External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip ()
1. 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] studied the qualitative properties of solitons of a dKdV equation,
(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
to make system (1.1) be
(1.2)
then integrated (1.2) and got
(1.3)
where g is the integral constant, and system (1.3) is equivalent to the following regular plane system with
(1.4)
Clearly the Hamiltonian of system (1.4) is
(1.5)
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. [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
system to be studied is
(1.6)
where
. Specially, when
, the unperturbed system associated with (1.6) is
(1.7)
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
(1.8)
and as
(1.9)
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 [19] ).
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 [17], 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
,
(2.1)
and
(2.2)
Based on the above hypotheses about system (1.7), system (2.1) has exponential dichotomies in
and
(see [12] ). We can obtain the following properties.
Lemma 2.1. System (2.1) has the fundamental solution matrices
which satisfy, respectively, for
that is
(2.3)
where
.
And for
that is,
(2.4)
where
.
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
(2.5)
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
(2.6)
and
(2.7)
for
(2.8)
and
(2.9)
Then, under transformation (2.5), system (1.6) has the following form by
and
:
(2.10)
where
is the partial derivation of
with respect to
. To integrate (2.10), we get
(2.11)
where
are called Melnikov vectors respect to
.
Which are defined as the global maps
with the expression by (2.11) given
(2.12)
as follows
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:
(2.13)
(2.14)
(2.15)
(2.16)
Thus, by (2.6), (2.12) (2.13), we obtain the first Poincaré map
as follows
(2.17)
by (2.8), (2.12), (2.14), we obtain the Poincaré map
as follows
(2.18)
by (2.8), (2.12), (2.15), we obtain the Poincaré map
as follows
(2.19)
by (2.6), (2.12), (2.16), we obtain the Poincaré map
as follows
(2.20)
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 ![]()
(2.21)
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.
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
(3.1)
Set
, rewrite the first Equation of (3.1) as
(3.2)
where
![]()
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
![]()
where
.
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
(3.3)
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
(3.4)
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
,
.
Set
, where
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
(3.5)
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
![]()
and
![]()
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
,
,
![]()
(3.6)
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,
![]()
(3.7)
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
![]()
and
![]()
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
![]()
and
![]()
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).
Acknowledgements
We gratefully acknowledge the reviewers for their patience in reading the first draft of this paper.
Funding
The authors were supported by National Natural Science Foundation of China (Grant No. 11871022).