In recent years, bifurcation theory has been widely concerned due to its importance in practical applications (see [1] [2] [3] [4] ) and in the study of traveling wave solutions for nonlinear partial differential equations. For example, in 2018, Zilburg and Rosenau [5] studied the qualitative properties of solitons of a dKdV equation,

∂ t u + ∂ x ( u ∂ x u ) + u 2 = 0. (1.1)

Then Zhang [6] analyzed (1.1) in the idea of bifurcation theory of dynamical system. Roughly to speak, he first set the variable transform u ( x , t ) = ϕ ( x − c t ) = ϕ ( ξ ) to make system (1.1) be

− c ϕ ′ + ( ϕ ( ϕ ϕ ′ ) ′ + ϕ 2 ) ′ = 0 , (1.2)

then integrated (1.2) and got

− c ϕ + ϕ ϕ ′ 2 + ϕ 2 ϕ ″ + ϕ 2 = g , (1.3)

where g is the integral constant, and system (1.3) is equivalent to the following regular plane system with d ξ = ϕ 2 d τ

{ d ϕ d τ = y ϕ 2 d y d τ = g + c ϕ − ϕ y 2 − ϕ 2 . (1.4)

Clearly the Hamiltonian of system (1.4) is

H ( ϕ , y ) = 1 2 y 2 ϕ 2 − g ϕ − 1 2 c ϕ 2 + 1 3 ϕ 3 = h . (1.5)

From H ( ϕ , y ) | S 1 = h 1 , a heteroclinic orbit is found as

y 2 = 2 h 1 + 2 g ϕ + c ϕ 2 − 2 3 ϕ 3 ϕ 2

for g = 0 , c > 0 , and the existing condition is given in some circumstances on two sides of the nonresonant heteroclinic bifurcation.

In fact, different kinds of high co-dimensional homoclinic or heteroclinic bifurcations have been discussed extensively. [7] described a phenomenon that occurred in the bifurcation theory of one-parameter families of diffeomorphisms. If all the equilibrium points of the orbit have the same dimension number of the stable manifold, the heteroclinic cycle is named as an equidimensional loop, otherwise, a heterodimensional. However, since different equilibrium points in n-dimensional systems do not necessarily have stable manifolds of the same dimension, the problem of heterodimensional loop is more general and practical than that of equidimensionals. Jens D.M in [8] considered a self-organized periodic replication process of travelling pulses which has been observed in reaction-Cdiffusion equations, and studied homoclinic orbits near codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions. Bykov analyzed the bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points (see [9] ). [10] studied the bifurcations of heterodimensional cycles with the connection of two hyperbolic saddle points and strong inclination flip in a four-dimensional system, they presented the conditions for the existence, coexistence and noncoexistence of the heterodimensional orbit, homoclinic orbit and periodic orbit, as well as the co-existence of heterodimensional orbit and homoclinic orbit and obtained some new features from the inclination flip in some bifurcation surfaces. Xu and Lu discussed heterodimensional loop bifurcation with orbit flip and inclination flip respectively in [11] [12] [13], and got the coexistence region of coexisting loop and periodic orbit. Meanwhile, they also constructed an example to provide a good reference for their main bifurcation problems. Specially, Liu’s team fabricated a model of heterodimensional cycles to verify their main bifurcation results (see [14] [15] [16] [17] ).

However in the study of systems with homoclinic loop or heteroclinic loop, few scholars focused on double heteroclinic bifurcation of three saddle points. We only found that [18] considered the bifurcation problem of rough heteroclinic loops connecting three saddle points, but not a “∞”-type, for a higher-dimensional system and [19] concerned “∞”-type double homoclinic loops, but not heteroclinic loops, with resonance characteristic roots in the common case and in a four-dimensional system to obtain the complete bifurcation diagram under different conditions. In this paper, we consider the bifurcation problem of double heteroclinic loops of ∞-type connecting three saddle points with four orbits. In addition, we also give an example model to demonstrate the existence of the bifurcation results.

It’s worth noting that, in the previous studies about homoclinic and heteroclinic loop bifurcations, few scholars focused on double heterodimensional cycles bifurcations of three saddle points. Jin and Zhu [18] considered the bifurcation problem of rough heteroclinic loop connecting three saddle points in a higher-dimensional system, but the loop is not a “∞”-type. [20] [21] [22] [23] discussed the heteroclinic loops with two saddle points, but the loops are not heterodimensional cycles. Lu and Liu et al. [10] [11] [13] studied the heterodimensional cycle, but the cycle is also neither a “∞”-type nor double. Jin et al. [19] [24] considered “∞”-type double homoclinic loops, but the loops are not heteroclinic or do not connect with three saddle points. Since heterodimensional or heteroclinic cycles are very normal and have applications in solitary wave problems and biology systems, see Kalyan Manna et al. [25] for example, and also for the completeness of theoretical research of heteroclinic bifurcation, in this paper, we focus on the double heterodimensional cycles in ∞-type with three saddle points.

The rest of the paper is structured as follows. In Section 2, through establishing a local moving frame system near the unperturbed heterodimensional cycle to obtain the Poincaré map and the successor function, we induce the bifurcation equations by using the implicit function theorem. Section 3 will show the bifurcation results on different parameter regions by analyzing the bifurcation equation.

The C r system to be studied is

Z ˙ = f ( z ) + g ( z , μ ) , (1.6)

where z ∈ R 3 , μ ∈ R l , l > 4 , | μ | ≪ 1 , f , g ∈ C r , r ≥ 4 . Specially, when μ = 0 , the unperturbed system associated with (1.6) is

Z ˙ = f ( z ) . (1.7)

satisfies the following hypotheses.

(H₁) (Hyperbolic) z = p i ( i = 1 , 2 , 3 ) are hyperbolic critical points of (1.7) such that g ( p i , μ ) = g ( p i , 0 ) = 0 for all i, and dim ( W 1 s ) = dim ( W 2 u ) = dim ( W 3 u ) = 2 where 0 means a zero vector. In addition, the linearization matrix D z f ( p i ) has a simple real eigenvalues: − ρ i 2 , − ρ i 1 , λ i 1 ( i = 1 , 3 ) , − ρ 2 1 , λ 2 1 , λ 2 2 satisfying

− ρ i 2 < − ρ i 1 < 0 < λ i 1 , − ρ 2 1 < 0 < λ 2 1 < λ 2 2

Throughout the paper we assume that system (1.7) is of at least C 3 uniformly linearizable. What’s more, there is a small neighborhood U i ( i = 1 , 2 , 3 ) of the equilibrium p i and a C 3 diffeomorphism depending on the parameter in C 3 manner, then we can use successively straightening transformations including the straightening of some orbit segments such that system (1.7) has the following C k normal in U i : as z = ( x , y , v ) * ∈ U i , i = 1 , 3

x ˙ = [ λ i 1 ( μ ) + o ( 1 ) ] x , y ˙ = [ − ρ i 1 ( μ ) + o ( 1 ) ] y + O ( v ) ( O ( x ) + O ( v ) ) , v ˙ = [ − ρ i 2 ( μ ) + o ( 1 ) ] v + O ( y ) ( O ( x ) + O ( y ) ) , (1.8)

and as z = ( x , y , u ) * ∈ U 2

x ˙ = [ λ 2 1 ( μ ) + o ( 1 ) ] x + O ( u ) ( O ( y ) + O ( u ) ) y ˙ = [ − ρ 2 1 ( μ ) + o ( 1 ) ] y , u ˙ = [ λ 2 1 ( μ ) + o ( 1 ) ] u + O ( x ) ( O ( x ) + O ( y ) ) , (1.9)

where k ≥ r − 2 , the sign “ ∗ ” stands for transposition. For ‖ u ‖ sufficiently small, where λ i 1 ( μ ) = λ i 1 , ρ i 1 ( μ ) = ρ i 1 , ρ i 2 ( μ ) = ρ i 2 ( i = 1 , 3 ) , λ 2 1 ( μ ) = λ 2 1 , λ 2 1 ( μ ) = λ 2 1 , ρ 2 1 ( μ ) = ρ 2 1 is the corresponding eigenvalues of the linearization matrix of perturbed system (1.6).

(H₂) (non-degeneration) System (1.7) has a double heterodimensional cycles γ = γ 1 ( t ) ∪ γ 2 ( t ) ∪ γ 3 ( t ) ∪ γ 4 ( t ) , where Γ i = { z = γ i ( t ) : t ∈ R } , γ 1 ( + ∞ ) = r 2 ( − ∞ ) = p 1 , γ 1 ( + ∞ ) = γ 2 ( − ∞ ) = γ 3 ( − ∞ ) = γ 4 ( + ∞ ) = p 2 , γ 3 ( + ∞ ) = γ 4 ( − ∞ ) = p 3 , and

dim ( T γ 1 ( t ) W 1 u ∩ T γ 1 ( t ) W 2 s ) = 1 ,   dim ( T γ 2 ( t ) W 2 u ∩ T γ 2 ( t ) W 1 s ) = 1 ,

dim ( T γ 3 ( t ) W 2 u ∩ T γ 3 ( t ) W 3 s ) = 1 ,   dim ( T γ 4 ( t ) W 3 u ∩ T γ 4 ( t ) W 2 s ) = 1 ,

Here γ i ( t ) represents the flow of system (17), t ∈ R and by T q M we denote the tangent space of the manifoldM at q.

(H₃) (Orbit flip) Let e i ∓ = lim t → ∓ ∞ γ ˙ i ( − t ) / | γ ˙ i ( − t ) | , then

e 1 + ∈ T p 1 W 1 u ,   e 2 + , e 3 + ∈ T p 2 W 2 u ,   e 4 + ∈ T p 3 W 3 u ,

e 1 − , e 4 − ∈ T p 2 W 2 s ,   e 2 − ∈ T p 1 W 1 s ,   e 3 − ∈ T p 3 W 3 u ,

where e i + , e i − ( i = 1,2,3 ) are unit eigenvectors corresponding to λ i 1 and ρ i 1 ( i = 1 , 2 , 3 ) respectively. Furthermore they satisfy the equation e 1 − = − e 4 − , e 3 + = − e 2 + (for details see [19] ).

Here, e 2 + and e 2 − are the unit eigenvectors corresponding to λ 2 1 and − ρ 1 2 which responds Γ 2 enters the equilibrium p 1 along the strong stable manifold W 1 s s (as t → + ∞ , enters the equailibruium p 2 along the unstable manifold W 2 u (as t → − ∞ ), that is, from [17], the heteroclinic orbit Γ 2 has orbits flips when t → + ∞ (see Figure 1).

(H₄) (Strong inclination)

lim t → + ∞ T γ 1 ( t ) W 1 u = s p a n { e 1 − , e 2 + } ,   lim t → − ∞ T γ 1 ( t ) W 2 s = s p a n { e 1 + , e 2 − } ; lim t → + ∞ T γ 2 ( t ) W 2 u = s p a n { e 2 − } ,   lim t → − ∞ T γ 2 ( t ) W 1 s = s p a n { e 2 + } ; lim t → + ∞ T γ 3 ( t ) W 2 u = s p a n { e 3 − } ,   lim t → − ∞ T γ 3 ( t ) W 3 s = s p a n { e 3 + } ; lim t → + ∞ T γ 4 ( t ) W 3 u = s p a n { e 4 − , e 3 + } ,   lim t → − ∞ T γ 4 ( t ) W 2 s = s p a n { e 4 + , e 3 − } .

Remark 1.1. Under the assumption H₁, p 1 and p 3 have a 1-dimensional unstable manifold and a 2-dimensional stable manifold, while p 2 has a 2-dimensional unstable manifold and a 1-dimensional stable manifold, hence Γ is double heterodimensional cycles.

Remark 1.2. Hypothesis (H₄) shows that W p i u and W p i s have strong inclination property. Due to the assumption (H₂), p 2 has a 2-dimensional unstable manifold, p 3 has a 2-dimensional stable manifold, and dim ( T γ 3 ( t ) W 3 u ∩ T γ 2 ( t ) W 2 s ) = 1 ,we can know the codimension of the heteroclinic orbit Γ 3 is 0. Then the orbits Γ 3 is transversal, that is, they can be preserved even under small perturbations.

In this section, we need first to take fundamental solutions of linear variational Equation (see Equation (1.6) as below) and use them as an active coordinate system along the heteroclinic orbits. Then using the new coordinates, we construct the global map spanned by the flow of (1.6) between the sections along the orbits. Next, we set up local maps near equilibriums. Finally the whole Poincaré map can be obtained by composing these maps. The implicit function theorem reveals the bifurcation equation.

By the stable and unstable manifolds theorem and up to two local linear transformations, we see that there are three open neighborhoods U i of p i = ( 0,0,0 ) *

such that p i have C r − 1 local manifolds W i , l o c s and W i , l o c s ( i = 1 , 2 , 3 ) which are expressed as below: for j = 1 , 3 ,

W j , l o c u = { z = ( x , y , v ) * ∈ U j | ( y , v ) = ( y , v ) ( x ) , ( y , v ) ( 0 ) = 0 , ∂ ( y , v ) ∂ x ( 0 ) = 0 } ,

W j , l o c s = { z = ( x , y , v ) * ∈ U j | x = x ( y , v ) , x ( 0 , 0 ) = 0 , ∂ x ∂ ( y , v ) ( 0 , 0 ) = 0 } ,

W 2 , l o c u = { z = ( x , y , u ) * ∈ U 2 | y = y ( x , u ) , y ( 0 , 0 ) = 0 , ∂ y ∂ ( x , u ) ( 0 , 0 ) = 0 } ,

W 2 , l o c s = { z = ( x , y , u ) * ∈ U 2 | ( x , u ) = ( x , u ) ( y ) , ( x , u ) ( 0 ) = 0 , ∂ ( x , u ) ∂ y ( 0 ) = 0 } .

Let the coordinate expression of γ k ( t ) be γ k ( t ) = ( γ k x ( t ) , γ k y ( t ) , γ k v ( t ) ) * in the small neighborhood U i of p i , ( i = 1 , 3 ) , and γ k ( t ) = ( γ k x ( t ) , γ k y ( t ) , γ k u ( t ) ) * in the small neighborhood U 2 of p 2 . Since T k > 0   ( k = 1 , 2 , 3 , 4 ) is large enough so that γ 1 ( − T 1 ) , γ 2 ( T 2 ) ∈ U 1 , γ 3 ( T 3 ) , γ 4 ( − T 4 ) ∈ U 3 , γ 1 ( T 1 ) , γ 2 ( − T 2 ) , γ 3 ( − T 3 ) , γ 4 ( T 4 ) ∈ U 2 and for k = 1 , 3 , 4 , γ k ( − T k ) = ( δ ,0,0 ) * , γ 2 ( − T 2 ) = ( − δ ,0,0 ) * , for k = 3 , 4 , γ k ( T k ) = ( 0 , δ , 0 ) , r 1 ( T 1 ) = ( 0 , 0 , δ ) , r 2 ( T 2 ) = ( 0 , − δ , 0 ) , where δ > 0 is small enough.

Now we take into account the linearly variational system and its corresponding adjoint system of (1.7) formed respectively by: let A k ( t ) = D f ( γ k ( t ) ) ,

z ˙ = A k ( t ) z (2.1)

and

ϕ ˙ = − A k ( t ) * ϕ (2.2)

Based on the above hypotheses about system (1.7), system (2.1) has exponential dichotomies in R + and R − (see [12] ). We can obtain the following properties.

Lemma 2.1. System (2.1) has the fundamental solution matrices

Z k ( t ) = ( z k 1 ( t ) , z k 2 ( t ) , z k 3 ( t ) ) ( k = 1 , 2 , 3 , 4 )

which satisfy, respectively, for k = 1 , 4

z k 1 ( t ) , z k 3 ( t ) ∈ ( T γ k ( t ) Γ k ( μ ) ) c , z k 2 ( t ) = γ k ( t ) / | γ k ( T k ) | ∈ T γ k ( t ) W k u ∩ T γ k ( t ) W k − ( − 1 ) k s

that is

Z k ( − T k ) = ( 0 w k 21 0 0 0 1 1 0 0 ) ,   Z k ( T k ) = ( w k 11 0 w k 31 w k 12 ( − 1 ) k w k 32 w k 13 0 w k 33 ) (2.3)

where w k = | w k 11 w k 31 w k 13 w k 33 | ≠ 0 , | w k i 2 ⋅ w k − 1 | ≪ 1 , i = 1 , 3 , W 4 u = W 2 u .

And for k = 2 , 3

z k 1 ( t ) ∈ ( T γ k ( t ) W 2 u ) c ∩ T γ k ( t ) W k − ( − 1 ) k s , z k 2 ( t ) = r ˙ k ( t ) / | γ ˙ k ( − T k ) | ∈ T γ k ( t ) W 2 u ∩ T γ k ( t ) W k − ( − 1 ) k s , z k 3 ( t ) ∈ T γ k ( t ) W 2 u ∩ ( T γ k ( t ) W k − ( − 1 ) k s ) c ,

that is,

Z k ( − T k ) = ( 0 w k 21 0 w ¯ k 12 0 1 1 0 w k 33 ) Z 2 ( T 2 ) = ( 1 0 w 2 31 w 2 12 0 1 0 1 0 ) Z 3 ( T 3 ) = ( 1 0 0 0 w 3 22 w 3 32 w 3 13 0 1 ) (2.4)

where w 2 21 < 0 , w 3 22 ≠ 0 , | w 2 12 ⋅ w 2 31 | ≪ 1 , | w 3 13 ⋅ w 3 32 | ≪ 1 , | w k 33 ⋅ ( w k 21 ) − 1 | ≪ 1 , W 4 s = W 3 u .

In what follows, we select ( Z k 2 ( t ) , Z k 2 ( t ) , Z k 3 ( t ) ) ( k = 1 , 2 , 3 , 4 ) as a new local coordinate system along Γ k . Let θ k ( t ) = ( ϕ 1 ( t ) , ϕ 2 ( t ) , ϕ 3 ( t ) ) = ( Z − 1 ( t ) ) * be the fundamental solution matrix of (2.2). By the [1, ?], we can know that the ϕ k 1 ( t ) is bounded and tends to zero exponentially as t → ± ∞ .

Take a coordinate transformation

z ( t ) = h k ( t ) = γ k ( t ) + Z k ( t ) N k ( t ) , t ∈ [ − T k , T k ] , 0 < ε ≪ δ . (2.5)

in a small neighborhood of Γ k , where N k ( t ) = ( n k 1 ( t ) , 0 , n k 3 ( t ) ) * , k = 1 , 2 , 3 , 4 , and n k 1 ( t ) , n k 3 ( t ) represents the coordinate decomposition of (1.6) in the new local coordinate system corresponding to Z k 1 ( t ) and Z k 3 ( t ) , Then we can take eight transverse sections vertical to the tangency T γ k ( t ) to each orbit γ k ( t ) (see Figure 2)

S 1 0 = { z = h 1 ( − T 1 ) : | x | , | y − δ | , | v − δ | < ε } , S 2 0 = { z = h 2 ( − T 2 ) : − | x | , | y − δ | , | u − δ | < ε } ,

S 3 0 = { z = h 3 ( − T 1 ) : | x | , | y − δ | , | u − δ | < ε } , S 4 0 = { z = h 4 ( − T 4 ) : | x | , | y − δ | , | v − δ | < ε } ,

S 1 1 = { z = h 1 ( T 1 ) : | x − δ | , − | y | , | u − δ | < ε } , S 2 1 = { z = h 2 ( T 2 ) : | x − δ | , | y − δ | , | v | < ε } ,

S 3 1 = { z = h 3 ( T 3 ) : | x − δ | , | y | , | v − δ | < ε } , S 4 1 = { z = h 4 ( T 4 ) : | x − δ | , | y | , | u − δ | < ε }

In order to obtain the corresponding bifurcation equation, we need to restrict our attention to set up the Poincaré return map of system (1.6). Firstly, we find the relationship between the old coordinates

q k 0 ( x k 0 , y k 0 , u ¯ k 0 ) ,   q j 1 ( x j 0 , y j 0 , u ¯ k o )

and new coordinates

q i 0 ( n i 0,1 ,0, n i 0,3 ) ,   q i 1 ( n i 1,1 ,0, n i 1,3 )

where k = 2 , 3 , j = 1 , 4 ,   u ¯ k 0 = u k 0 ; k = 1 , 4 , j = 2 , 3 ,   u ¯ k 0 = v k 0 . Then, combining with the Equations (2.3), (2.4), we obtain for k = 1 , 4

{ n k 0 , 1 = v k 0 n k 0 , 3 = y k 0 x k 0 = δ (2.6)

and

{ n k 1 , 1 = w k − 1 ( w k 33 x k 1 − w k 31 u k 1 ) n k 1 , 3 = w k − 1 ( w k 11 u k 1 − w k 13 x k 1 ) y k 1 = δ + w k − 1 ( w k 12 w k 33 − w k 32 w k 13 ) x k 1 + w − 1 ( w k 31 w k 11 − w k 12 w k 31 ) u k 1 ≈ δ (2.7)

for k = 2 , 3

{ n k 0 , 1 = u k 0 − w k 33 y k 0 n k 0 , 3 = y k 0 − w k 12 u k 0 x k 0 = ( − 1 ) k − 1 δ (2.8)

and

{ n 2 1 , 1 = x 2 1 − w 2 31 y 2 1 n 2 1 , 3 = y 2 1 − w 2 12 x 2 1 v 2 1 = δ , { n 3 1 , 1 = x 3 1 − w 3 32 v 3 1 n 3 1 , 3 = v 3 1 − w 3 13 x 3 1 y 3 1 = δ (2.9)

Then, under transformation (2.5), system (1.6) has the following form by γ ˙ ( t ) = f ( γ ( t ) ) and Z ˙ k ( t ) = D f ( γ ( t ) ) Z k ( t ) :

N ˙ k ( t ) = θ k * ( t ) g μ ( γ k ( t ) , 0 ) μ + h . o . t , (2.10)

where g μ is the partial derivation of g ( z , μ ) with respect to μ . To integrate (2.10), we get

N k ( T k ) = N k ( − T k ) + ∫ − T k T k     θ k * ( t ) g μ ( γ k ( t ) , 0 ) μ d t + h . o . t . ≜ N k ( − T k ) + M k j μ + h . o . t . (2.11)

where M k j ( μ ) = ∫ − T k T k     θ k j * ( t ) g μ ( γ k ( t ) , 0 ) μ d t   ( j = 1 , 3 ; k = 1 , 2 , 3 , 4 ) are called Melnikov vectors respect to μ .

Which are defined as the global maps F k 1 : S k 0 → S k 1 ( k = 1 , 2 , 3 , 4 ) with the expression by (2.11) given

n ¯ k 1 , 1 = n k 0 , 1 + M k 1 μ + h . o . t . , n ¯ k 1 , 3 = n k 0 , 3 + M k 3 μ + h . o . t .. (2.12)

as follows

F k 1 ( n k 0 , 1 , 0 , n k 0 , 3 ) = ( n ¯ k 1 , 1 , 0 , n ¯ k 1 , 3 ) .

Next we consider the local maps,

F 1 0 : q 2 1 ∈ S 2 1 ↦ q 1 0 ∈ S 1 0 ,   F 2 0 : q 1 1 ∈ S 1 1 ↦ q 3 0 ∈ S 3 0 , F 3 0 : q 3 1 ∈ S 3 1 ↦ q 4 0 ∈ S 4 0 ,   F 4 0 : q 4 1 ∈ S 4 1 ↦ q 2 0 ∈ S 2 0

induced by flows confined in the neighborhood U i ( i = 1 , 2 , 3 ) .

Let τ 1 , τ 3 be the time spent from q 2 1 to q 1 0 and from q 3 1 to q 4 0 respectively, corresponding their Shilnikov time. Let be the time spent from to and from to, then their Shilnikov time are.

Then under the assumptions among the eigenvalues, by the normal forms (1.8)-(1.9), and the formula of variation of constants, we obtain the local maps:

Thus, by (2.6), (2.12) (2.13), we obtain the first Poincaré map as follows

by (2.8), (2.12), (2.14), we obtain the Poincaré map as follows

by (2.8), (2.12), (2.15), we obtain the Poincaré map as follows

by (2.6), (2.12), (2.16), we obtain the Poincaré map as follows

Then, by (2.7), (2.9), (2.17), (2.18), (2.19), (2.20), we induce the successor functions

where

By the implicit function theorem, solving the equation, we have

Substituting them into, we obtain the bifurcation equations, for

Remark 2.1. In fact, is independent of the choice of for, which can be verified similarly as in [13]

Remark 2.2. Generally, in two-dimensional plane system, when we study bifurcations of singular cycle, Poincaré mapping can only be established on one side of the singular cycle. Therefore, there are no other types of orbits except the one with infinite approaching to saddle point on the left side of and the right side of. However, in high-dimensional system, it remains to be verified whether other types of orbits can bypass different surfaces for connection. To make the study go on, we assume that, that is, the orbit starting from to just be a singular orbit which is infinitely approaching when; for the orbit starting from to is similar near.

Remark 2.3. Basing on remark 2.2, it can be seen that (2.13) and (2.14) become and .

Remark 2.4. Shilnikov variables were introduced by Shilnikov in 1968 to compute the local transition map near equilibria to leading order. Instead of solving an initial-value problem, solutions near the equilibrium are found using an appropriate boundary-value problem.

In this section, we analyze the bifurcation of system (1.6) under hypotheses (A₁)-(A₄). The existence of “∞”-shape double heterodimensional cycles, the heteroclinic cycle composed of three orbits and connecting with three saddle points, and large 1-heteroclinic connecting with and are studied by discussing the corresponding bifurcation equation. Clearly if the double heterodimensional cycle (“∞”) of system (1.6) is persistent; if, , system (1.6) has a heterodimensional cycle consisting of two saddles of (1.2) type and one saddle of (2.1) type composed of one big orbit linking and two orbits linking and respectively, which is called the second shape heterodimensional cycle in later of this paper; if, , system (1.6) has another heterodimensional cycle consisting of two saddles of (2.1) type and one saddle of (1.2) type composed of one big orbit linking and two orbits linking and respectively, which is called another second shape heterodimensional cycle in later of this paper; if and, system (1.6) has the large 1-heteroclinic cycle consisting of two saddles and of (2.1) type composed with two big orbits linking and respectively. What is noteworthy is that if the conditions make untenable and set tenable, the conditions make untenable and tenable, system (1.6) has the third heterodimensional cycle consisting of one saddle of (2.1) type and one saddle of (1.2) type and composed of one orbit starting from to and another orbit starting from to under the assumption (H₂). So in the following, we need to consider solutions and of the bifurcation Equation (2.21).

We gratefully acknowledge the reviewers for their patience in reading the first draft of this paper.

The authors were supported by National Natural Science Foundation of China (Grant No. 11871022).

The authors declare that they have no competing interests.

Dong, H.M. and Zhang, T.S. (2021) External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip. Applied Mathematics, 12, 348-369. https://doi.org/10.4236/am.2021.124025