^{1}

^{*}

^{1}

In this paper, external bifurcations of heterodimensional cycles connecting three saddle points with one orbit flip, in the shape of “∞”, are studied in three-dimensional vector field. We construct a poincaré return map between returning points in a transverse section by establishing a locally active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the bifurcation equations are obtained under different conditions. Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with
*P*
_{1} and
*P*
_{3}. The coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.

In recent years, bifurcation theory has been widely concerned due to its importance in practical applications (see [

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

Then Zhang [

− 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. [

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 [

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 [

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.

(H_{1}) (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).

(H_{2}) (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.

(H_{3}) (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 [

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 [

(H_{4}) (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 H_{1}, 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 (H_{4}) shows that W p i u and W p i s have strong inclination property. Due to the assumption (H_{2}), 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 [

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

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

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

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

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

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

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

Substituting them into

Remark 2.1. In fact,

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

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

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 (A_{1})-(A_{4}). 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 _{2}). So in the following, we need to consider solutions

Corresponding results about the existence of the second heterodimensional cycle, the third heterodimensional cycle and large-1 heteroclinic cycle, as well as the coexistence of double heterodimensional cycle and the large 1-heteroclinic cycle are contained in the next theorems. For convenience to discuss, we set eight regions:

From the discussion of Theorem 1, if one of

Since the first two equations of Equation (2.11) have the same structure as the last two, we only analyze the first and second equations as following

Set

where

Then we have

If

1) If

2) If

Without loss of generality, we discuss the case

where

When

with a normal surface

3) If

a) As

It has a solution

for

b) As

there is a small positive solution

So system (1.6) has a heteroclinic orbit consisting of

4) If

Set

when

When

for

When

5) If

To solve the first equation of (3.5), there is

we can get two solutions

Putting the expression

Remark 3.1. The analysis of the third and fourth equations of (20) is similar to the above analysis process, so it will not be repeated here.

With the analysis above, we can get the following theorems about existence of the second and the third shape heterodimensional cycle and the large-1 heteroclinic cycle under small perturbation.

Theorem 3.1. Under (H_{1})-(H_{4}) and Rank

1) If

2)If

3) If

with normal vector

4)If

and

such that the system (1.6)has the second shape heterodimensional cycle near

An alternative explanation for the existence of the second heterodimensional cycle is as follows. If there is an orbit starting from the section

Theorem 3.2. Suppose that (H_{1})-(H_{4}) hold and Rank

with a normal plane

Proof. As we explained above,

If

when

Corresponding, some new orbits

where

Where

Remark 3.2. The second heterodimensional cycle consists of two saddles of (1.2) type and one saddle of (2.1) type and is composed of one big orbit linking

Remark 3.3. As for the other theorem of the similar second shape heterdimensional cycle which consists of two saddles of (1.2) type and one saddle of (2.1) type and is composed of one big orbit linking

Theorem 3.3. Suppose (H_{1})-(H_{4}) are valid and

1) If

with normal vector

2)If

and

both with normal vector

and

3) If

4) If

then the system (1.6)has one 2-fold large-1heteroclinic cycles near

For the alternative explanation from the gaps for the existence of the large-1 heteroclinic cycle is the following. If

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