 ¼ 0; dv ¼ 0. By solving ðv 0 ; v 2 ; v 4 ; v 6 Þ from ðG13 ; G23 ; G33 ; G43 Þ ¼ 0 and substituting it into ðG11 ; G21 ; G31 ; G41 Þ, we obtain the bifurcation equations: a ðx11 Þ1 ds2 þ ðx11 Þ1 x13 ds1þ þ ðx32 Þ1 dsa1 sb4 þ M3 lsa1 þ M 1 l þ h:o:t: ¼ 0; 1 1
1
1þa 13 ds2
ðx11 Þ ds3 þ ðx11 Þ x
b a 32 Þ ds1 s2 1 b a 32 Þ ds2 s3 1 b a 32 Þ ds3 s4
þ ðx
a þ ðx ðx11 Þ1 ds4 þ ðx11 Þ1 x13 ds1þ 3 1
1
1þa 13 ds4
ðx11 Þ ds1 þ ðx11 Þ x
1
þ ðx
a
ð20Þ
þ M3 ls2 þ M 1 l þ h:o:t: ¼ 0;
ð21Þ
þ M3 lsa3 þ M 1 l þ h:o:t: ¼ 0;
ð22Þ
a
þ M3 ls4 þ M 1 l þ h:o:t: ¼ 0:
ð23Þ
So, we just need to consider the above equations that admit s1 ¼ 0; s2 > 0; s3 > 0; s4 > 0 as their solutions. Correspondingly, from (20), we have s2 ¼ x11 d1 M 1 l þ h:o:t. for x11 M 1 l > 0. Then (21) yields
h i a s3 ¼ x11 d1 M3 l x11 d1 M1 l þ M 1 l þ h:o:t: ¼ x11 d1 M 1 l þ h:o:t:
for
l 2 fl : x11 d1 M1 l > 0g \ fl : M3 l ¼ oðjM1 lj1a Þg.
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Fig. 3. Bifurcation diagram in the case of 1 > a > b > 0; a þ b > 1; dv ¼ 0; x11 > 0; x33 ¼ 0.
If a þ b > 1, we obtain s4 ¼ x11 d1 M 1 l þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Then, (23) gives the 4-homoclinic bifurcation surface H4 :
a M3 l x11 d1 M1 l þ M 1 l þ h:o:t: ¼ 0:
ð24Þ
Here, H4 is defined on fl : M 3 l ¼ OðjM1 lj1a Þg. If 0 < a þ b < 1, we get s4 ¼ x11 ðx32 Þ1 ðx11 d1 M 1 lÞaþb þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Consequently, from (23), the 4-homoclinic bifurcation surface H4 should be:
b h aþb ia ðx32 Þ1 d x11 d1 M 1 l x11 ðx32 Þ1 x11 d1 M1 l h aþb ia þ M 3 l x11 ðx32 Þ1 x11 d1 M 1 l þ h:o:t: ¼ 0:
ð25Þ
In this case, H4 is defined on fl : M 3 l ¼ OðjM1 ljb Þg. If a þ b ¼ 1, then s4 ¼ x11 d1 ½1 þ ðx32 Þ1 M 1 l þ h:o:t. for l 2 fl : x11 M 1 l > 0g \ fl : M 3 l ¼ oðjM 1 lj1a Þg. Then, we get the 4-homoclinic bifurcation surface H4 :
n h i oa h ia M3 l x11 d1 1 þ ðx32 Þ1 M 1 l þ M 1 l þ ðx32 Þ1 x11 1 þ ðx32 Þ1 M1 l þ h:o:t: ¼ 0:
ð26Þ
Owing to the bifurcation surface equation, H4 is defined on fl : M 3 l ¼ OðjM1 lj1a Þg as well. Summing up, we get the 4-homoclinic bifurcation surface H4 : (24)–(26) on the parameter surface fl : M 3 l ¼ oðjM 1 lj1a Þg. n Repeating the above procedure, we can also get the 2n -homoclinic bifurcation surface H2 and the period-doubling bifurcation 2n surface P for arbitrary n 2 N. To well illustrate our main theorem, we give the bifurcation diagrams under the assumptions 1 > a > b > 0; x33 ¼ 0; dv ¼ 0 and rankðM1 ; M 3 Þ ¼ 2, where O represents that there is no periodic orbits, while P (respectively, Pk ) represents that there exists a 1-periodic (respectively, k-periodic) orbit in the corresponding region (see Fig. 3).
4. Conclusion By employing the local active coordinates approach, we have established the complete bifurcation results for non-resonance 3D homoclinic orbit with an inclination flip. More precisely, when the homoclinic orbit takes place an inclination flip but not a strong inclination flip or it occurs a strong inclination flip but not of the ‘weak’ type, then the bifurcation result is unique, i.e. either a homoclinic orbit or a periodic orbit persists. Otherwise, the uniqueness of solution cannot be guaranteed, which is quite challenging for the bifurcation discussion. However, with the help of induction, we manage to establish the 1n homoclinic bifurcation surface H1 , twofold periodic orbit bifurcation surface SN 1 , a period-doubling bifurcation surface P2 of n1 n 2n periodic orbit and a 2 -homoclinic bifurcation surface H for 8 n 2 N. Thus, complete bifurcation results are obtained 2 and complete bifurcation diagram is given. Acknowledgements The author is grateful to Professor Deming Zhu for having introduced to her the subject and for his many helpful suggestions. She also wishes to thank Dr. Guoting Chen for a careful reading of the article and the anonymous referees for their comments and suggestions which improved the results.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Chow SN, Deng B, Fiedler B. Homoclinic bifurcation at resonant eigenvalues. J Dyn Syst Diff Eqs 1990;2:177–244. Deng B. Homoclinic twisting bifurcation and cusp horseshoe maps. J Dyn Diff Eqs 1993;5:417–67. Dumortier F, Kokubu H, Oka H. A degenerate singularity generating geometric Lorenz attractors. Ergod Theory Dyn Syst. 1995;15:833–56. Geng FJ, Zhu D, Xu Y. Bifurcations of heterodimensional cycles with two saddle points. Chaos, Solitons & Fractals 2009;39:2063–75. He JH. Limit cycle and bifurcation of nonlinear problems. Chaos, Solitons & Fractals 2005;26:827–33. Homburg AJ, Kokubu H, Krupa M. The cusp horse-shoe and its bifurcation in the unfolding of an inclination-flip homoclinic orbit. Ergod Theory Dyn Syst 1994;14:667–93. Jin Y, Zhu D. Bifurcations of rough heteroclinic loops with three saddle points. Acta Math Sin Eng Ser 2002;18:199–208. Kisaka M, Kokubu H, Oka H. Bifurcations to N-homoclinic orbit and N-periodic orbits in vector fields. J Dyn Diff Eqs 1993;5:305–57. Kisaka M, Kokubu H, Oka H. Supplement to homoclinic doubling bifurcation in vector fields, Dynamical systems (Santiago, 1990), Pitman Res. Notes Math. Ser., 285, Longman Sci. Tech., Harlow; 1993. p. 92–116. Lau YT. Global aspects of homoclinic bifurcations of three-dimensional saddles. Chaos, Solitons & Fractals 1993;3:369–82. Lu Q. Codimension 2 bifurcation of twisted double homoclinic loops. Comput Math Appl 2009;57:1127–41. Naudot V. Strange attractor in the unfolding of an inclination-flip homoclinic orbit. Ergod Theory Dyn. Syst 1996;16:1071–86. Shil’nikov LP. A case of the existence of a denumerable set of periodic motions. Sov Math Dockl 1965;6:163–6. Shil’nikov LP. On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium of state of saddle type. Math USSR Sbornik 1968;6:427–37. Sternberg S. On the local homeomorphisms of Euclidean n-space, Am. J. Math, 80 (1958), 623-631 and 81 (1959), 578-604. Tian Q, Zhu D. Bifurcations of nontwisted heteroclinic loops. Sci Chin Ser A 2000;30:193–202. Yanagida E. Branching of double pulse solutions from single pulse solutions in nerve axon equations. J Diff Eqs 1987;66:243–62. Zhang W, Wang FX, Zu JW. Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation. Chaos, Solitons & Fractals 2005;24:977–98. Zhang T, Zhu D. Codimension 3 homoclinic bifurcation of orbit flip with resonant eigenvalues corresponding to the tangent directions. Int J Bifurcat Chaos 2004;14:4161–75. Zhang WP, Zhu D. Codimension 2 bifurcations of double homoclinic loops. Chaos, Solitons & Fractals 2009;39:295–303. Zhu D. Problems in homoclinic bifurcation with higher dimensions. Acta Math Sin Engl Ser 1998;14:341–52.