In recent years, bifurcation theory has been widely concerned due to its importance in practical applications (see     ) and in the study of traveling wave solutions for nonlinear partial differential equations. For example, in 2018, Zilburg and Rosenau  studied the qualitative properties of solitons of a dKdV equation,
Then Zhang  analyzed (1.1) in the idea of bifurcation theory of dynamical system. Roughly to speak, he first set the variable transform to make system (1.1) be
then integrated (1.2) and got
where g is the integral constant, and system (1.3) is equivalent to the following regular plane system with
Clearly the Hamiltonian of system (1.4) is
From , a heteroclinic orbit is found as
for , 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.  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  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  ).  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   , 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     ).
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  considered the bifurcation problem of rough heteroclinic loops connecting three saddle points, but not a “∞”-type, for a higher-dimensional system and  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  considered the bifurcation problem of rough heteroclinic loop connecting three saddle points in a higher-dimensional system, but the loop is not a “∞”-type.     discussed the heteroclinic loops with two saddle points, but the loops are not heterodimensional cycles. Lu and Liu et al.    studied the heterodimensional cycle, but the cycle is also neither a “∞”-type nor double. Jin et al.   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.  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 system to be studied is
where . Specially, when , the unperturbed system associated with (1.6) is
satisfies the following hypotheses.
(H1) (Hyperbolic) are hyperbolic critical points of (1.7) such that for all i, and where means a zero vector. In addition, the linearization matrix has a simple real eigenvalues: satisfying
Throughout the paper we assume that system (1.7) is of at least uniformly linearizable. What’s more, there is a small neighborhood of the equilibrium and a diffeomorphism depending on the parameter in manner, then we can use successively straightening transformations including the straightening of some orbit segments such that system (1.7) has the following normal in : as
where , the sign “ ” stands for transposition. For sufficiently small, where , , , , , is the corresponding eigenvalues of the linearization matrix of perturbed system (1.6).
(H2) (non-degeneration) System (1.7) has a double heterodimensional cycles , where , , , , and
Here represents the flow of system (17), and by we denote the tangent space of the manifoldM at q.
(H3) (Orbit flip) Let , then
where are unit eigenvectors corresponding to and respectively. Furthermore they satisfy the equation , (for details see  ).
Here, and are the unit eigenvectors corresponding to and which responds enters the equilibrium along the strong stable manifold (as , enters the equailibruium along the unstable manifold (as ), that is, from , the heteroclinic orbit has orbits flips when (see Figure 1).
(H4) (Strong inclination)
Remark 1.1. Under the assumption H1, and have a 1-dimensional unstable manifold and a 2-dimensional stable manifold, while has a 2-dimensional unstable manifold and a 1-dimensional stable manifold, hence is double heterodimensional cycles.
Remark 1.2. Hypothesis (H4) shows that and have strong inclination property. Due to the assumption (H2), has a 2-dimensional unstable manifold, has a 2-dimensional stable manifold, and ,we can know the codimension of the heteroclinic orbit is 0. Then the orbits is transversal, that is, they can be preserved even under small perturbations.
2. Local Coordinates and Bifurcation Equations
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 of
Figure 1. Double heterodimensional cycles of three saddle points with four orbits .
such that have local manifolds and which are expressed as below: for ,
Let the coordinate expression of be in the small neighborhood of , , and in the small neighborhood of . Since is large enough so that , , and for , , , for , , , , where is small enough.
Now we take into account the linearly variational system and its corresponding adjoint system of (1.7) formed respectively by: let ,
Based on the above hypotheses about system (1.7), system (2.1) has exponential dichotomies in and (see  ). We can obtain the following properties.
Lemma 2.1. System (2.1) has the fundamental solution matrices
which satisfy, respectively, for
In what follows, we select as a new local coordinate system along . Let be the fundamental solution matrix of (2.2). By the [1, ?], we can know that the is bounded and tends to zero exponentially as .
Take a coordinate transformation
in a small neighborhood of , where , and represents the coordinate decomposition of (1.6) in the new local coordinate system corresponding to and , Then we can take eight transverse sections vertical to the tangency to each orbit (see Figure 2)
Figure 2. The cross-section and Poincaré map.
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
and new coordinates
where ; . Then, combining with the Equations (2.3), (2.4), we obtain for
Then, under transformation (2.5), system (1.6) has the following form by and :
where is the partial derivation of with respect to . To integrate (2.10), we get
where are called Melnikov vectors respect to .
Which are defined as the global maps with the expression by (2.11) given
Next we consider the local maps,
induced by flows confined in the neighborhood .
Let be the time spent from to and from to 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
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 
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.
3. Heterodimensional Cycle Bifurcation of “∞” Type
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).
3.1. Analysis Procedure
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 and is 0, the second heterodimensional cycle will appear. And if, , a large 1-heteroclinic cycle connecting with and will exist. As well as, if there are conditions that make be invalid and or be invalid and, the third heterodimensional cycle will arise. Therefore it is enough to discuss the solutions of the Equation (2.21).
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, rewrite the first Equation of (3.1) as
Then we have
If, the equation has a unique small positive solution If, it makes be untenable.
1) If, or, , the straight line L and the curve N cannot intersect in the half plane for, so Equation (3.2) has not any positive solutions, that is, system (1.6) only has the transversal heteroclinic orbit in the region.
2) If, or, , the straight line and the curve intersect at one positive point, that is, (3.2) has one positive solution.
Without loss of generality, we discuss the case,. There are
When,. It is clear that (3.2) has a unique solution satisfying. Putting it into the second equation of (3.1), there is, it defines a surface
with a normal surface at for. That is to say, system (1.6) has the only one heteroclinic orbit consisting of and near for.
3) If, or, , there are two special cases:
a) As, Equation (3.2) can be simplified to be
It has a solution. Substituting into the second equation of (2.11), we get immediately a surface tangent to,
for. So system (1.6) has a heteroclinic orbit consisting of and near for. Next putting the expression of into the verification condition, it is equivalently.
b) As, Equation (3.2) is then
there is a small positive solution. In the same way, we can get the surface which is tangent to with the condition, where
So system (1.6) has a heteroclinic orbit consisting of and in the region for.
4) If, or, , without loss of generality, we discuss the case,. There are, .
is the solution of and
when, the straight line intersects the curve exactly at two points, which means Equation (3.2) has two positive solutions. Therefore, system (1.6) has two heteroclinic orbits connecting and near.
When, the equations and have the solution, therefore the straight line must be tangent to the curve at the point. Putting it into the second equation of (3.1) yields a surface with a normal surface at, where
for. Then, system (1.6) has a 2-fold heteroclinic orbit connecting and near.
When, the straight line does not intersect the curve in the half plane, then there is only the transversal heteroclnic orbit connecting and near.
5) If, Equation (3.1) is
To solve the first equation of (3.5), there is
we can get two solutions and for
. However, if , the above equation has only one zero solution. Equation (3.5) finally defines a surface.
Putting the expression into the second equation of (3.5) obtains the set of as, that means the system of (6) coexists two types of heteroclinic orbit: a large-1 heteroclinc orbit connecting with and, a heteroclinic orbit composed of two orbits which one orbit connects with and and the other orbit connects with and in the region as, where
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.
3.2. Bifurcation Conclusions
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 (H1)-(H4) and Rank, as well as, there are the following conclusions:
1) If or, the system (1.6) exists the third shape heterodimensional cycle in the -dimensional surface with normal vector at,where
2)If, the system (1.6) exists the third shape heterodimensional cycle near as and,where
3) If or, there exists an -dimensional surface
with normal vector at,which is tangent to the surface at,such that the system (1.6) has the second shape hetrodimensional cycle near as and.
4)If or, there exist two -dimensional surfaces
such that the system (1.6)has the second shape heterodimensional cycle near as,,respectively,and.
An alternative explanation for the existence of the second heterodimensional cycle is as follows. If there is an orbit starting from the section and arriving at the section that passes through the sections and with finite time without orienting to the saddle point, we denote it by. Similarly, we can define in this way. Set the time of the orbit from to to be and the time of from to to be; and from to to be, , respectively. Moreover, system (1.6) still has solutions, ,
Theorem 3.2. Suppose that (H1)-(H4) hold and Rank there is an -dimensional surface
with a normal plane,such that system (1.6)has a unique double heteroclinic loop (“∞”) in the tubular neighborhood of as,.
Proof. As we explained above, in Equation (2.11) means the flying time of an orbit starting from to is infinite, that is, the orbit must go into the equilibrium and then leave, which corresponds to a heteroclinic orbit; and for, it is similar. Hence, set in Equation (2.11), we have
If, there is a codimension-4 surface with a normal plane spanned by as below
when, system (1.6) has four heteroclinic orbits connecting the equilibriums, , and they form an “∞”-type double heterodimensional cycle, or it says that the original heterodimensional cycle is preserved. □
Corresponding, some new orbits (resp.) appear from unstable (resp. stable) manifold of the equilibrium of system (1.6) with the following properties,
where and are the stable and unstable manifolds of the equilibrium, because the original heteroclinic trajectory is obtained as a transversal intersection of 2-dimensional manifolds, which is a structurally stable situation. After a small perturbation, such an intersection is preserved. That is, the gap. As well as, if the gap in, it means that the original double heterodimensional cycles are kept (see Figure 3).
Where , and still meet Equation (3.1). Clearly system (1.6) has the second shape heterodimensional cycle, if the gaps, (see Figure 4).
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 and two orbits linking and respectively (see Figure 3).
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 and two orbits linking and respectively is analogous to theorem 3.1, so it will not be repeated here.
Theorem 3.3. Suppose (H1)-(H4) are valid and, there are the following conclusions:
Figure 3. The gap in the figure, the original double heterodimensional cycles exists.
Figure 4. The gap, in the figure, there is the second heterodimensional cycle.
1) If or ,there exists an -dimensional surface
with normal vector at,which is tangent to the surface,then the system (1.6)has a 1-fold large-1heteroclinic cycle near as and,where
2)If or ,there exists two -dimensional surface
both with normal vector at,which both are tangent to the surface,then the system (1.6)has a 1-fold large-1heteroclinic cycle near as,,respectively,and,where
3) If,and,the system (1.6)has two 1-fold large-1heteroclinic cycles near,where
Figure 5. The gap, in the figure, there is the large 1-heteroclinic cycle.
4) If,there exists a -dimensional surface with normal vector,which is tangent to the surface at,where
then the system (1.6)has one 2-fold large-1heteroclinic cycles near for,where
For the alternative explanation from the gaps for the existence of the large-1 heteroclinic cycle is the following. If, and still meet Equation (3.2) and (3.1). Clearly system (1.6) has a large 1-heteroclinic cycle composed of two big orbits linking and of (1.2) type respectively, if the gaps, (see Figure 5).
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).
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