External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip

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 1 p and 3 p . 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.


Introduction
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]  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 heterodi-mensional 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 r C system to be studied is and as ( ) where 2 k r ≥ − , the sign " * " stands for transposition. For u sufficiently small, where ( ) (H 2 ) (non-degeneration) System (1.7) has a double heterodimensional cycles , that is, from [17], the heteroclinic orbit 2 Γ has orbits flips when t → +∞ (see Figure 1). p has a 2-dimensional unstable manifold, 3 p has a 2-dimensional stable manifold, and we can know the codimension of the heteroclinic orbit 3 Γ is 0. Then the orbits 3 Γ is transversal, that is, they can be preserved even under small perturbations.

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.
δ 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 + R and − R (see [12]). We can obtain the following properties.
Take a coordinate transformation 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 ( ) ( ) Then, under transformation (2.5), system (1.6) has the following form by where g µ is the partial derivation of ( ) Melnikov vectors respect to µ .

Heterodimensional Cycle Bifurcation of "∞" Type
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 1 p and 3 p are studied by discussing the corresponding bifurcation equation. Clearly if 2 4 0 s s = = the double heterodimensional cycle ("∞") of system (1.6) is persistent; if 4 0 s = , 2 0 s > , 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 3 1 , p p and two orbits linking 3 2 , p p and 2 1 , p p respectively, which is called the second shape heterodimensional cycle in later of this paper; if 2 0 s = , 4 0 s > , 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 1 3 , p p and two orbits linking 1 2 , p p and 2 3 , p p respectively, which is called another second shape heterodimensional cycle in later of this paper; if 2 0 s > and 4 0 s > , system (1.6) has the large 1-heteroclinic cycle consisting of two saddles , p p respectively. What is noteworthy is that if the conditions make 2 0 s > untenable and set 4 0 s = tenable, the conditions make 4 0 s > untenable and 2 0 s = tenable, system (1.6) has the third heterodimensional cycle consisting of one saddle 2 p of (2.1) type and one saddle 3 p of (1.2) type and composed of one orbit starting from 2 p to 3 p and another orbit starting from 3 p to 2 p under the assumption (H 2 ). So in the following, we need to consider solutions 2 s and 4 s of the bifurcation Equation (2.21).
there is a small positive solution ( ) : .
So system (1.6) has a heteroclinic orbit consisting of 3 p and 1 p in the re-   (20) is similar to the above analysis process, so it will not be repeated here.