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Bifurcations of heterodimensional cycles with two saddle points
Chaos, Solitons and Fractals 39 (2009) 2063–2075 www.elsevier.com/locate/chaos
Bifurcations of heterodimensional cycles with two saddle points Fengjie Geng a
a,*
, Deming Zhu b, Yancong Xu
b
School of Information Technology, China University of Geosciences (Beijing), Beijing 100083, China b Department of Mathematics, East China Normal University, Shanghai 200062, China Accepted 21 June 2007
Abstract The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The bifurcations of dynamical system have been studied extensively in the literature. In recent years, there has been much research work concerning the high-codimensional bifurcations of homoclinic and heteroclinic cycles which include the bifurcations of heterodimensional cycles, resonant cycles, orbit flip cycles, inclination flip cycles, etc. The readers are referred to the papers [1–5,9–14, and references cited therein]. Moreover, the bifurcations of heterodimensional cycles have attracted special attention these years (see [1,3,9,11]). Comparatively speaking, to study the bifurcations of heterodimensional cycles is much more difficult and challenging. A heteroclinic loop is said to be a heterodimensional cycle if it is formed by two or more normally hyperbolic invariant manifolds which have different dimensions of the unstable manifolds and two or more heteroclinic orbits connecting them. As we know, the existence of heterodimensional cycles connecting saddle-foci always yields extremely complex dynamics behavior. It is demonstrated in [9] that the reversible vector fields with heterodimensional cycles are dense near Hopf-zero bifurcation. Rademacher [11] investigated the bifurcations of heterodimensional cycles connecting one hyperbolic equilibrium and one hyperbolic periodic orbit. And he derived the conditions for the existence and uniqueness of countably infinite families of curve segments of homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. One may see that bifurcations of heterodimensional cycles play a very important role in the theoretic study of high-codimensional bifurcations of homoclinic and heteroclinic cycles. In this paper, we are concerned with the bifurcations of heterodimensional cycles connecting two hyperbolic saddles. By utilizing the method originally established by [15,16] and then improved in [6–8], we obtain the existence of one *
Corresponding author. E-mail addresses: [email protected] (F. Geng), [email protected] (D. Zhu), [email protected] (Y. Xu).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.077
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Fig. 1. Heterodimensional Cycle.
homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional cycles are revealed. Consider the following Cr system z_ ¼ f ðzÞ þ gðz; lÞ;
ð1:1Þ
and its unperturbed system z_ ¼ f ðzÞ;
ð1:2Þ
where r P 7, z 2 R3 , l 2 Rl , l > 2, 0 < jlj 1, gðz; 0Þ ¼ 0. Assume that f ðpi Þ ¼ 0, gðpi ; lÞ ¼ 0 for i ¼ 1; 2. In addition, throughout the paper, the following conditions hold: ðH 1 Þp1 ; p2 are hyperbolic critical points of (1.2). dimðW sp1 Þ ¼ dimðW up2 Þ ¼ 1, dimðW up1 Þ ¼ dimðW sp2 Þ ¼ 2, where W spi , W upi denote the Cr stable manifold and C r unstable manifold of pi, respectively. Moreover, the linearization Df ðp1 Þ has simple real eigenvalues k11 ; k21 , q11 satisfying q11 < 0 < k11 < k21 ; Df ðp2 Þ has simple real eigenvalues q12 ; q22 , k12 fulfilling q22 < q12 < 0 < k12 . (H2) System (1.2) has a heteroclinic loop C ¼ C1 [ C2 , where Ci ¼ fz ¼ ri ðtÞ : t 2 Rg, ri ðþ1Þ ¼ riþ1 ð1Þ ¼ piþ1 , i ¼ 1; 2, r3 ðtÞ ¼ r1 ðtÞ, p3 ¼ p1 . r_ i ðtÞ u s u u þ þ (H3) Define e i ¼ limt!1 j_ri ðtÞj, i ¼ 1; 2. Then e1 2 T p1 W p1 , e1 2 T p2 W p2 , e2 2 T p2 W p2 , e2 2 T p1 W p1 are the unit eigenvectors corresponding to k11 , q12 , k12 , q11 , respectively. See Fig. 1. ðH 4 ÞW up1 intersects W sp2 transversely. It is easy to see that C is a heterodimensional cycle. 2. Local coordinates and bifurcation equations Now, we shall establish the normal forms of (1.1) in the neighborhood of the saddles p1, p2 in four steps. Step 1. Suppose the neighborhood U i of pi is small enough, by a linear transformation system (1.1) turns out to be the following form in U1 x_ ¼ k11 ðlÞx þ Oð2Þ;
y_ ¼ q11 ðlÞy þ Oð2Þ;
u_ ¼ k21 ðlÞu þ Oð2Þ:
ð2:1Þ
v_ ¼ q22 ðlÞv þ Oð2Þ;
ð2:2Þ
And in U2 system (1.1) takes the form x_ ¼ k12 ðlÞx þ Oð2Þ; kj1 ð0Þ
kj1 ,
qj2 ð0Þ
y_ ¼ q12 ðlÞy þ Oð2Þ; qj2 ,
q11 ð0Þ
q11 ,
k12 ð0Þ
k12 :
where ¼ ¼ j ¼ 1; 2, ¼ ¼ Step 2. We know that system (2.1) and (2.2) have Cr local manifolds W spi ;loc and W upi ;loc , i ¼ 1; 2, where W sp1 ;loc ¼ fz ¼ ðx; y; uÞ jx ¼ xðyÞ; u ¼ uðyÞ; xð0Þ ¼ uð0Þ ¼ 0; oðx;uÞ ¼ 0; ð0; y; 0Þ 2 U s1 g, W up1 ;loc ¼ fz ¼ ðx; y; uÞ jy ¼ yðx; uÞ; oy oy u s ox ¼ 0; ð0; y; vÞ 2 U s2 g, W up2 ;loc ¼ yð0; 0Þ ¼ 0; oðx;uÞ ¼ 0; ðx; 0; uÞ 2 U 1 g, W p2 ;loc ¼ fz ¼ ðx; y; vÞ jx ¼ xðy; vÞ; xð0; 0Þ ¼ 0; oðy;vÞ oðy;vÞ u s fz ¼ ðx; y; vÞ jy ¼ yðxÞ; v ¼ vðxÞ; yð0Þ ¼ vð0Þ ¼ 0; ox ¼ 0; ðx; 0; 0Þ 2 U 2 g, where U 1 fzjx ¼ u ¼ 0g U 1 , U u1 fzjy ¼ 0g U 1 , U s2 fzjx ¼ 0g U 2 , U u2 fzjy ¼ v ¼ 0g U 2 . Now we straighten the local manifold W spi ;loc and W upi ;loc such that W sp1 ;loc ¼ fz 2 U 1 jx ¼ u ¼ 0g, W up1 ;loc ¼ fz 2 U 1 jy ¼ 0g, W sp2 ;loc ¼ fz 2 U 2 jx ¼ 0g, W up2 ;loc ¼ fz 2 U 2 jy ¼ v ¼ 0g. So system (1.1) has the following normal form
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
x_ ¼ k11 ðlÞ þ x þ OðuÞ½OðyÞ þ OðuÞ ; 1 y_ ¼ q1 ðlÞ þ y; u_ ¼ k21 ðlÞ þ u þ OðxÞ½OðxÞ þ OðyÞ
2065
ð2:3Þ
in U1 and x_ ¼ k12 ðlÞ þ x; y_ ¼ q12 ðlÞ þ y þ OðvÞ½OðxÞ þ OðvÞ ; 2 v_ ¼ q2 ðlÞ þ v þ OðyÞ½OðxÞ þ OðyÞ
ð2:4Þ
in U2, where kj1 ð0Þ ¼ kj1 , qj2 ð0Þ ¼ qj2 , j ¼ 1; 2, q11 ð0Þ ¼ q11 , k12 ð0Þ ¼ k12 . System (2.3) and (2.4) is at least C r1 . ss Step 3. Continue to straighten the local strong unstable manifold W uu p1 ;loc of p1 and local strong stable manifold W p2 ;loc uu ss of p2 such that W p1 ;loc ¼ fz 2 U 1 jx ¼ y ¼ 0g, W p2 ;loc ¼ fz 2 U 2 jx ¼ y ¼ 0g. Then by the invariant manifold theorem one can deduce that system (1.1) now has the following normal form x_ ¼ k11 ðlÞ þ x þ OðuÞOðyÞ; 1 ð2:5Þ y_ ¼ q1 ðlÞ þ y; 2 u_ ¼ k1 ðlÞ þ u þ OðxÞ½OðxÞ þ OðyÞ in U1 and x_ ¼ k12 ðlÞ þ x; 1 y_ ¼ q2 ðlÞ þ y þ OðvÞOðxÞ; v_ ¼ q22 ðlÞ þ v þ OðyÞ½OðxÞ þ OðyÞ
ð2:6Þ
in U2. System (2.5) and (2.6) is at least C r2 . Step 4. At last, we straighten the heteroclinic orbit Ci in the small neighborhood Ui such that C1 \ U 1 ¼ fz 2 U 1 jy ¼ u ¼ 0g; System (1.1) then takes the form x_ ¼ k11 ðlÞ þ x þ OðuÞOðyÞ; y_ ¼ q11 ðlÞ þ y; 2 u_ ¼ k1 ðlÞ þ u þ OðxÞOðyÞ
C1 \ U 2 ¼ fz 2 U 2 jx ¼ v ¼ 0g:
ð2:5 Þ
in U1 and
x_ ¼ k12 ðlÞ þ x; 1 y_ ¼ q2 ðlÞ þ y þ OðvÞOðxÞ; v_ ¼ q22 ðlÞ þ v þ OðyÞOðxÞ:
ð2:6 Þ
Denote ri ðtÞ ¼ ðrxi ðtÞ; ryi ðtÞ; rui ðtÞÞ in U1 and ri ðtÞ ¼ ðrxi ðtÞ; ryi ðtÞ; rvi ðtÞÞ in U2 for i ¼ 1; 2. Due to (2.5*) and (2.6*), one may choose Ti and Ti, jT i j 1 such that ri ðT i Þ ¼ ðd; 0; 0Þ , ri ðT i Þ ¼ ð0; d; 0Þ , i ¼ 1; 2, where d > 0 is small enough such that fðx; y; uÞ : jxj; jyj; juj < 2dg U 1 , fðx; y; vÞ : jxj; jyj; jvj < 2dg U 2 . Take into account the linear variational system z_ ¼ Df ðri ðtÞÞz
ð2:7i Þ
and its adjoint system /_ ¼ ðDf ðri ðtÞÞÞ /;
ðz1i ðtÞ; z2i ðtÞ; z3i ðtÞÞ
ð2:8i Þ
be a fundamental solution matrix of **(2.7i), t 2 ½T i ; T i : We need the following i ¼ 1; 2. Let Z i ðtÞ ¼ additional generic hypothesis: uu þ ðH 5 ÞZ 2 ðT 2 ÞZ 1 2 ðT 2 ÞT p1 W p1 R spanfe2 ; e1 g. 1 ðtÞ Based on (H4), one can select z11 ðtÞ ¼ j_r1r_ðT 2 T r1 ðtÞ W up1 \ T r1 ðtÞ W sp2 , z21 ðtÞ 2 T r1 ðtÞ W up1 \ ðT r1 ðtÞ W sp2 Þc , z31 ðtÞ 2 1 Þj u c s ðT r1 ðtÞ W p1 Þ \ T r1 ðtÞ W p2 such that
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0
1 0 B Z 1 ðT 1 Þ ¼ @ 0 0 0 1
1 w31 1 C w32 1 A;
0
0 B 12 Z 1 ðT 1 Þ ¼ @ w1 0
w33 1
w21 1 w22 1
1 0 C 0 A;
w23 1
1
3j 2j 32 21 32 1 21 1 where w12 1 < 0, w1 –0; w1 –0, jw1 ðw1 Þ j 1, for j ¼ 1; 3, jw1 ðw1 Þ j 1, for j ¼ 2; 3. 1 2 2 Due to (H5) and Liouville formula, one can select z2 ðtÞ; z2 ðtÞ 2 ðT r2 ðtÞ C2 Þc , z12 ðT 2 Þ 2 T p1 W uu p1 , z2 ðT 2 Þ ¼ e1 , u s 2 ðtÞ 2 T W \ T W such that z32 ðtÞ ¼ j_r2r_ðT r2 ðtÞ r2 ðtÞ p2 p1 2 Þj 1 1 0 11 0 w2 0 1 0 w21 0 2 C C B B Z 2 ðT 2 Þ ¼ @ w12 1 0 A; Z 2 ðT 2 Þ ¼ @ 0 w22 w32 2 2 2 A; w13 0 0 1 w23 0 2 2 1j 21 32 13 1 22 21 1 where w13 2 –0, w2 –0, w2 < 0, jw2 ðw2 Þ j 1, for j ¼ 1; 2, jw2 ðw2 Þ j 1. as a local coordinate system along Ci . Let Now, we may choose ðz1i ðtÞ; z2i ðtÞ; z3i ðtÞÞ Ui ðtÞ ¼ ð/1i ðtÞ; /2i ðtÞ; /3i ðtÞÞ ¼ ðZ 1 ðtÞÞ , then U ðtÞ is the fundamental solution matrix of (2.8 ), i ¼ 1; 2. i i i Take a coordinate change as z ¼ ri ðtÞ þ Z i ðtÞN i ðtÞ , hi ðtÞ, where N 1 ðtÞ ¼ ð0; n21 ; n31 Þ , N 2 ðtÞ ¼ ðn12 ; n22 ; 0Þ , i ¼ 1; 2. Define the sections S 01 ¼ fz ¼ h1 ðT 1 Þ : jxj; jyj; juj < 2dg, S 02 ¼ fz ¼ h2 ðT 2 Þ : jxj; jyj; jvj < 2dg, S 11 ¼ fz ¼ h1 ðT 1 Þ : jxj; jyj; jvj < 2dg, S 12 ¼ fz ¼ h2 ðT 2 Þ : jxj; jyj; juj < 2dg, it is clear that S 0i and S 1i cross Ci at t = Ti and t = Ti, respectively, for i ¼ 1; 2. Consider the maps F 0i : q1i1 2 S 1i1 #q0i 2 S 0i and F 1i : q0i 2 S 0i #q1i 2 S 1i induced by the flow of (1.1), where S 10 ¼ S 12 , 1 q0 ¼ q12 . First, we may find the relationship between q01 ðx01 ; y 01 ; u01 Þ , q11 ðx11 ; y 11 ; v11 Þ and their new coordinates 0;3 1;2 1;3 1 q01 ð0; n0;2 1 ; n1 Þ , q1 ð0; n1 ; n1 Þ . Notice that
q01 ¼ ðx01 ; y 01 ; u01 Þ ¼ r1 ðT 1 Þ þ Z 1 ðT 1 ÞN 1 ðT 1 Þ; q11 ¼
ðx11 ; y 11 ; v11 Þ
¼ r1 ðT 1 Þ þ Z 1 ðT 1 ÞN 1 ðT 1 Þ;
0;3 N 1 ðT 1 Þ ¼ ð0; n0;2 1 ; n1 Þ ; 1;3 N 1 ðT 1 Þ ¼ ð0; n1;2 1 ; n1 Þ :
Based on the expressions of Z i ðT i Þ and Z i ðT i Þ, i ¼ 1; 2, one derives that the new coordinate of q01 as ( 0 33 32 1 0 n0;2 1 ¼ u1 w1 ðw1 Þ y 1 ; 32 1 0 n0;3 1 ¼ ðw1 Þ y 1 ; 0;3 1 x01 ¼ d þ w31 1 n1 d, and the new coordinate of q1 as ( 21 1 1 n1;2 1 ¼ ðw1 Þ x1 ;
ð2:9Þ
ð2:10Þ
1 23 21 1 1 n1;3 1 ¼ v1 w1 ðw1 Þ x1 ; 1;2 y 11 ¼ d þ w22 1 n1 d. Similarly, for
q02 ¼ ðx02 ; y 02 ; v02 Þ ¼ r2 ðT 2 Þ þ Z 2 ðT 2 ÞN 2 ðT 2 Þ; q12
¼
ðx12 ; y 12 ; u12 Þ
¼ r2 ðT 2 Þ þ Z 2 ðT 2 ÞN 2 ðT 2 Þ;
0;2 N 2 ðT 2 Þ ¼ ðn0;1 2 ; n2 ; 0Þ ; 1;2 N 2 ðT 2 Þ ¼ ðn1;1 2 ; n2 ; 0Þ ;
one gets the new coordinate of q02 is ( 13 1 0 n0;1 2 ¼ ðw2 Þ v2 ;
ð2:11Þ
0 12 13 1 0 n0;2 2 ¼ y 2 w2 ðw2 Þ v2 ; 0;1 1 x02 ¼ d þ w11 2 n2 d, and the new coordinate of q2 is ( 1 23 21 1 1 1 23 21 1 1 n1;1 2 ¼ u2 w2 ðw2 Þ x2 ¼ u0 w2 ðw2 Þ x0 ; 21 1 1 21 1 1 n1;2 2 ¼ ðw2 Þ x2 ¼ ðw2 Þ x0 :
ð2:12Þ
Putting z ¼ hi ðtÞ into (1.1), then r_ i ðtÞ þ Z_ i ðtÞN i ðtÞ þ Z i ðtÞN_ i ðtÞ ¼ ¼
f ðri ðtÞ þ Z i ðtÞN i ðtÞÞ þ gðri ðtÞ þ Z i ðtÞN i ðtÞ; lÞ f ðri ðtÞÞ þ Df ðri ðtÞÞZ i ðtÞN i ðtÞ þ gðri ðtÞ; 0Þ þ gz ðri ðtÞ; 0ÞZ i ðtÞN i ðtÞ þ gl ðri ðtÞ; 0Þl þ h:o:t::
By way of the fact r_ i ðtÞ ¼ f ðri ðtÞÞ, Z_ i ðtÞ ¼ Df ðri ðtÞÞZ i ðtÞ and gðz; 0Þ ¼ 0, it then follows that N_ i ðtÞ ¼ Z 1 i ðtÞg l ðri ðtÞ; 0Þl þ h:o:t:
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
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Integrating the above equation from Ti to Ti, we arrive at Z Ti N i ðT i Þ ¼ N i ðT i Þ þ Z 1 i ðtÞgl ðri ðtÞ; 0Þl dt þ h:o:t; T i
N i ðT i Þ; N i ðT i Þ are defined as before. It then produces that 0;j j n1;j 1 ¼ n1 þ M 1 l þ h:o:t;
j ¼ 2; 3;
n1;k 2
k ¼ 1; 2;
¼
n0;k 2
þ
M k2 l
þ h:o:t;
ð2:13Þ
where M j1 ¼ M k2
¼
R T1
/j 1 gl ðr1 ðtÞ; 0Þ dt;
j ¼ 2; 3;
/k g ðr ðtÞ; 0Þ dt; T 2 2 l 2
k ¼ 1; 2:
T 1
R T2
Thus the maps 0;3 1;2 1;3 ð0; n0;2 1 ; n1 Þ ! ð0; n1 ; n1 Þ ; 0;1 0;2 1;1 1;2 ðn2 ; n2 ; 0Þ ! ðn2 ; n2 ; 0Þ ;
F 11 : S 01 ! S 11 ; F 12 : S 02 ! S 12 ; are defined by
0;j j n1;j 1 ¼ n1 þ M 1 l þ h:o:t;
j ¼ 2; 3;
0;k k n1;k 2 ¼ n2 þ M 2 l þ h:o:t;
k ¼ 1; 2
ð2:14Þ
respectively. q1 ðlÞ q1 ðlÞ Denoted by b1 ðlÞ ¼ k11 ðlÞ, b2 ðlÞ ¼ k12 ðlÞ. For notational convenience, denote bi ¼ bi ðlÞ, i ¼ 1; 2. Let si , i ¼ 1; 2 be the 1
1
2
1
flying time from q1i1 to q0i , s1 ¼ ek1 ðlÞs1 , s2 ¼ eq2 ðlÞs2 . Using linearization of (1.1) at pi, we can easily obtain the map F 01 : S 10 ! S 01 ;
q10 ðx10 ; y 10 ; u10 Þ ! q01 ðx01 ; y 01 ; u01 Þ
x10
sb1 1 d;
as s1 d;
y 01
u10
k2 ðlÞ 1 k1 ðlÞ 1
s1 u01
ð2:15Þ
and the map F 02 : S 11 ! S 02 ;
q11 ðx11 ; y 11 ; v11 Þ ! q02 ðx02 ; y 02 ; v02 Þ
defined by x11
1 b2
s2 d;
y 02
s2 d;
v02
q2 ðlÞ 2 q1 ðlÞ 2
s2 v11
ð2:16Þ
ðs1 ; u01 Þ; ðs2 ; v11 Þ
if we neglect the higher order terms. are called Shilnikov coordinates. Consequently, combine with Eqs. (2.9), (2.11), (2.14), (2.15) and (2.16), the Poincare´ map F 1 ¼ F 11 F 01 : S 10 #S 11 can be defined as 0 33 32 1 b1 2 n1;2 1 ¼ u1 w1 ðw1 Þ s1 d þ M 1 l þ h:o:t; 32 1 b1 3 n1;3 1 ¼ ðw1 Þ s1 d þ M 1 l þ h:o:t
ð2:17Þ
and F 2 ¼ F 12 F 02 : S 11 #S 12 as n1;1 2
¼
q2 ðlÞ 2 1
1 q2 ðlÞ 1 ðw13 2 Þ s2 v1
þ M 12 l þ h:o:t; q2 ðlÞ 2 q1 ðlÞ 2
12 13 1 1 2 n1;2 2 ¼ s2 d w2 ðw2 Þ s2 v1 þ M 2 l þ h:o:t:
See Fig. 2.
ð2:18Þ
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
Fig. 2. Sections and Poincare Map.
Let Gi ¼ F i ðq1i1 Þ q1i , i ¼ 1; 2. Owing to (2.10), (2.12), (2.15), (2.16), (2.17) and (2.18), we have the successor functions Gi as follows 1 b
G21
¼
2 32 1 b1 21 1 2 u01 w33 1 ðw1 Þ s1 d ðw1 Þ s2 d þ M 1 l þ h:o:t;
G31
¼
1 b1 3 1 23 21 1 2 ðw32 1 Þ s1 d v1 þ w1 ðw1 Þ s2 d þ M 1 l þ h:o:t;
¼
1 q2 ðlÞ 1 ðw13 v1 2 Þ s2
G12 G22
1 b
q2 ðlÞ 2 1
¼
s2 d
k2 ðlÞ 1 k1 ðlÞ 1
1 21 1 s1 u01 þ w23 2 ðw2 Þ s1 d þ M 2 l þ h:o:t; q2 ðlÞ 2 1
13 1 q2 ðlÞ 1 w12 2 ðw2 Þ s2 v1
1 2 ðw21 2 Þ s1 d þ M 2 l þ h:o:t:
Clearly, there is an 1–1 correspondence between the solution of the equation ðG21 ; G31 ; G12 ; G22 Þ ¼ 0
ð2:19Þ
with s1 P 0, s2 P 0 and the heteroclinic loop, homoclinic loop and periodic orbit of (1.1).
3. Main results To study the bifurcations near C, we should consider the solution of (2.19). It is easy to see that equation ðG21 ; G31 Þ ¼ 0 always has the unique solution 1 b
2 32 1 b1 21 1 2 u01 ¼ w33 1 ðw1 Þ s1 d þ ðw1 Þ s2 d M 1 l þ h:o:t; 1 b
1 b1 3 23 21 1 2 v11 ¼ ðw32 1 Þ s1 d þ w1 ðw1 Þ s2 d þ M 1 l þ h:o:t:
This fact is essentially guaranteed by the transversality given in (H4). Putting u01 ; v11 in the above equations into ðG12 ; G22 Þ ¼ 0, then we obtain w23 2 s1
1
¼d
k2 1 1
k 33 32 1 b1 1 w21 2 s1 ðw1 ðw1 Þ s1 d
1
þ
1 b2 ðw21 1 Þ s2 d
M 21 lÞ
1
d
1 w21 2 M 2l
1
d
q2 2 1
13 1 q2 32 1 b1 w21 2 ðw2 Þ s2 ððw1 Þ s1 d
1 b2
21 1 3 þ w23 1 ðw1 Þ s2 d þ M 1 lÞ þ h:o:t; 1 1 2 s2 ¼ ðw21 2 Þ s1 d M 2 l þ h:o:t:
ð3:1Þ
and (3.1) is called the bifurcation equation. To investigate the existence of heteroclinic loop, homoclinic loop and periodic orbit, what we need to do is pondering on the existence of solutions to Eq. (3.1) such as s1 ¼ s2 ¼ 0, s1 ¼ 0; s2 > 0 or s1 > 0; s2 ¼ 0, and s1 > 0; s2 > 0, respectively. Theorem 3.1. Suppose conditions ðH 1 Þ–ðH 5 Þ are satisfied. Then (1) If rank ðM 12 ; M 22 Þ ¼ 2, then there exists an (l 2)-dimensional surface L12 ¼ fl : M 12 l þ oðjljÞ ¼ M 22 lþ oðjljÞ ¼ 0g with a normal plane span fM 12 ; M 22 g at l = 0, such that system (1.1) has a unique heteroclinic loop Cl ¼ Cl1 [ Cl2 near C if l 2 L12 and 0 < jlj 1. (2) If w23 2 –0 and l 2 L12 , 0 < jlj 1, then system (1.1) has no periodic orbit. That is to say, the heteroclinic loop and periodic orbit of system (1.1) cannot coexist.
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
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Proof 1. Assume (3.1) has a solution s1 ¼ s2 ¼ 0, then M j2 l þ h:o:t ¼ 0;
j ¼ 1; 2:
(1) If rank ðM 12 ; M 22 Þ ¼ 2, then L12 ¼ fl : M 12 l þ oðjljÞ ¼ M 22 l þ oðjljÞ ¼ 0g is an (l 2)-dimensional surface such that (3.1) has a solution s1 ¼ s2 ¼ 0 as l 2 L12 and 0 < jlj 1, i.e., system (1.1) has a unique heteroclinic loop near C. Clearly, L12 has a normal plane spanned by M 12 and M 22 at l = 0. (2) In the case w23 2 –0 and l 2 L12 , (3.1) produces s1 ¼
13 1 21 ðw23 2 w2 Þ ðw2 Þ
2 q2 q1 q2 2 2 1 q 1 q 2 2
s1
1 b1 ðw32 1 Þ s1
þ
21 w23 1 w1
h ib1 2 1 3 21 1 ðw2 Þ s1 þ d M 1 l þ h:o:t:
q2
Since q21 > 1 and 0 < jlj 1, it is obviously that the above equation has not any solution satisfying 0 < s1 1. Which 2 implies system (1.1) has no periodic orbit near C. This completes the proof. h Theorem 3.2. Assume the hypotheses ðH 1 Þ–ðH 5 Þ are valid, then the following propositions are true. (1) If M 12 –0 and M 22 –0, then there exists an (l 1)-dimensional surface 9 8 q2 q2 = < 2þ 1 2 1 1 b2 1 1 1 q q 21 13 1 2 2 3 2 2 2 þ ðw13 L21 ¼ l : W 21 , M 12 l þ dw23 1 ðw1 w2 Þ ðd M 2 lÞ 2 Þ ðd M 2 lÞ M 1 l þ h:o:t ¼ 0; M 2 l < 0 ; :
ð3:2Þ
with a normal vector M 12 and tangent to L12 at l = 0, such that system (1.1) has a unique homoclinic loop C21 connecting p1 when l 2 L21 and 0 < jlj 1. 2 (2) If M 22 –0 and M 12 þ w23 2 M 2 –0, then there exists an (l 1)-dimensional surface 9 8 k2 k2 = < 1 þb 1 1 1 21 k1 k1 1 1 1 23 2 33 32 1 1 21 2 2 2 21 2 L2 ¼ l : W 2 , M 2 l þ w2 M 2 l dw1 ðw1 Þ ðd w2 M 2 lÞ 1 þ ðd w2 M 2 lÞ 1 M 1 l þ h:o:t ¼ 0; w2 M 2 l > 0 ; : ð3:3Þ 1 2 1 with a normal M 12 þ w23 2 M 2 at l = 0, such that system (1.1) has a unique homoclinic loop C2 joining p2 as l 2 L2 and 0 < jlj 1.
Proof 2 (1) Suppose (3.1) has a solution s1 ¼ 0; s2 > 0. Then one obtains q2 2þ 1 1 b2
21 1 23 q2 dðw13 2 w1 Þ w1 s2
þ
q2 2 1
1 q2 3 ðw13 2 Þ s2 M 1 l
þ M 12 l þ h:o:t ¼ 0;
ð3:4Þ
s2 ¼ d1 M 22 l þ h:o:t: Thus under the conditions M 12 –0 and M 22 –0, one deduces from (3.4) and the implicit function theorem that there exists an (l 1)-dimensional surface L21 defined by (3.2), such that (3.1) has a solution s1 ¼ 0; s2 ¼ s2 ðlÞ > 0 as l 2 L21 and 0 < jlj 1. That is, system (1.1) has a loop C21 homoclinic to p1 as l 2 L21 and 0 < jlj 1. Obviously, L21 has a normal vector M 12 at l = 0, and hence it is tangent to L12 at l = 0. (2) Now if (3.1) has solution s1 > 0, s2 ¼ 0, one thus derives 2 w23 2 s1
1
¼d
k2 1 1
4sk11 ðw33 ðw32 Þ1 sb1 1 d w21 2 1 1
2 s1 ¼ d1 w21 2 M 2 l þ h:o:t:
3
M 21 lÞ
M 12 l5
þ h:o:t;
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
It then follows that there exists an (l 1)-dimensional surface L12 given by (3.3) such that (3.1) has a solution s1 ¼ s1 ðlÞ > 0; s2 ¼ 0 as l 2 L12 and 0 < jlj 1. Equivalently, system (1.1) has a homoclinic loop C12 connecting p2 2 as l 2 L12 and 0 < jlj 1. There is no difficulty to see that L12 has a normal M 12 þ w23 2 M 2 at l = 0. h
The proof is then finished.
2 Remark 3.1. In the above theorem, the condition M 22 –0 guarantees that the regions of M 22 l < 0 and w21 2 M 2 l > 0 are 1 1 2 23 not empty. The conditions M 2 –0 and M 2 þ w2 M 2 –0 assure the existence of normal vectors.
Theorem 3.3. Assume that ðH 1 Þ–ðH 5 Þ hold, w23 2 –0 and 0 < jlj 1, then 2 2 23 (1) If M 12 –0 and M 22 –0, then system (1.1) has a unique periodic orbit near C as w21 2 > 0; M 2 l < 0, w2 W 1 < 0 (that is, 2 1 2 2 23 21 l is in the side of L1 pointed by the direction w2 M 2 ), and has no periodic orbit as w2 > 0; M 2 l < 0, w23 2 W1 > 0 2 1 2 23 (that is, l is in the side of L1 pointed by the direction w2 M 2 ), where W 1 is defined by (3.2). 2 2 21 (2) If M 22 –0 and M 12 þ w23 2 M 2 –0, then system (1.1) has a unique periodic orbit near C for w2 > 0; M 2 l > 0, 1 1 1 2 23 23 23 w2 W 2 < 0 (that is, l is in the side of L2 pointed by the direction w2 ðM 2 þ w2 M 2 ÞÞ, and has not any periodic 2 1 1 1 2 23 23 23 orbit for w21 2 > 0; M 2 l > 0, w2 W 2 > 0 (that is, l is in the side of L2 pointed by the direction w2 ðM 2 þ w2 M 2 ÞÞ, where W 12 is given by (3.3). 2 (3) If w21 2 < 0, M 2 l > 0, then system (1.1) has no periodic orbit.
Proof 3 2 (1) For the case w21 2 > 0; M 2 l < 0, we see that 1 1 2 s2 ¼ gðs1 Þ , ðw21 2 Þ s1 d M 2 l þ h:o:t > 0
if 0 6 s1 1. Substituting s2 into the first equation of (3.1), we attain F ðs1 ; lÞ ,
w23 2 s1
k2 1 1
k 1 w21 2 s1
32 1 b1 w33 1 ðw1 Þ s1
þ
1 ðw21 1 Þ
h
1 ðw21 2 Þ s1
1
d
ib1
M 22 l
2
1
þd
k2 1 1
k 2 1 w21 2 s1 M 1 l
2
h iq21 h i q 1 13 1 21 1 2 32 1 b1 2 ðw Þ ðw Þ ðw Þ s d M l s1 þ d1 M 31 l þ w21 1 2 2 2 2 1 2
h iq21 þb1 q 1 2 21 21 13 1 21 1 2 1 2 w ðw w Þ ðw Þ s d M l þ d1 w21 þ w23 1 1 2 1 2 2 2 2 M 2 l þ h:o:t ¼ 0: k21 k11
ð3:5Þ
q22 q12
1 21 2 23 Notice that > 1; > 1, we get F 0s1 ðs1 ; lÞ w23 2 . Suppose that w2 > 0. Then based on the fact F ð0; lÞ ¼ d w2 W 1 and 0 2 F s1 ðs1 ; lÞ w23 2 > 0, F ðs1 ; lÞ ¼ 0 has a unique small positive solution s1 ¼ s1 ðlÞ if W 1 < 0 and has no positive solution 2 2 as W 21 > 0, which shows that system (1.1) has a unique periodic orbit as w21 2 > 0; M 2 l < 0 and W 1 < 0 and has no peri2 2 21 odic orbit as w2 > 0; M 2 l < 0 and W 1 > 0. With similar arguments one may attain the results for the case w23 2 < 0. 2 (2) If w21 2 > 0; M 2 l > 0, it is clear that 1 2 s1 ¼ g1 ðs2 Þ , w21 2 ðs2 þ d M 2 lÞ þ h:o:t > 0
for 0 6 s2 1. Eliminating s1 in (3.1) one reaches k2
21 k11 þb1 1 23 1 1 2 1 32 1 33 2 1 Gðs2 ; lÞ , w23 2 s2 þ d w2 M 2 l þ d M 2 l ðw1 Þ w1 w2 ðs2 þ d M 2 lÞ
þ q2
k2
q2 2þ 1 1 1 b 1 b2 1 q2 2 1 21 2 23 21 13 1 ðw Þ s2 d M 1 l þ w1 ðw1 w2 Þ s2 1
k2
w21 2 ðs2
1
þd q2 2 1
1 q2 ðw13 2 Þ s2
n
M 22 lÞ
k11
o b1 1 21 ðw32 w2 ðs2 þ d1 M 22 lÞ þ d1 M 31 l þ h:o:t ¼ 0: 1 Þ
1 1 Due to q21 > 1, k11 > 1, one has G0s2 ðs2 ; lÞ w23 2 . Using Gð0; lÞ ¼ d W 2 and some similar arguments to the proof of (1), 2 1 we may complete the proof of (2).
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
2071
1 2 2 21 1 (3) Obviously, if w21 2 < 0, M 2 l > 0, then s2 ¼ ðw2 Þ s1 d M 2 l þ h:o:t shows that s1 > 0, s2 > 0 can not be the solution of (3.1). Therefore system (1.1) has no periodic orbit in this case. The proof is then completed. h
1 2 1 Theorem 3.4. Suppose the hypotheses ðH 1 Þ–ðH 5 Þ are valid, w23 2 ¼ 0, rank ðM 2 ; M 2 Þ ¼ 2 and b1 < b2 . Then for l 2 L12 , 0 < jlj 1, we have the following propositions.
(1) If w21 2 < 0, then system (1.1) has no periodic orbit. q22 k21 2 33 33 32 (2) For w21 2 > 0, w1 –0 and k1 þ b1 < q1 , system (1.1) has a unique (resp. no) periodic orbit if w1 w1 M 1 l > 0 (resp. 1 2 2 33 32 w1 w1 M 1 l < 0). a1 q22 k21 k21 3 2 33 a (3) For w21 2 > 0, w1 –0 and k1 þ b1 > q1 > k1 , system (1.1) has a unique periodic orbit in the region jM 1 lj jM 1 lj , 1
2
1
b q1 k1
a1
1 2 1 2 3 3 2 2 32 33 a w13 2 M 1 lM 1 l < 0 or in the region jM 1 lj jM 1 lj , w1 w1 M 1 l > 0, where a ¼ q2 k1 q1 k2 .
k21 k11
2 1
q22 q12
2 1
2 33 (4) For w21 þ b12 < , system (1.1) has a unique (resp. no) periodic orbit if w21 2 > 0, w1 ¼ 0 and 1 M 1 l > 0 (resp. 2 21 w1 M 1 l < 0). q22 q22 k21 33 1 (5) For w21 2 > 0, w1 ¼ 0 and q1 < k1 þ b2 < q1 þ b1 ; system (1.1) has a unique periodic orbit in regions 2
a1
1
2
q1 k 1
a1
2 3 3 2 2 21 2 1 a jM 31 lj jM 21 lj a , w13 2 M 1 lM 1 l < 0 and jM 1 lj jM 1 lj , w1 M 1 l > 0, where a ¼ ðq2 k1 q1 k2 Þb . k2
2 1
q2
2 1
2
3 3 32 32 1 2 (6) For w21 2 > 0 and k1 > q1 þ b1 , system (1.1) has a unique (resp. no) periodic orbit if w1 M 1 l < 0 (resp. w1 M 1 l > 0). 1 q2 2 k2 a1 q22 2 21 2 1 (7) For w2 > 0 and q1 < k1 < q1 þ b1 , system (1.1) has a unique periodic orbit in regions jM 1 lj jM 31 lj a , 1 1 2 1 2 a1 q2 k 1 b1 2 3 2 3 3 32 a w13 2 M 1 lM 1 l < 0 and jM 1 lj jM 1 lj , w1 M 1 l < 0, where a ¼ q1 k2 q2 k1 . 2 1
2 1
Proof 4. For w23 2 ¼ 0 and l 2 L12 , (3.1) becomes that k2 1 k1 1
s1
32 1 b1 w33 1 ðw1 Þ s1
1
þ
1 b2 ðw21 1 Þ s2
q2 2 1 q1 1 1 32 1 b1 23 21 1 b2 3 2 d M 21 l ðw13 Þ s ðw Þ s þ w ðw Þ s þ d M l ¼ 0; 2 1 2 2 1 1 1 1
1
ð3:6Þ
1 s2 ¼ ðw21 2 Þ s1 :
(1) One may easily get the result based on the second equation of (3.6). 21 1 When w21 2 > 0, we may see that s2 P 0 for s1 P 0. Putting s2 ¼ ðw2 Þ s1 into the first equation of (3.6), one attains k2 1 þb 1 1
32 1 k1 w33 1 ðw1 Þ s1
þ
1
k2 1þ 1 1 b2
1 21 b2 k1 ðw21 s1 1 Þ ðw2 Þ
13 21 1 21 w23 1 ðw2 w1 Þ ðw2 Þ
q2 q2 2 1 ð 21 þb1 Þ q1 þb2 q 2 2 2
s1
2
1
d
s1 M 21 l
1 þb 1 1
q q2 k1 2 @sk1 (2) If w33 1 þ b1 < q1 , then s1 ¼ o 1 1 –00and k1 0 1 1 2 2 2 2 2 k k q k 1þ 1 k1 b2 1
s1
1 þb 1 k1
¼ o@s11
k2 1 þb 1 1
32 1 k1 w33 1 ðw1 Þ s1
2þ 1 q1 b2 2
1 þb 1 k1
A; s 1
1
k2 1 k1 1
1
k2 1 k1 1
¼ o@s11
1 21 ðw13 2 dÞ ðw2 Þ 0 1 q2 k2 2 1
2
k2 1 k1 1
32 1 21 ðw13 2 w1 Þ ðw2 Þ
2 q2 q2 2 1 þb1 q1 q2 2
s1
2 q2 q2 21 q1 q 2 2
s1 M 31 l ¼ 0: 0 k2 q2 2 þb 1 1
A; sq2 1
1 þb 1 1
k ¼ o@s11
ð3:7Þ 1 A. And the condition b1 < 1 shows that b2
A: Therefore, (3.7) becomes
d s1 M 21 l þ h:o:t ¼ 0; 32 2 b1 w M l 2 32 which has a unique solution s1 ¼ 1dw331 1 þ h:o:t > 0 as w33 1 w1 M 1 l > 0. This means that the heteroclinic loop and a 1 simple periodic orbit of system2 (1.1) can2 coexist. q2 k1 k21 (3) In the case w33 1 –0 and k1 þ b1 > q1 > k1 , it then follows from (3.7) that 1
k2 1 þb 1 k1 1
32 1 w33 1 ðw1 Þ s1
d
s1 M 21 l
2
1
1 21 ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q1 q2 2
s1 M 31 l þ h:o:t ¼ 0:
Now we want to look for the positive solution of (3.8). Multiplying (3.8) by s1 1 32 1 b1 2 13 1 21 w33 1 ðw1 Þ s1 d M 1 l ðw2 dÞ ðw2 Þ
2 2 q2 q2 k1 21 q1 k1 q 2 1 2
s1
M 31 l þ h:o:t ¼ 0:
k2 1 k1 1
ð3:8Þ
results in
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075 q2 k2 2 1 q1 k1 2 1
Denote t ¼ s1
b q1 k1
2 1 and a ¼ q2 k11 q 1 k2 > 1, it then gives rise to 2 1 2 1 2 3 2 q
32 1 a 4 13 1 21 w33 1 ðw1 Þ t ðw2 dÞ ðw2 Þ
2 q1 2
M 31 l5t d1 M 21 l þ h:o:t ¼ 0:
ð3:9Þ
a1
2 3 In the region jM 31 lj jM 21 lj a and w13 2 M 1 lM 1 l < 0, (3.9) has a small positive solution t ¼ fying 0 < t 1. One may see that
2 a q2
2
33 32 1a
33 32 1 13 21 q12 a
M 1 l
w1 ðw1 Þ t w1 ðw1 Þ ðw2 ðw2 Þ Þ
3 d1 M 21 l ;
M 1l
2
2
q2 2 1 21 Þq2 M 2 l w13 ðw 2 1 2 M 31 l
2
þ h:o:t satis-
a1
q k k 3 2 a which means t is reasonable. This shows that when k11 < q21 < k11 þ b1 and w33 and 1 –0, if jM 1 lj jM 1 lj 1 2 1 2 3 13 w2 M 1 lM 1 l < 0, then the heteroclinic loop and periodic obit of system (1.1) coexist. 32 2 1a a1 w M l 2 33 þ h:o:t > 0 as w32 Now suppose jM 31 lj jM 21 lj a , then (3.9) has a solution t ¼ 1dw331 1 w1 M 1 l > 0. Clearly, 1 0 < t 1, t is reasonable since
32 2 1a q2 q2
13 1 21 q21
2 a1
13 1 21 q21 3
ðw dÞ ðw Þ 2 M lt ðw dÞ ðw Þ 2
M l a w1 M 1 l d1 M 2 l : 2 1
2 1
1
2
2
dw33
1 k2
q2
k2
3 2 Consequently, for k11 < q21 < k11 þ b1 and w33 1 –0, if jM 1 lj jM 1 lj 1 2 1 periodic obit of system (1.1) can coexist.
(4) For w33 1 ¼ 0 and 1 21 ðw21 1 Þ ðw2 Þ
b1 2
k21
a1 a
q2
þ b12 < q21 , it follows from (3.7) that
k11
k2 1þ 1 k1 b2 1
s1
2 k2 1 k1 1
d1 s1 M 21 l þ h:o:t ¼ 0:
ð3:10Þ 1
w21 ðw21 Þb2 M 21 l 1 2 d
Thus, (3.10) has a unique small positive solution s1 ¼
eroclinic loop and periodic orbit of system (1.1) coexists if (5) If
w33 1
¼ 0 and
q22 q12
2 33 and w32 1 w1 M 1 l > 0; hence the heteroclinic loop and
<
k2 1þ 1 k1 b2 1
1 1 21 b2 s1 ðw21 1 Þ ðw2 Þ
k21 k11
þ
1 b2
<
q22 q12
k21 k11
!b2 2 þ h:o:t if w21 1 M 1 l > 0. In another word, the hetq2
2 21 þ b12 < q21 , w33 1 ¼ 0 and w1 M 1 l > 0 hold. 2
þ b1 ; (3.7) yields
k2 1 k1 1
1
21 d1 s1 M 21 l ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q 1 q 2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:11Þ
k2 1 k1 1
In order to find positive solutions of (3.11), multiplying (3.11) by s1 , one derives 1
1 1 21 b2 b2 ðw21 s1 1 Þ ðw2 Þ q2 k 2 2 1 q1 k 1 2 1
Let t ¼ s1
,a¼
1 21 ðw21 1 Þ ðw2 Þ
1
21 d1 M 21 l ðw13 2 dÞ ðw2 Þ
q12 k11
ðq22 k11 q12 k21 Þb2 2 b1 2
2 2 q2 q2 k 1 2 1 k1 q 1 q 2 1 2
s1
M 31 l þ h:o:t ¼ 0:
ð3:12Þ
> 1, (3.12) thus becomes
1 21 ta 4ðw13 2 dÞ ðw2 Þ
q2 2 q1 2
3 M 31 l5t d1 M 21 l þ h:o:t ¼ 0:
With a similar way to discuss (3.9), we may easily get the results. q2 k2 (6) If k11 > q21 þ b1 , then (3.7) takes the form 1
2
32 1 21 ðw13 2 w1 Þ ðw2 Þ
q2 2 q1 2
q2 2 þb 1 q1 2
s1
1
21 þ ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q 1 q 2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:13Þ
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
w32 M 3 l
b1
2073 q2
k2
3 2 1 (3.13) has a unique positive solution s1 ¼ 1 d 1 1 þ h:o:t if w32 1 M 1 l < 0. Which tells that when q1 þ b1 < k1 , the het2 1 3 32 eroclinic and periodic orbit coexists if w1 M 1 l < 0. q22 q22 k21 (7) If q1 < k1 < q1 þ b1 , then (3.7) turns out to be 2
1
2
32 1 21 ðw13 2 w1 Þ ðw2 Þ
2 q2 q2 2 1 þb1 q1 q2 2
s1
q2 21 q 2
Multiplying (3.14) by s1 32 1 21 ðw13 2 w1 Þ ðw2 Þ
q2 2 q1 2
1
k2 1 k1 1
1
21 þ d s1 M 21 l þ ðw13 2 dÞ ðw2 Þ k2 q2 1 2 k1 q1 1 2
and denote by t ¼ s1
2 q2 q2 2 1 q1 q2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:14Þ
q1 k 1 b
1 1 , a ¼ q1 k22 q 2 k1 > 1, it then becomes 2 1
2 1
q2
2 1 21 q12 ta þ d1 M 21 l t þ ðw13 M 31 l þ h:o:t ¼ 0 2 dÞ ðw2 Þ
The claim now can be confirmed like doing for (3.9).
h
Remark 3.2. We only consider the case b1 < b12 in Theorem 3.4, in fact, one may achieve the similar results for b1 P b12 . It then shows that, comparing to the non-heterodimensional heteroclinic cycles, heterodimensional cycles have different bifurcation behaviors, since it is well known that, in the former case, a persistent heteroclinic loop can coexist with a bifurcated periodic orbit only when the original cycle is resonant, that is, b1 b2 ¼ 1. At last, we consider the existence of 2-fold periodic orbit for system (1.1). In case w23 2 –0 and b1 > 1, (3.1) reduces to w23 2 s1
¼
s2 ¼ w21 2
q2 2
1 21 q12 dw13 w 2 s2 2
1
21 1 b12 3 1 23 w1 w1 s2 d þ M 1 l d1 w21 2 M 2 l þ h:o:t , f ðs2 Þ
s1 d1 M 22 l þ h:o:t:
Then q2 þk1 q1 2 2 2 q1 2
23 ðq2 þ k1 Þw21 2 w1 s2 f ðs2 Þ ¼ 2 1 213 21 q2 w2 w1
q2 q1 2 2
q2 w21 M 3 l q1 2 21 131 s2 2 þ h:o:t: dq2 w2 b2 q22 w21 M 31 l 0 1 In case w23 þ h:o:t , s, which in turn produces 1 –0, f ðs2 Þ ¼ 0 has the solution s2 ¼ dðq2 þk1 Þw23 0
2
q2 2 q1 2
2
1
21 1 1 1 21 1 sb2 d þ M 31 l d1 w21 s w23 f ðsÞ ¼ ðdw13 2 Þ w2 1 w1 2 M 2 l þ h:o:t ! ! q2 2 w21 w23 d q22 w21 M 31 l q1 3 1 2 1 1 þ M 1 l d1 w21 ¼ 13 s 2 2 2 M 2 l þ h:o:t w21 dw2 d q2 þ k12 w23 1 1 q2
2 w21 k1 M 3 l 1 1 ¼ 22 2 1 1 13 sq2 d1 w21 2 M 2 l þ h:o:t d q2 þ k2 w2 q2 þk1
k1 w23 w21 2 1 2 1 ¼ 22 123 221 s q2 d1 w21 2 M 2 l þ h:o:t: q2 w2 w1 2 1 1 2 q2 2q1 3 23 q2 þk2 2q2 2 2 q þ k12 q22 þ k12 q12 w21 q22 q22 q12 w21 2 w1 2 M 1l q1 q1 2 2 s s þ h:o:t f 00 ðsÞ ¼ 2 13 1 2 13 21 1 2 ðq2 Þ w2 w1 ðq2 Þ dw2 ! ! 2 2 q2 2q1 1 1 1 2 2 2 3 w23 w21 q22 w21 M 31 l 1 d q2 þ k2 q2 þ k2 q2 q1 2 1 2 1 s 2 þ q2 q2 q2 M 1 l þ h:o:t 2 ¼ 2 w21 d q2 þ k12 w23 ðq12 Þ dw13 1 1 2 ¼
3 2 1 w21 2 q2 k2 M 1 l
q2 2q1 2 2 q1 2
s þ h:o:t 2 ðq12 Þ dw13 2 2 1 1 2 21 q2 þk2 2q2 q2 þ k12 k12 w23 1 w2 q1 s 2 þ h:o:t: ¼ 2 13 ðq12 Þ w21 1 w2 Hence, we may write f(s2) as
ð3:15Þ
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
1 f ðs2 Þ ¼ f ðsÞ þ f 00 ðsÞðs2 sÞ2 þ h:o:t: 2 Putting the second equation of (3.1) into the above equation, one has for w23 2 –0 that h i2 1 1 1 2 sÞ þ f 00 ðsÞ ðw21 s þ h:o:t; w23 2 s1 ¼ f ð 2 Þ s1 d M 2 l 2 which in turn yields AðlÞ þ BðlÞs1 þ s21 þ h:o:t ¼ 0;
ð3:16Þ
where 2 2 2 00 AðlÞ ¼ 2ðw21 sÞÞ1 f ðsÞ þ w21 s þ d1 M 22 l 2 Þ ðf ð 2 and 21 2 00 BðlÞ ¼ 2w23 sÞÞ1 2w21 s þ d1 M 22 l : 2 ðw2 Þ ðf ð 2 1 2 2 21 1 In the next theorem, we assume w21 2 > 0; M 2 l < 0, which guarantees s2 ¼ ðw2 Þ s1 d M 2 l þ h:o:t > 0 for 0 6 s1 1. It is enough to consider the nonnegative solution of (3.16) for investigating the homoclinic loop and periodic orbit bifurcated from C. 1 3 2 21 23 21 2 1 Theorem 3.5. Assume ðH 1 Þ–ðH 6 Þ hold, b1 > 1, w23 2 –0, w1 w1 M 1 l < 0, w2 > 0, M 2 l < 0. If q2 þ k2 < 2q2 , then the following propositions are true for 0 < jlj 1.
(1) If AðlÞ > 0; BðlÞ > 0, then system (1.1) has no periodic orbit and homoclinic orbit. (2) If AðlÞ < 0, then system (1.1) has a unique periodic orbit. (3) For AðlÞ > 0, then (i) if D , B2 ðlÞ 4AðlÞ > 0, BðlÞ < 0, then system (1.1) has two simple periodic orbits; (ii) if D , B2 ðlÞ 4AðlÞ ¼ 0, BðlÞ < 0, then system (1.1) has a 2-fold periodic orbit; (iii) if D , B2 ðlÞ 4AðlÞ < 0, system (1.1) has no periodic orbit. Proof 5. The results may be achieved from (3.16) directly.
h
Remark 3.3. The condition q22 þ k12 < 2q12 in Theorem 3.5 guarantees that 0 < jAðlÞj; jBðlÞj 1, which in turn means the positive solutions for (3.16) are small enough. 4. Conclusions In this paper, we discuss the bifurcations of heterodimensional cycles with two saddles. By using the method initiated by [15,16], we obtain the existence of one heteroclinic loop, one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbits, and the coexistence of one periodic orbit and heteroclinic loop. The most interesting is the coexistence of one periodic orbit and heteroclinic loop, since it is well known that, in the case of non-heterodimensional heteroclinic cycles, a persistent heteroclinic loop can not coexist with a bifurcated periodic orbit unless the original cycle is resonant. So, heterodimensional cycles have different bifurcation behaviors to the case of non-heterodimensional cycles. Acknowledgement The authors were supported by National Natural Science Foundation of China (]10671069).
References [1] Bonatti C, Diaz LJ, Pujals E, Rocha J. Robust transitivity and heterodimensional cycles. Asterisque 2003;286:187–222. [2] Chow SN, Hale JK. Methods of bifurcation theory. New York: Springer-Verlag; 1982. [3] Diaz LJ, Rocha J. Heterodimensional cycles, partial hyperbolity and limit dynamics. Fundat Math 2002;174:127–86.
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[4] Han MA, Bi P. Existence and bifurcation of periodic solutions of high-dimensional delay differential equations. Chaos, Solitons & Fractals 2004;20:1027–36. [5] He JH. Limit cycle and bifurcation of nonlinear problems. Chaos, Solitons & Fractals 2005;26:827–33. [6] Jin YL, Zhu DM. Degenerated homoclinic bifurcations with higher dimensions. Chin Ann Math 2000;21 B:201–10. [7] Jin YL, Zhu DM. Bifurcations of rough heteroclinic loops with three saddle points. Acta Math Sin Eng Ser 2002;18:199–208. [8] Jin YL, Zhu DM. Bifurcations of rough heteroclinic loop with two saddle points. Sci Chin Ser A 2003;46:459–68. [9] Lamb JSW, Teixeira MA, Webster KN. Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fileds in R3 . J Differ Equat 2005;219:78–115. [10] Pomeau Y. Remarks on bifurcations with symmetry. Chaos, Solitons & Fractals 1995;5:1755–61. [11] Rademacher JDM. Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit. J Differ Equat 2005;218:390–443. [12] Shui SL, Zhu DM. Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips. Sci Chin Ser A 2005;48:248–60. [13] Wiggins S. Introduction to applied nonlinear dynamical system and chaos. New York: Springer-Verlag; 1990. [14] Zhang TS, Zhu DM. Codimension 3 homoclinic bifurcation of orbit flip with resonant eigenvalues corresponding to the tangent directions. Int J Bifurcat Chaos Appl Sci Eng 2004;14:4161–75. [15] Zhu DM. Problems in homoclinic bifurcation with higher dimensions. Acta Math Sin Eng Ser 1998;14:341–52. [16] Zhu DM, Xia Z H. Bifurcation of heteroclinic loops. Sci Chin Ser A 1998;41:837–48.
Bifurcations of heterodimensional cycles with two saddle points Fengjie Geng a
a,*
, Deming Zhu b, Yancong Xu
b
School of Information Technology, China University of Geosciences (Beijing), Beijing 100083, China b Department of Mathematics, East China Normal University, Shanghai 200062, China Accepted 21 June 2007
Abstract The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The bifurcations of dynamical system have been studied extensively in the literature. In recent years, there has been much research work concerning the high-codimensional bifurcations of homoclinic and heteroclinic cycles which include the bifurcations of heterodimensional cycles, resonant cycles, orbit flip cycles, inclination flip cycles, etc. The readers are referred to the papers [1–5,9–14, and references cited therein]. Moreover, the bifurcations of heterodimensional cycles have attracted special attention these years (see [1,3,9,11]). Comparatively speaking, to study the bifurcations of heterodimensional cycles is much more difficult and challenging. A heteroclinic loop is said to be a heterodimensional cycle if it is formed by two or more normally hyperbolic invariant manifolds which have different dimensions of the unstable manifolds and two or more heteroclinic orbits connecting them. As we know, the existence of heterodimensional cycles connecting saddle-foci always yields extremely complex dynamics behavior. It is demonstrated in [9] that the reversible vector fields with heterodimensional cycles are dense near Hopf-zero bifurcation. Rademacher [11] investigated the bifurcations of heterodimensional cycles connecting one hyperbolic equilibrium and one hyperbolic periodic orbit. And he derived the conditions for the existence and uniqueness of countably infinite families of curve segments of homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. One may see that bifurcations of heterodimensional cycles play a very important role in the theoretic study of high-codimensional bifurcations of homoclinic and heteroclinic cycles. In this paper, we are concerned with the bifurcations of heterodimensional cycles connecting two hyperbolic saddles. By utilizing the method originally established by [15,16] and then improved in [6–8], we obtain the existence of one *
Corresponding author. E-mail addresses: [email protected] (F. Geng), [email protected] (D. Zhu), [email protected] (Y. Xu).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.077
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Fig. 1. Heterodimensional Cycle.
homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional cycles are revealed. Consider the following Cr system z_ ¼ f ðzÞ þ gðz; lÞ;
ð1:1Þ
and its unperturbed system z_ ¼ f ðzÞ;
ð1:2Þ
where r P 7, z 2 R3 , l 2 Rl , l > 2, 0 < jlj 1, gðz; 0Þ ¼ 0. Assume that f ðpi Þ ¼ 0, gðpi ; lÞ ¼ 0 for i ¼ 1; 2. In addition, throughout the paper, the following conditions hold: ðH 1 Þp1 ; p2 are hyperbolic critical points of (1.2). dimðW sp1 Þ ¼ dimðW up2 Þ ¼ 1, dimðW up1 Þ ¼ dimðW sp2 Þ ¼ 2, where W spi , W upi denote the Cr stable manifold and C r unstable manifold of pi, respectively. Moreover, the linearization Df ðp1 Þ has simple real eigenvalues k11 ; k21 , q11 satisfying q11 < 0 < k11 < k21 ; Df ðp2 Þ has simple real eigenvalues q12 ; q22 , k12 fulfilling q22 < q12 < 0 < k12 . (H2) System (1.2) has a heteroclinic loop C ¼ C1 [ C2 , where Ci ¼ fz ¼ ri ðtÞ : t 2 Rg, ri ðþ1Þ ¼ riþ1 ð1Þ ¼ piþ1 , i ¼ 1; 2, r3 ðtÞ ¼ r1 ðtÞ, p3 ¼ p1 . r_ i ðtÞ u s u u þ þ (H3) Define e i ¼ limt!1 j_ri ðtÞj, i ¼ 1; 2. Then e1 2 T p1 W p1 , e1 2 T p2 W p2 , e2 2 T p2 W p2 , e2 2 T p1 W p1 are the unit eigenvectors corresponding to k11 , q12 , k12 , q11 , respectively. See Fig. 1. ðH 4 ÞW up1 intersects W sp2 transversely. It is easy to see that C is a heterodimensional cycle. 2. Local coordinates and bifurcation equations Now, we shall establish the normal forms of (1.1) in the neighborhood of the saddles p1, p2 in four steps. Step 1. Suppose the neighborhood U i of pi is small enough, by a linear transformation system (1.1) turns out to be the following form in U1 x_ ¼ k11 ðlÞx þ Oð2Þ;
y_ ¼ q11 ðlÞy þ Oð2Þ;
u_ ¼ k21 ðlÞu þ Oð2Þ:
ð2:1Þ
v_ ¼ q22 ðlÞv þ Oð2Þ;
ð2:2Þ
And in U2 system (1.1) takes the form x_ ¼ k12 ðlÞx þ Oð2Þ; kj1 ð0Þ
kj1 ,
qj2 ð0Þ
y_ ¼ q12 ðlÞy þ Oð2Þ; qj2 ,
q11 ð0Þ
q11 ,
k12 ð0Þ
k12 :
where ¼ ¼ j ¼ 1; 2, ¼ ¼ Step 2. We know that system (2.1) and (2.2) have Cr local manifolds W spi ;loc and W upi ;loc , i ¼ 1; 2, where W sp1 ;loc ¼ fz ¼ ðx; y; uÞ jx ¼ xðyÞ; u ¼ uðyÞ; xð0Þ ¼ uð0Þ ¼ 0; oðx;uÞ ¼ 0; ð0; y; 0Þ 2 U s1 g, W up1 ;loc ¼ fz ¼ ðx; y; uÞ jy ¼ yðx; uÞ; oy oy u s ox ¼ 0; ð0; y; vÞ 2 U s2 g, W up2 ;loc ¼ yð0; 0Þ ¼ 0; oðx;uÞ ¼ 0; ðx; 0; uÞ 2 U 1 g, W p2 ;loc ¼ fz ¼ ðx; y; vÞ jx ¼ xðy; vÞ; xð0; 0Þ ¼ 0; oðy;vÞ oðy;vÞ u s fz ¼ ðx; y; vÞ jy ¼ yðxÞ; v ¼ vðxÞ; yð0Þ ¼ vð0Þ ¼ 0; ox ¼ 0; ðx; 0; 0Þ 2 U 2 g, where U 1 fzjx ¼ u ¼ 0g U 1 , U u1 fzjy ¼ 0g U 1 , U s2 fzjx ¼ 0g U 2 , U u2 fzjy ¼ v ¼ 0g U 2 . Now we straighten the local manifold W spi ;loc and W upi ;loc such that W sp1 ;loc ¼ fz 2 U 1 jx ¼ u ¼ 0g, W up1 ;loc ¼ fz 2 U 1 jy ¼ 0g, W sp2 ;loc ¼ fz 2 U 2 jx ¼ 0g, W up2 ;loc ¼ fz 2 U 2 jy ¼ v ¼ 0g. So system (1.1) has the following normal form
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
x_ ¼ k11 ðlÞ þ x þ OðuÞ½OðyÞ þ OðuÞ ; 1 y_ ¼ q1 ðlÞ þ y; u_ ¼ k21 ðlÞ þ u þ OðxÞ½OðxÞ þ OðyÞ
2065
ð2:3Þ
in U1 and x_ ¼ k12 ðlÞ þ x; y_ ¼ q12 ðlÞ þ y þ OðvÞ½OðxÞ þ OðvÞ ; 2 v_ ¼ q2 ðlÞ þ v þ OðyÞ½OðxÞ þ OðyÞ
ð2:4Þ
in U2, where kj1 ð0Þ ¼ kj1 , qj2 ð0Þ ¼ qj2 , j ¼ 1; 2, q11 ð0Þ ¼ q11 , k12 ð0Þ ¼ k12 . System (2.3) and (2.4) is at least C r1 . ss Step 3. Continue to straighten the local strong unstable manifold W uu p1 ;loc of p1 and local strong stable manifold W p2 ;loc uu ss of p2 such that W p1 ;loc ¼ fz 2 U 1 jx ¼ y ¼ 0g, W p2 ;loc ¼ fz 2 U 2 jx ¼ y ¼ 0g. Then by the invariant manifold theorem one can deduce that system (1.1) now has the following normal form x_ ¼ k11 ðlÞ þ x þ OðuÞOðyÞ; 1 ð2:5Þ y_ ¼ q1 ðlÞ þ y; 2 u_ ¼ k1 ðlÞ þ u þ OðxÞ½OðxÞ þ OðyÞ in U1 and x_ ¼ k12 ðlÞ þ x; 1 y_ ¼ q2 ðlÞ þ y þ OðvÞOðxÞ; v_ ¼ q22 ðlÞ þ v þ OðyÞ½OðxÞ þ OðyÞ
ð2:6Þ
in U2. System (2.5) and (2.6) is at least C r2 . Step 4. At last, we straighten the heteroclinic orbit Ci in the small neighborhood Ui such that C1 \ U 1 ¼ fz 2 U 1 jy ¼ u ¼ 0g; System (1.1) then takes the form x_ ¼ k11 ðlÞ þ x þ OðuÞOðyÞ; y_ ¼ q11 ðlÞ þ y; 2 u_ ¼ k1 ðlÞ þ u þ OðxÞOðyÞ
C1 \ U 2 ¼ fz 2 U 2 jx ¼ v ¼ 0g:
ð2:5 Þ
in U1 and
x_ ¼ k12 ðlÞ þ x; 1 y_ ¼ q2 ðlÞ þ y þ OðvÞOðxÞ; v_ ¼ q22 ðlÞ þ v þ OðyÞOðxÞ:
ð2:6 Þ
Denote ri ðtÞ ¼ ðrxi ðtÞ; ryi ðtÞ; rui ðtÞÞ in U1 and ri ðtÞ ¼ ðrxi ðtÞ; ryi ðtÞ; rvi ðtÞÞ in U2 for i ¼ 1; 2. Due to (2.5*) and (2.6*), one may choose Ti and Ti, jT i j 1 such that ri ðT i Þ ¼ ðd; 0; 0Þ , ri ðT i Þ ¼ ð0; d; 0Þ , i ¼ 1; 2, where d > 0 is small enough such that fðx; y; uÞ : jxj; jyj; juj < 2dg U 1 , fðx; y; vÞ : jxj; jyj; jvj < 2dg U 2 . Take into account the linear variational system z_ ¼ Df ðri ðtÞÞz
ð2:7i Þ
and its adjoint system /_ ¼ ðDf ðri ðtÞÞÞ /;
ðz1i ðtÞ; z2i ðtÞ; z3i ðtÞÞ
ð2:8i Þ
be a fundamental solution matrix of **(2.7i), t 2 ½T i ; T i : We need the following i ¼ 1; 2. Let Z i ðtÞ ¼ additional generic hypothesis: uu þ ðH 5 ÞZ 2 ðT 2 ÞZ 1 2 ðT 2 ÞT p1 W p1 R spanfe2 ; e1 g. 1 ðtÞ Based on (H4), one can select z11 ðtÞ ¼ j_r1r_ðT 2 T r1 ðtÞ W up1 \ T r1 ðtÞ W sp2 , z21 ðtÞ 2 T r1 ðtÞ W up1 \ ðT r1 ðtÞ W sp2 Þc , z31 ðtÞ 2 1 Þj u c s ðT r1 ðtÞ W p1 Þ \ T r1 ðtÞ W p2 such that
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
0
1 0 B Z 1 ðT 1 Þ ¼ @ 0 0 0 1
1 w31 1 C w32 1 A;
0
0 B 12 Z 1 ðT 1 Þ ¼ @ w1 0
w33 1
w21 1 w22 1
1 0 C 0 A;
w23 1
1
3j 2j 32 21 32 1 21 1 where w12 1 < 0, w1 –0; w1 –0, jw1 ðw1 Þ j 1, for j ¼ 1; 3, jw1 ðw1 Þ j 1, for j ¼ 2; 3. 1 2 2 Due to (H5) and Liouville formula, one can select z2 ðtÞ; z2 ðtÞ 2 ðT r2 ðtÞ C2 Þc , z12 ðT 2 Þ 2 T p1 W uu p1 , z2 ðT 2 Þ ¼ e1 , u s 2 ðtÞ 2 T W \ T W such that z32 ðtÞ ¼ j_r2r_ðT r2 ðtÞ r2 ðtÞ p2 p1 2 Þj 1 1 0 11 0 w2 0 1 0 w21 0 2 C C B B Z 2 ðT 2 Þ ¼ @ w12 1 0 A; Z 2 ðT 2 Þ ¼ @ 0 w22 w32 2 2 2 A; w13 0 0 1 w23 0 2 2 1j 21 32 13 1 22 21 1 where w13 2 –0, w2 –0, w2 < 0, jw2 ðw2 Þ j 1, for j ¼ 1; 2, jw2 ðw2 Þ j 1. as a local coordinate system along Ci . Let Now, we may choose ðz1i ðtÞ; z2i ðtÞ; z3i ðtÞÞ Ui ðtÞ ¼ ð/1i ðtÞ; /2i ðtÞ; /3i ðtÞÞ ¼ ðZ 1 ðtÞÞ , then U ðtÞ is the fundamental solution matrix of (2.8 ), i ¼ 1; 2. i i i Take a coordinate change as z ¼ ri ðtÞ þ Z i ðtÞN i ðtÞ , hi ðtÞ, where N 1 ðtÞ ¼ ð0; n21 ; n31 Þ , N 2 ðtÞ ¼ ðn12 ; n22 ; 0Þ , i ¼ 1; 2. Define the sections S 01 ¼ fz ¼ h1 ðT 1 Þ : jxj; jyj; juj < 2dg, S 02 ¼ fz ¼ h2 ðT 2 Þ : jxj; jyj; jvj < 2dg, S 11 ¼ fz ¼ h1 ðT 1 Þ : jxj; jyj; jvj < 2dg, S 12 ¼ fz ¼ h2 ðT 2 Þ : jxj; jyj; juj < 2dg, it is clear that S 0i and S 1i cross Ci at t = Ti and t = Ti, respectively, for i ¼ 1; 2. Consider the maps F 0i : q1i1 2 S 1i1 #q0i 2 S 0i and F 1i : q0i 2 S 0i #q1i 2 S 1i induced by the flow of (1.1), where S 10 ¼ S 12 , 1 q0 ¼ q12 . First, we may find the relationship between q01 ðx01 ; y 01 ; u01 Þ , q11 ðx11 ; y 11 ; v11 Þ and their new coordinates 0;3 1;2 1;3 1 q01 ð0; n0;2 1 ; n1 Þ , q1 ð0; n1 ; n1 Þ . Notice that
q01 ¼ ðx01 ; y 01 ; u01 Þ ¼ r1 ðT 1 Þ þ Z 1 ðT 1 ÞN 1 ðT 1 Þ; q11 ¼
ðx11 ; y 11 ; v11 Þ
¼ r1 ðT 1 Þ þ Z 1 ðT 1 ÞN 1 ðT 1 Þ;
0;3 N 1 ðT 1 Þ ¼ ð0; n0;2 1 ; n1 Þ ; 1;3 N 1 ðT 1 Þ ¼ ð0; n1;2 1 ; n1 Þ :
Based on the expressions of Z i ðT i Þ and Z i ðT i Þ, i ¼ 1; 2, one derives that the new coordinate of q01 as ( 0 33 32 1 0 n0;2 1 ¼ u1 w1 ðw1 Þ y 1 ; 32 1 0 n0;3 1 ¼ ðw1 Þ y 1 ; 0;3 1 x01 ¼ d þ w31 1 n1 d, and the new coordinate of q1 as ( 21 1 1 n1;2 1 ¼ ðw1 Þ x1 ;
ð2:9Þ
ð2:10Þ
1 23 21 1 1 n1;3 1 ¼ v1 w1 ðw1 Þ x1 ; 1;2 y 11 ¼ d þ w22 1 n1 d. Similarly, for
q02 ¼ ðx02 ; y 02 ; v02 Þ ¼ r2 ðT 2 Þ þ Z 2 ðT 2 ÞN 2 ðT 2 Þ; q12
¼
ðx12 ; y 12 ; u12 Þ
¼ r2 ðT 2 Þ þ Z 2 ðT 2 ÞN 2 ðT 2 Þ;
0;2 N 2 ðT 2 Þ ¼ ðn0;1 2 ; n2 ; 0Þ ; 1;2 N 2 ðT 2 Þ ¼ ðn1;1 2 ; n2 ; 0Þ ;
one gets the new coordinate of q02 is ( 13 1 0 n0;1 2 ¼ ðw2 Þ v2 ;
ð2:11Þ
0 12 13 1 0 n0;2 2 ¼ y 2 w2 ðw2 Þ v2 ; 0;1 1 x02 ¼ d þ w11 2 n2 d, and the new coordinate of q2 is ( 1 23 21 1 1 1 23 21 1 1 n1;1 2 ¼ u2 w2 ðw2 Þ x2 ¼ u0 w2 ðw2 Þ x0 ; 21 1 1 21 1 1 n1;2 2 ¼ ðw2 Þ x2 ¼ ðw2 Þ x0 :
ð2:12Þ
Putting z ¼ hi ðtÞ into (1.1), then r_ i ðtÞ þ Z_ i ðtÞN i ðtÞ þ Z i ðtÞN_ i ðtÞ ¼ ¼
f ðri ðtÞ þ Z i ðtÞN i ðtÞÞ þ gðri ðtÞ þ Z i ðtÞN i ðtÞ; lÞ f ðri ðtÞÞ þ Df ðri ðtÞÞZ i ðtÞN i ðtÞ þ gðri ðtÞ; 0Þ þ gz ðri ðtÞ; 0ÞZ i ðtÞN i ðtÞ þ gl ðri ðtÞ; 0Þl þ h:o:t::
By way of the fact r_ i ðtÞ ¼ f ðri ðtÞÞ, Z_ i ðtÞ ¼ Df ðri ðtÞÞZ i ðtÞ and gðz; 0Þ ¼ 0, it then follows that N_ i ðtÞ ¼ Z 1 i ðtÞg l ðri ðtÞ; 0Þl þ h:o:t:
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
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Integrating the above equation from Ti to Ti, we arrive at Z Ti N i ðT i Þ ¼ N i ðT i Þ þ Z 1 i ðtÞgl ðri ðtÞ; 0Þl dt þ h:o:t; T i
N i ðT i Þ; N i ðT i Þ are defined as before. It then produces that 0;j j n1;j 1 ¼ n1 þ M 1 l þ h:o:t;
j ¼ 2; 3;
n1;k 2
k ¼ 1; 2;
¼
n0;k 2
þ
M k2 l
þ h:o:t;
ð2:13Þ
where M j1 ¼ M k2
¼
R T1
/j 1 gl ðr1 ðtÞ; 0Þ dt;
j ¼ 2; 3;
/k g ðr ðtÞ; 0Þ dt; T 2 2 l 2
k ¼ 1; 2:
T 1
R T2
Thus the maps 0;3 1;2 1;3 ð0; n0;2 1 ; n1 Þ ! ð0; n1 ; n1 Þ ; 0;1 0;2 1;1 1;2 ðn2 ; n2 ; 0Þ ! ðn2 ; n2 ; 0Þ ;
F 11 : S 01 ! S 11 ; F 12 : S 02 ! S 12 ; are defined by
0;j j n1;j 1 ¼ n1 þ M 1 l þ h:o:t;
j ¼ 2; 3;
0;k k n1;k 2 ¼ n2 þ M 2 l þ h:o:t;
k ¼ 1; 2
ð2:14Þ
respectively. q1 ðlÞ q1 ðlÞ Denoted by b1 ðlÞ ¼ k11 ðlÞ, b2 ðlÞ ¼ k12 ðlÞ. For notational convenience, denote bi ¼ bi ðlÞ, i ¼ 1; 2. Let si , i ¼ 1; 2 be the 1
1
2
1
flying time from q1i1 to q0i , s1 ¼ ek1 ðlÞs1 , s2 ¼ eq2 ðlÞs2 . Using linearization of (1.1) at pi, we can easily obtain the map F 01 : S 10 ! S 01 ;
q10 ðx10 ; y 10 ; u10 Þ ! q01 ðx01 ; y 01 ; u01 Þ
x10
sb1 1 d;
as s1 d;
y 01
u10
k2 ðlÞ 1 k1 ðlÞ 1
s1 u01
ð2:15Þ
and the map F 02 : S 11 ! S 02 ;
q11 ðx11 ; y 11 ; v11 Þ ! q02 ðx02 ; y 02 ; v02 Þ
defined by x11
1 b2
s2 d;
y 02
s2 d;
v02
q2 ðlÞ 2 q1 ðlÞ 2
s2 v11
ð2:16Þ
ðs1 ; u01 Þ; ðs2 ; v11 Þ
if we neglect the higher order terms. are called Shilnikov coordinates. Consequently, combine with Eqs. (2.9), (2.11), (2.14), (2.15) and (2.16), the Poincare´ map F 1 ¼ F 11 F 01 : S 10 #S 11 can be defined as 0 33 32 1 b1 2 n1;2 1 ¼ u1 w1 ðw1 Þ s1 d þ M 1 l þ h:o:t; 32 1 b1 3 n1;3 1 ¼ ðw1 Þ s1 d þ M 1 l þ h:o:t
ð2:17Þ
and F 2 ¼ F 12 F 02 : S 11 #S 12 as n1;1 2
¼
q2 ðlÞ 2 1
1 q2 ðlÞ 1 ðw13 2 Þ s2 v1
þ M 12 l þ h:o:t; q2 ðlÞ 2 q1 ðlÞ 2
12 13 1 1 2 n1;2 2 ¼ s2 d w2 ðw2 Þ s2 v1 þ M 2 l þ h:o:t:
See Fig. 2.
ð2:18Þ
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
Fig. 2. Sections and Poincare Map.
Let Gi ¼ F i ðq1i1 Þ q1i , i ¼ 1; 2. Owing to (2.10), (2.12), (2.15), (2.16), (2.17) and (2.18), we have the successor functions Gi as follows 1 b
G21
¼
2 32 1 b1 21 1 2 u01 w33 1 ðw1 Þ s1 d ðw1 Þ s2 d þ M 1 l þ h:o:t;
G31
¼
1 b1 3 1 23 21 1 2 ðw32 1 Þ s1 d v1 þ w1 ðw1 Þ s2 d þ M 1 l þ h:o:t;
¼
1 q2 ðlÞ 1 ðw13 v1 2 Þ s2
G12 G22
1 b
q2 ðlÞ 2 1
¼
s2 d
k2 ðlÞ 1 k1 ðlÞ 1
1 21 1 s1 u01 þ w23 2 ðw2 Þ s1 d þ M 2 l þ h:o:t; q2 ðlÞ 2 1
13 1 q2 ðlÞ 1 w12 2 ðw2 Þ s2 v1
1 2 ðw21 2 Þ s1 d þ M 2 l þ h:o:t:
Clearly, there is an 1–1 correspondence between the solution of the equation ðG21 ; G31 ; G12 ; G22 Þ ¼ 0
ð2:19Þ
with s1 P 0, s2 P 0 and the heteroclinic loop, homoclinic loop and periodic orbit of (1.1).
3. Main results To study the bifurcations near C, we should consider the solution of (2.19). It is easy to see that equation ðG21 ; G31 Þ ¼ 0 always has the unique solution 1 b
2 32 1 b1 21 1 2 u01 ¼ w33 1 ðw1 Þ s1 d þ ðw1 Þ s2 d M 1 l þ h:o:t; 1 b
1 b1 3 23 21 1 2 v11 ¼ ðw32 1 Þ s1 d þ w1 ðw1 Þ s2 d þ M 1 l þ h:o:t:
This fact is essentially guaranteed by the transversality given in (H4). Putting u01 ; v11 in the above equations into ðG12 ; G22 Þ ¼ 0, then we obtain w23 2 s1
1
¼d
k2 1 1
k 33 32 1 b1 1 w21 2 s1 ðw1 ðw1 Þ s1 d
1
þ
1 b2 ðw21 1 Þ s2 d
M 21 lÞ
1
d
1 w21 2 M 2l
1
d
q2 2 1
13 1 q2 32 1 b1 w21 2 ðw2 Þ s2 ððw1 Þ s1 d
1 b2
21 1 3 þ w23 1 ðw1 Þ s2 d þ M 1 lÞ þ h:o:t; 1 1 2 s2 ¼ ðw21 2 Þ s1 d M 2 l þ h:o:t:
ð3:1Þ
and (3.1) is called the bifurcation equation. To investigate the existence of heteroclinic loop, homoclinic loop and periodic orbit, what we need to do is pondering on the existence of solutions to Eq. (3.1) such as s1 ¼ s2 ¼ 0, s1 ¼ 0; s2 > 0 or s1 > 0; s2 ¼ 0, and s1 > 0; s2 > 0, respectively. Theorem 3.1. Suppose conditions ðH 1 Þ–ðH 5 Þ are satisfied. Then (1) If rank ðM 12 ; M 22 Þ ¼ 2, then there exists an (l 2)-dimensional surface L12 ¼ fl : M 12 l þ oðjljÞ ¼ M 22 lþ oðjljÞ ¼ 0g with a normal plane span fM 12 ; M 22 g at l = 0, such that system (1.1) has a unique heteroclinic loop Cl ¼ Cl1 [ Cl2 near C if l 2 L12 and 0 < jlj 1. (2) If w23 2 –0 and l 2 L12 , 0 < jlj 1, then system (1.1) has no periodic orbit. That is to say, the heteroclinic loop and periodic orbit of system (1.1) cannot coexist.
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
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Proof 1. Assume (3.1) has a solution s1 ¼ s2 ¼ 0, then M j2 l þ h:o:t ¼ 0;
j ¼ 1; 2:
(1) If rank ðM 12 ; M 22 Þ ¼ 2, then L12 ¼ fl : M 12 l þ oðjljÞ ¼ M 22 l þ oðjljÞ ¼ 0g is an (l 2)-dimensional surface such that (3.1) has a solution s1 ¼ s2 ¼ 0 as l 2 L12 and 0 < jlj 1, i.e., system (1.1) has a unique heteroclinic loop near C. Clearly, L12 has a normal plane spanned by M 12 and M 22 at l = 0. (2) In the case w23 2 –0 and l 2 L12 , (3.1) produces s1 ¼
13 1 21 ðw23 2 w2 Þ ðw2 Þ
2 q2 q1 q2 2 2 1 q 1 q 2 2
s1
1 b1 ðw32 1 Þ s1
þ
21 w23 1 w1
h ib1 2 1 3 21 1 ðw2 Þ s1 þ d M 1 l þ h:o:t:
q2
Since q21 > 1 and 0 < jlj 1, it is obviously that the above equation has not any solution satisfying 0 < s1 1. Which 2 implies system (1.1) has no periodic orbit near C. This completes the proof. h Theorem 3.2. Assume the hypotheses ðH 1 Þ–ðH 5 Þ are valid, then the following propositions are true. (1) If M 12 –0 and M 22 –0, then there exists an (l 1)-dimensional surface 9 8 q2 q2 = < 2þ 1 2 1 1 b2 1 1 1 q q 21 13 1 2 2 3 2 2 2 þ ðw13 L21 ¼ l : W 21 , M 12 l þ dw23 1 ðw1 w2 Þ ðd M 2 lÞ 2 Þ ðd M 2 lÞ M 1 l þ h:o:t ¼ 0; M 2 l < 0 ; :
ð3:2Þ
with a normal vector M 12 and tangent to L12 at l = 0, such that system (1.1) has a unique homoclinic loop C21 connecting p1 when l 2 L21 and 0 < jlj 1. 2 (2) If M 22 –0 and M 12 þ w23 2 M 2 –0, then there exists an (l 1)-dimensional surface 9 8 k2 k2 = < 1 þb 1 1 1 21 k1 k1 1 1 1 23 2 33 32 1 1 21 2 2 2 21 2 L2 ¼ l : W 2 , M 2 l þ w2 M 2 l dw1 ðw1 Þ ðd w2 M 2 lÞ 1 þ ðd w2 M 2 lÞ 1 M 1 l þ h:o:t ¼ 0; w2 M 2 l > 0 ; : ð3:3Þ 1 2 1 with a normal M 12 þ w23 2 M 2 at l = 0, such that system (1.1) has a unique homoclinic loop C2 joining p2 as l 2 L2 and 0 < jlj 1.
Proof 2 (1) Suppose (3.1) has a solution s1 ¼ 0; s2 > 0. Then one obtains q2 2þ 1 1 b2
21 1 23 q2 dðw13 2 w1 Þ w1 s2
þ
q2 2 1
1 q2 3 ðw13 2 Þ s2 M 1 l
þ M 12 l þ h:o:t ¼ 0;
ð3:4Þ
s2 ¼ d1 M 22 l þ h:o:t: Thus under the conditions M 12 –0 and M 22 –0, one deduces from (3.4) and the implicit function theorem that there exists an (l 1)-dimensional surface L21 defined by (3.2), such that (3.1) has a solution s1 ¼ 0; s2 ¼ s2 ðlÞ > 0 as l 2 L21 and 0 < jlj 1. That is, system (1.1) has a loop C21 homoclinic to p1 as l 2 L21 and 0 < jlj 1. Obviously, L21 has a normal vector M 12 at l = 0, and hence it is tangent to L12 at l = 0. (2) Now if (3.1) has solution s1 > 0, s2 ¼ 0, one thus derives 2 w23 2 s1
1
¼d
k2 1 1
4sk11 ðw33 ðw32 Þ1 sb1 1 d w21 2 1 1
2 s1 ¼ d1 w21 2 M 2 l þ h:o:t:
3
M 21 lÞ
M 12 l5
þ h:o:t;
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It then follows that there exists an (l 1)-dimensional surface L12 given by (3.3) such that (3.1) has a solution s1 ¼ s1 ðlÞ > 0; s2 ¼ 0 as l 2 L12 and 0 < jlj 1. Equivalently, system (1.1) has a homoclinic loop C12 connecting p2 2 as l 2 L12 and 0 < jlj 1. There is no difficulty to see that L12 has a normal M 12 þ w23 2 M 2 at l = 0. h
The proof is then finished.
2 Remark 3.1. In the above theorem, the condition M 22 –0 guarantees that the regions of M 22 l < 0 and w21 2 M 2 l > 0 are 1 1 2 23 not empty. The conditions M 2 –0 and M 2 þ w2 M 2 –0 assure the existence of normal vectors.
Theorem 3.3. Assume that ðH 1 Þ–ðH 5 Þ hold, w23 2 –0 and 0 < jlj 1, then 2 2 23 (1) If M 12 –0 and M 22 –0, then system (1.1) has a unique periodic orbit near C as w21 2 > 0; M 2 l < 0, w2 W 1 < 0 (that is, 2 1 2 2 23 21 l is in the side of L1 pointed by the direction w2 M 2 ), and has no periodic orbit as w2 > 0; M 2 l < 0, w23 2 W1 > 0 2 1 2 23 (that is, l is in the side of L1 pointed by the direction w2 M 2 ), where W 1 is defined by (3.2). 2 2 21 (2) If M 22 –0 and M 12 þ w23 2 M 2 –0, then system (1.1) has a unique periodic orbit near C for w2 > 0; M 2 l > 0, 1 1 1 2 23 23 23 w2 W 2 < 0 (that is, l is in the side of L2 pointed by the direction w2 ðM 2 þ w2 M 2 ÞÞ, and has not any periodic 2 1 1 1 2 23 23 23 orbit for w21 2 > 0; M 2 l > 0, w2 W 2 > 0 (that is, l is in the side of L2 pointed by the direction w2 ðM 2 þ w2 M 2 ÞÞ, where W 12 is given by (3.3). 2 (3) If w21 2 < 0, M 2 l > 0, then system (1.1) has no periodic orbit.
Proof 3 2 (1) For the case w21 2 > 0; M 2 l < 0, we see that 1 1 2 s2 ¼ gðs1 Þ , ðw21 2 Þ s1 d M 2 l þ h:o:t > 0
if 0 6 s1 1. Substituting s2 into the first equation of (3.1), we attain F ðs1 ; lÞ ,
w23 2 s1
k2 1 1
k 1 w21 2 s1
32 1 b1 w33 1 ðw1 Þ s1
þ
1 ðw21 1 Þ
h
1 ðw21 2 Þ s1
1
d
ib1
M 22 l
2
1
þd
k2 1 1
k 2 1 w21 2 s1 M 1 l
2
h iq21 h i q 1 13 1 21 1 2 32 1 b1 2 ðw Þ ðw Þ ðw Þ s d M l s1 þ d1 M 31 l þ w21 1 2 2 2 2 1 2
h iq21 þb1 q 1 2 21 21 13 1 21 1 2 1 2 w ðw w Þ ðw Þ s d M l þ d1 w21 þ w23 1 1 2 1 2 2 2 2 M 2 l þ h:o:t ¼ 0: k21 k11
ð3:5Þ
q22 q12
1 21 2 23 Notice that > 1; > 1, we get F 0s1 ðs1 ; lÞ w23 2 . Suppose that w2 > 0. Then based on the fact F ð0; lÞ ¼ d w2 W 1 and 0 2 F s1 ðs1 ; lÞ w23 2 > 0, F ðs1 ; lÞ ¼ 0 has a unique small positive solution s1 ¼ s1 ðlÞ if W 1 < 0 and has no positive solution 2 2 as W 21 > 0, which shows that system (1.1) has a unique periodic orbit as w21 2 > 0; M 2 l < 0 and W 1 < 0 and has no peri2 2 21 odic orbit as w2 > 0; M 2 l < 0 and W 1 > 0. With similar arguments one may attain the results for the case w23 2 < 0. 2 (2) If w21 2 > 0; M 2 l > 0, it is clear that 1 2 s1 ¼ g1 ðs2 Þ , w21 2 ðs2 þ d M 2 lÞ þ h:o:t > 0
for 0 6 s2 1. Eliminating s1 in (3.1) one reaches k2
21 k11 þb1 1 23 1 1 2 1 32 1 33 2 1 Gðs2 ; lÞ , w23 2 s2 þ d w2 M 2 l þ d M 2 l ðw1 Þ w1 w2 ðs2 þ d M 2 lÞ
þ q2
k2
q2 2þ 1 1 1 b 1 b2 1 q2 2 1 21 2 23 21 13 1 ðw Þ s2 d M 1 l þ w1 ðw1 w2 Þ s2 1
k2
w21 2 ðs2
1
þd q2 2 1
1 q2 ðw13 2 Þ s2
n
M 22 lÞ
k11
o b1 1 21 ðw32 w2 ðs2 þ d1 M 22 lÞ þ d1 M 31 l þ h:o:t ¼ 0: 1 Þ
1 1 Due to q21 > 1, k11 > 1, one has G0s2 ðs2 ; lÞ w23 2 . Using Gð0; lÞ ¼ d W 2 and some similar arguments to the proof of (1), 2 1 we may complete the proof of (2).
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
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1 2 2 21 1 (3) Obviously, if w21 2 < 0, M 2 l > 0, then s2 ¼ ðw2 Þ s1 d M 2 l þ h:o:t shows that s1 > 0, s2 > 0 can not be the solution of (3.1). Therefore system (1.1) has no periodic orbit in this case. The proof is then completed. h
1 2 1 Theorem 3.4. Suppose the hypotheses ðH 1 Þ–ðH 5 Þ are valid, w23 2 ¼ 0, rank ðM 2 ; M 2 Þ ¼ 2 and b1 < b2 . Then for l 2 L12 , 0 < jlj 1, we have the following propositions.
(1) If w21 2 < 0, then system (1.1) has no periodic orbit. q22 k21 2 33 33 32 (2) For w21 2 > 0, w1 –0 and k1 þ b1 < q1 , system (1.1) has a unique (resp. no) periodic orbit if w1 w1 M 1 l > 0 (resp. 1 2 2 33 32 w1 w1 M 1 l < 0). a1 q22 k21 k21 3 2 33 a (3) For w21 2 > 0, w1 –0 and k1 þ b1 > q1 > k1 , system (1.1) has a unique periodic orbit in the region jM 1 lj jM 1 lj , 1
2
1
b q1 k1
a1
1 2 1 2 3 3 2 2 32 33 a w13 2 M 1 lM 1 l < 0 or in the region jM 1 lj jM 1 lj , w1 w1 M 1 l > 0, where a ¼ q2 k1 q1 k2 .
k21 k11
2 1
q22 q12
2 1
2 33 (4) For w21 þ b12 < , system (1.1) has a unique (resp. no) periodic orbit if w21 2 > 0, w1 ¼ 0 and 1 M 1 l > 0 (resp. 2 21 w1 M 1 l < 0). q22 q22 k21 33 1 (5) For w21 2 > 0, w1 ¼ 0 and q1 < k1 þ b2 < q1 þ b1 ; system (1.1) has a unique periodic orbit in regions 2
a1
1
2
q1 k 1
a1
2 3 3 2 2 21 2 1 a jM 31 lj jM 21 lj a , w13 2 M 1 lM 1 l < 0 and jM 1 lj jM 1 lj , w1 M 1 l > 0, where a ¼ ðq2 k1 q1 k2 Þb . k2
2 1
q2
2 1
2
3 3 32 32 1 2 (6) For w21 2 > 0 and k1 > q1 þ b1 , system (1.1) has a unique (resp. no) periodic orbit if w1 M 1 l < 0 (resp. w1 M 1 l > 0). 1 q2 2 k2 a1 q22 2 21 2 1 (7) For w2 > 0 and q1 < k1 < q1 þ b1 , system (1.1) has a unique periodic orbit in regions jM 1 lj jM 31 lj a , 1 1 2 1 2 a1 q2 k 1 b1 2 3 2 3 3 32 a w13 2 M 1 lM 1 l < 0 and jM 1 lj jM 1 lj , w1 M 1 l < 0, where a ¼ q1 k2 q2 k1 . 2 1
2 1
Proof 4. For w23 2 ¼ 0 and l 2 L12 , (3.1) becomes that k2 1 k1 1
s1
32 1 b1 w33 1 ðw1 Þ s1
1
þ
1 b2 ðw21 1 Þ s2
q2 2 1 q1 1 1 32 1 b1 23 21 1 b2 3 2 d M 21 l ðw13 Þ s ðw Þ s þ w ðw Þ s þ d M l ¼ 0; 2 1 2 2 1 1 1 1
1
ð3:6Þ
1 s2 ¼ ðw21 2 Þ s1 :
(1) One may easily get the result based on the second equation of (3.6). 21 1 When w21 2 > 0, we may see that s2 P 0 for s1 P 0. Putting s2 ¼ ðw2 Þ s1 into the first equation of (3.6), one attains k2 1 þb 1 1
32 1 k1 w33 1 ðw1 Þ s1
þ
1
k2 1þ 1 1 b2
1 21 b2 k1 ðw21 s1 1 Þ ðw2 Þ
13 21 1 21 w23 1 ðw2 w1 Þ ðw2 Þ
q2 q2 2 1 ð 21 þb1 Þ q1 þb2 q 2 2 2
s1
2
1
d
s1 M 21 l
1 þb 1 1
q q2 k1 2 @sk1 (2) If w33 1 þ b1 < q1 , then s1 ¼ o 1 1 –00and k1 0 1 1 2 2 2 2 2 k k q k 1þ 1 k1 b2 1
s1
1 þb 1 k1
¼ o@s11
k2 1 þb 1 1
32 1 k1 w33 1 ðw1 Þ s1
2þ 1 q1 b2 2
1 þb 1 k1
A; s 1
1
k2 1 k1 1
1
k2 1 k1 1
¼ o@s11
1 21 ðw13 2 dÞ ðw2 Þ 0 1 q2 k2 2 1
2
k2 1 k1 1
32 1 21 ðw13 2 w1 Þ ðw2 Þ
2 q2 q2 2 1 þb1 q1 q2 2
s1
2 q2 q2 21 q1 q 2 2
s1 M 31 l ¼ 0: 0 k2 q2 2 þb 1 1
A; sq2 1
1 þb 1 1
k ¼ o@s11
ð3:7Þ 1 A. And the condition b1 < 1 shows that b2
A: Therefore, (3.7) becomes
d s1 M 21 l þ h:o:t ¼ 0; 32 2 b1 w M l 2 32 which has a unique solution s1 ¼ 1dw331 1 þ h:o:t > 0 as w33 1 w1 M 1 l > 0. This means that the heteroclinic loop and a 1 simple periodic orbit of system2 (1.1) can2 coexist. q2 k1 k21 (3) In the case w33 1 –0 and k1 þ b1 > q1 > k1 , it then follows from (3.7) that 1
k2 1 þb 1 k1 1
32 1 w33 1 ðw1 Þ s1
d
s1 M 21 l
2
1
1 21 ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q1 q2 2
s1 M 31 l þ h:o:t ¼ 0:
Now we want to look for the positive solution of (3.8). Multiplying (3.8) by s1 1 32 1 b1 2 13 1 21 w33 1 ðw1 Þ s1 d M 1 l ðw2 dÞ ðw2 Þ
2 2 q2 q2 k1 21 q1 k1 q 2 1 2
s1
M 31 l þ h:o:t ¼ 0:
k2 1 k1 1
ð3:8Þ
results in
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075 q2 k2 2 1 q1 k1 2 1
Denote t ¼ s1
b q1 k1
2 1 and a ¼ q2 k11 q 1 k2 > 1, it then gives rise to 2 1 2 1 2 3 2 q
32 1 a 4 13 1 21 w33 1 ðw1 Þ t ðw2 dÞ ðw2 Þ
2 q1 2
M 31 l5t d1 M 21 l þ h:o:t ¼ 0:
ð3:9Þ
a1
2 3 In the region jM 31 lj jM 21 lj a and w13 2 M 1 lM 1 l < 0, (3.9) has a small positive solution t ¼ fying 0 < t 1. One may see that
2 a q2
2
33 32 1a
33 32 1 13 21 q12 a
M 1 l
w1 ðw1 Þ t w1 ðw1 Þ ðw2 ðw2 Þ Þ
3 d1 M 21 l ;
M 1l
2
2
q2 2 1 21 Þq2 M 2 l w13 ðw 2 1 2 M 31 l
2
þ h:o:t satis-
a1
q k k 3 2 a which means t is reasonable. This shows that when k11 < q21 < k11 þ b1 and w33 and 1 –0, if jM 1 lj jM 1 lj 1 2 1 2 3 13 w2 M 1 lM 1 l < 0, then the heteroclinic loop and periodic obit of system (1.1) coexist. 32 2 1a a1 w M l 2 33 þ h:o:t > 0 as w32 Now suppose jM 31 lj jM 21 lj a , then (3.9) has a solution t ¼ 1dw331 1 w1 M 1 l > 0. Clearly, 1 0 < t 1, t is reasonable since
32 2 1a q2 q2
13 1 21 q21
2 a1
13 1 21 q21 3
ðw dÞ ðw Þ 2 M lt ðw dÞ ðw Þ 2
M l a w1 M 1 l d1 M 2 l : 2 1
2 1
1
2
2
dw33
1 k2
q2
k2
3 2 Consequently, for k11 < q21 < k11 þ b1 and w33 1 –0, if jM 1 lj jM 1 lj 1 2 1 periodic obit of system (1.1) can coexist.
(4) For w33 1 ¼ 0 and 1 21 ðw21 1 Þ ðw2 Þ
b1 2
k21
a1 a
q2
þ b12 < q21 , it follows from (3.7) that
k11
k2 1þ 1 k1 b2 1
s1
2 k2 1 k1 1
d1 s1 M 21 l þ h:o:t ¼ 0:
ð3:10Þ 1
w21 ðw21 Þb2 M 21 l 1 2 d
Thus, (3.10) has a unique small positive solution s1 ¼
eroclinic loop and periodic orbit of system (1.1) coexists if (5) If
w33 1
¼ 0 and
q22 q12
2 33 and w32 1 w1 M 1 l > 0; hence the heteroclinic loop and
<
k2 1þ 1 k1 b2 1
1 1 21 b2 s1 ðw21 1 Þ ðw2 Þ
k21 k11
þ
1 b2
<
q22 q12
k21 k11
!b2 2 þ h:o:t if w21 1 M 1 l > 0. In another word, the hetq2
2 21 þ b12 < q21 , w33 1 ¼ 0 and w1 M 1 l > 0 hold. 2
þ b1 ; (3.7) yields
k2 1 k1 1
1
21 d1 s1 M 21 l ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q 1 q 2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:11Þ
k2 1 k1 1
In order to find positive solutions of (3.11), multiplying (3.11) by s1 , one derives 1
1 1 21 b2 b2 ðw21 s1 1 Þ ðw2 Þ q2 k 2 2 1 q1 k 1 2 1
Let t ¼ s1
,a¼
1 21 ðw21 1 Þ ðw2 Þ
1
21 d1 M 21 l ðw13 2 dÞ ðw2 Þ
q12 k11
ðq22 k11 q12 k21 Þb2 2 b1 2
2 2 q2 q2 k 1 2 1 k1 q 1 q 2 1 2
s1
M 31 l þ h:o:t ¼ 0:
ð3:12Þ
> 1, (3.12) thus becomes
1 21 ta 4ðw13 2 dÞ ðw2 Þ
q2 2 q1 2
3 M 31 l5t d1 M 21 l þ h:o:t ¼ 0:
With a similar way to discuss (3.9), we may easily get the results. q2 k2 (6) If k11 > q21 þ b1 , then (3.7) takes the form 1
2
32 1 21 ðw13 2 w1 Þ ðw2 Þ
q2 2 q1 2
q2 2 þb 1 q1 2
s1
1
21 þ ðw13 2 dÞ ðw2 Þ
2 q2 q2 2 1 q 1 q 2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:13Þ
F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
w32 M 3 l
b1
2073 q2
k2
3 2 1 (3.13) has a unique positive solution s1 ¼ 1 d 1 1 þ h:o:t if w32 1 M 1 l < 0. Which tells that when q1 þ b1 < k1 , the het2 1 3 32 eroclinic and periodic orbit coexists if w1 M 1 l < 0. q22 q22 k21 (7) If q1 < k1 < q1 þ b1 , then (3.7) turns out to be 2
1
2
32 1 21 ðw13 2 w1 Þ ðw2 Þ
2 q2 q2 2 1 þb1 q1 q2 2
s1
q2 21 q 2
Multiplying (3.14) by s1 32 1 21 ðw13 2 w1 Þ ðw2 Þ
q2 2 q1 2
1
k2 1 k1 1
1
21 þ d s1 M 21 l þ ðw13 2 dÞ ðw2 Þ k2 q2 1 2 k1 q1 1 2
and denote by t ¼ s1
2 q2 q2 2 1 q1 q2 2
s1 M 31 l þ h:o:t ¼ 0:
ð3:14Þ
q1 k 1 b
1 1 , a ¼ q1 k22 q 2 k1 > 1, it then becomes 2 1
2 1
q2
2 1 21 q12 ta þ d1 M 21 l t þ ðw13 M 31 l þ h:o:t ¼ 0 2 dÞ ðw2 Þ
The claim now can be confirmed like doing for (3.9).
h
Remark 3.2. We only consider the case b1 < b12 in Theorem 3.4, in fact, one may achieve the similar results for b1 P b12 . It then shows that, comparing to the non-heterodimensional heteroclinic cycles, heterodimensional cycles have different bifurcation behaviors, since it is well known that, in the former case, a persistent heteroclinic loop can coexist with a bifurcated periodic orbit only when the original cycle is resonant, that is, b1 b2 ¼ 1. At last, we consider the existence of 2-fold periodic orbit for system (1.1). In case w23 2 –0 and b1 > 1, (3.1) reduces to w23 2 s1
¼
s2 ¼ w21 2
q2 2
1 21 q12 dw13 w 2 s2 2
1
21 1 b12 3 1 23 w1 w1 s2 d þ M 1 l d1 w21 2 M 2 l þ h:o:t , f ðs2 Þ
s1 d1 M 22 l þ h:o:t:
Then q2 þk1 q1 2 2 2 q1 2
23 ðq2 þ k1 Þw21 2 w1 s2 f ðs2 Þ ¼ 2 1 213 21 q2 w2 w1
q2 q1 2 2
q2 w21 M 3 l q1 2 21 131 s2 2 þ h:o:t: dq2 w2 b2 q22 w21 M 31 l 0 1 In case w23 þ h:o:t , s, which in turn produces 1 –0, f ðs2 Þ ¼ 0 has the solution s2 ¼ dðq2 þk1 Þw23 0
2
q2 2 q1 2
2
1
21 1 1 1 21 1 sb2 d þ M 31 l d1 w21 s w23 f ðsÞ ¼ ðdw13 2 Þ w2 1 w1 2 M 2 l þ h:o:t ! ! q2 2 w21 w23 d q22 w21 M 31 l q1 3 1 2 1 1 þ M 1 l d1 w21 ¼ 13 s 2 2 2 M 2 l þ h:o:t w21 dw2 d q2 þ k12 w23 1 1 q2
2 w21 k1 M 3 l 1 1 ¼ 22 2 1 1 13 sq2 d1 w21 2 M 2 l þ h:o:t d q2 þ k2 w2 q2 þk1
k1 w23 w21 2 1 2 1 ¼ 22 123 221 s q2 d1 w21 2 M 2 l þ h:o:t: q2 w2 w1 2 1 1 2 q2 2q1 3 23 q2 þk2 2q2 2 2 q þ k12 q22 þ k12 q12 w21 q22 q22 q12 w21 2 w1 2 M 1l q1 q1 2 2 s s þ h:o:t f 00 ðsÞ ¼ 2 13 1 2 13 21 1 2 ðq2 Þ w2 w1 ðq2 Þ dw2 ! ! 2 2 q2 2q1 1 1 1 2 2 2 3 w23 w21 q22 w21 M 31 l 1 d q2 þ k2 q2 þ k2 q2 q1 2 1 2 1 s 2 þ q2 q2 q2 M 1 l þ h:o:t 2 ¼ 2 w21 d q2 þ k12 w23 ðq12 Þ dw13 1 1 2 ¼
3 2 1 w21 2 q2 k2 M 1 l
q2 2q1 2 2 q1 2
s þ h:o:t 2 ðq12 Þ dw13 2 2 1 1 2 21 q2 þk2 2q2 q2 þ k12 k12 w23 1 w2 q1 s 2 þ h:o:t: ¼ 2 13 ðq12 Þ w21 1 w2 Hence, we may write f(s2) as
ð3:15Þ
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F. Geng et al. / Chaos, Solitons and Fractals 39 (2009) 2063–2075
1 f ðs2 Þ ¼ f ðsÞ þ f 00 ðsÞðs2 sÞ2 þ h:o:t: 2 Putting the second equation of (3.1) into the above equation, one has for w23 2 –0 that h i2 1 1 1 2 sÞ þ f 00 ðsÞ ðw21 s þ h:o:t; w23 2 s1 ¼ f ð 2 Þ s1 d M 2 l 2 which in turn yields AðlÞ þ BðlÞs1 þ s21 þ h:o:t ¼ 0;
ð3:16Þ
where 2 2 2 00 AðlÞ ¼ 2ðw21 sÞÞ1 f ðsÞ þ w21 s þ d1 M 22 l 2 Þ ðf ð 2 and 21 2 00 BðlÞ ¼ 2w23 sÞÞ1 2w21 s þ d1 M 22 l : 2 ðw2 Þ ðf ð 2 1 2 2 21 1 In the next theorem, we assume w21 2 > 0; M 2 l < 0, which guarantees s2 ¼ ðw2 Þ s1 d M 2 l þ h:o:t > 0 for 0 6 s1 1. It is enough to consider the nonnegative solution of (3.16) for investigating the homoclinic loop and periodic orbit bifurcated from C. 1 3 2 21 23 21 2 1 Theorem 3.5. Assume ðH 1 Þ–ðH 6 Þ hold, b1 > 1, w23 2 –0, w1 w1 M 1 l < 0, w2 > 0, M 2 l < 0. If q2 þ k2 < 2q2 , then the following propositions are true for 0 < jlj 1.
(1) If AðlÞ > 0; BðlÞ > 0, then system (1.1) has no periodic orbit and homoclinic orbit. (2) If AðlÞ < 0, then system (1.1) has a unique periodic orbit. (3) For AðlÞ > 0, then (i) if D , B2 ðlÞ 4AðlÞ > 0, BðlÞ < 0, then system (1.1) has two simple periodic orbits; (ii) if D , B2 ðlÞ 4AðlÞ ¼ 0, BðlÞ < 0, then system (1.1) has a 2-fold periodic orbit; (iii) if D , B2 ðlÞ 4AðlÞ < 0, system (1.1) has no periodic orbit. Proof 5. The results may be achieved from (3.16) directly.
h
Remark 3.3. The condition q22 þ k12 < 2q12 in Theorem 3.5 guarantees that 0 < jAðlÞj; jBðlÞj 1, which in turn means the positive solutions for (3.16) are small enough. 4. Conclusions In this paper, we discuss the bifurcations of heterodimensional cycles with two saddles. By using the method initiated by [15,16], we obtain the existence of one heteroclinic loop, one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbits, and the coexistence of one periodic orbit and heteroclinic loop. The most interesting is the coexistence of one periodic orbit and heteroclinic loop, since it is well known that, in the case of non-heterodimensional heteroclinic cycles, a persistent heteroclinic loop can not coexist with a bifurcated periodic orbit unless the original cycle is resonant. So, heterodimensional cycles have different bifurcation behaviors to the case of non-heterodimensional cycles. Acknowledgement The authors were supported by National Natural Science Foundation of China (]10671069).
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