Bifurcation Analysis of Homoclinic Flips at Principal Eigenvalues Resonance ()
1. Introduction
Homoclinic bifurcations have been comprehensively investigated from the initial work of Silnikov in [1] who gave a detailed study of a system which permits an orbit homoclinic to a saddle-focus. After that many flips cases attract researcher’s interests, including resonant eigenvalues case in [2], orbit flips in [3,4], inclination flips in [5-7], and also resonant homoclinic flips in [8-11]. In these cases homoclinic-doubling bifurcation has been expensively studied, which is a codimension-two transition from an n-homoclinic to a 2n-homoclinic orbit. Some applications of these cases may be referred to a model for electro-chemical oscillators, the FitzHugh-Nagumo nerveaxon equations [12], a Shimitzu-Morioka equation for convection instabilities [13], and a Hodgkin-Huxley model of thermally sensitive neurons [14], etc.
More recently, the flip of heterodimensional cycles or accompanied by transcritical bifurcation is got attention, see [15-17], the double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for the homoclinic orbit and the periodic orbit. But the research is not concerned with multiple flips. While multiple cases may have more complicated bifurcation behaviors and even chaos, it is necessary to give a deep study. This paper produces mainly a theoretical study of homoclinic bifurcation with one orbit flip and two inclination flips, which can take place at least in a fourdimensional system. Compared with the above work mentioned, our problem has higher codimension with resonant, and we get not only the existence of 1-periodic orbit, 1-homoclinic orbit, and double periodic orbit, but also the
-homoclinic orbit and their corresponding bifurcation surfaces.
In the present context, we consider the following
system
(1.1)
and its unperturbed system
(1.2)
where
.
Hypothesis
We assume that system (1.2) has a homoclinic orbit
to an equilibrium
, which is hyperbolic and has two negative and two positive eigenvalues, denoted by
, and additionally
. Set
(resp.
) and
(resp.
) the stable (resp. strong stable) manifold and unstable (resp. strong unstable) manifold of the equilibrium
, respectively. Now we further make three assumptions:
(H1) (Resonance)
for
, where
and
.
(H2) (Orbit flip) Define
,
, then
and
are unit eigenvectors corresponding to
and
respectively, where
is the tangent space of the corresponding manifold
at the saddle
, and the similar meaning for
.
(H3) (Inclination flips) Denote by
and
the unit eigenvectors corresponding to
and
respectively, let
![](https://www.scirp.org/html/1-7401314\c2588299-c550-48f4-b088-2bd8c60d4ac0.jpg)
The paper is organized as follows. In Section 2 we will construct the Poincaré map by the method used in [18] to get the associated successor function. In Section 3, we first establish bifurcation equation. Then a delicate study shows our main results about the existence of double 1-periodic orbit, 1-homoclinic orbit and also
-homoclinic orbit. The last section gives a conclusion of the work.
2. Two Normal Forms and Successor Function
From the above hypotheses, the normal form theory provides a
system as follows after four successive
to
transformations in
(see [10,11,18])
(2.1)
with the assumption
(H4)
.
Indeed we have
![](https://www.scirp.org/html/1-7401314\19e1a0a5-b263-4b23-8a0d-c3ad04a7bbe1.jpg)
where
![](https://www.scirp.org/html/1-7401314\02f471de-7a5b-4869-8be2-f3a435461d1f.jpg)
and
![](https://www.scirp.org/html/1-7401314\085d0537-b9a8-49b0-95d3-1e8fab67c291.jpg)
and
![](https://www.scirp.org/html/1-7401314\cb3f83af-f906-40c5-88d6-c7b987696145.jpg)
and
![](https://www.scirp.org/html/1-7401314\f11c9950-568b-466b-931d-cef7cfef4237.jpg)
and
are parameters depending on
. Notice that we have straightened the corresponding invariant manifolds. So it is possible to choose some moment
, such that
and
, where
is small enough and
.
Now we turn to consider the linear variational system and its adjoint system
(2.2)
(2.3)
First we introduce a lemma, see [10,11]
Lemma 2.1 There exists a fundamental solution matrix
of system (2.2) satisfying
![](https://www.scirp.org/html/1-7401314\e9eff310-9845-40e7-8f11-ad3b51f9762c.jpg)
where
![](https://www.scirp.org/html/1-7401314\7bf9e348-8c54-4f53-9c61-11af16d70e2f.jpg)
and
, and
.
Remark 2.1 The matrix
is a fundamental solution matrix of system (2.3), denote by
then
![](https://www.scirp.org/html/1-7401314\61163057-7326-4c93-862d-6427af455589.jpg)
is bounded and tends to zero exponentially as
due to
and
tends exponentially to infinity.
Let
(2.4)
where
. We can well regard
as a new local coordinate system along
, and choose
![](https://www.scirp.org/html/1-7401314\e0ca2018-cfb5-43d2-8e14-20acafe7b860.jpg)
as the cross sections of
at
and
respectively. Under the transformation of (2.4), system (1.1) becomes
![](https://www.scirp.org/html/1-7401314\e464e9e3-15e1-457b-b2ab-8efc4d269240.jpg)
A simple integrating of both sides from
to
of the above equation, we further achieve
(2.5)
where
![](https://www.scirp.org/html/1-7401314\33d946cf-10b6-4708-8196-8d0069e421c9.jpg)
are the Melnikov vectors (see [18]).
Lemma 2.2
.
Actually a regular map is given by (2.5) as (see Figure 1(a))
![](https://www.scirp.org/html/1-7401314\d4e2a01a-a5fc-4afc-aab1-b4dbbee7840d.jpg)
But this map is established in the new coordinate system, so we should look for the relationship between two coordinate systems. Set
![](https://www.scirp.org/html/1-7401314\d673d29f-663d-41db-8701-5c50d2d928e8.jpg)
and
(a)
(b)
Figure 1. Transition maps. (a) F1: S1→S0; (b) F0: S0→S1.
Take
respectively in (2.4), we have
(2.6)
(2.7)
and
(2.8)
Next, we start to set up a singular map
![](https://www.scirp.org/html/1-7401314\0e22bdfd-fbeb-48f5-aa26-0e741fe5f6da.jpg)
(see Figure 1(b)) induced by the solutions of system (2.1) in the neighborhood
, for example
![](https://www.scirp.org/html/1-7401314\04308338-1cc8-4a50-8c99-5ca10502f4f6.jpg)
where
is the time going from
to
. Denote the Silnikov time
, then there is
.
Similarly, there are
(2.9)
With Equations (2.6)-(2.9), Equation (2.5) well defines the Poincaré map
,
![](https://www.scirp.org/html/1-7401314\94f4865d-d920-4fcc-8391-915c473d1e32.jpg)
The above fact enables one to achieve the associated successor function
as follows:
(2.10)
3. Main Results
To begin the bifurcation study,
and
first give
![](https://www.scirp.org/html/1-7401314\9bb58df8-e577-487a-8d5c-f6a5963b02df.jpg)
Substitute them into
, we get
(3.1)
this is the bifurcation equation. Here we have omitted the parameter
in
and
, and replaced the exponent
by one owing to (H1) for concision.
Set
, we find that, when
,
![](https://www.scirp.org/html/1-7401314\ee84f331-3acb-4b27-b591-364a5ad0476d.jpg)
The implicit function theorem reveals that
has a unique solution
![](https://www.scirp.org/html/1-7401314\7b793cb0-f243-404a-8cc3-c647b2639794.jpg)
satisfying
So system (1.1) has a unique periodic orbit as
or a unique homoclinic orbit as
, and they do not coexist. Furthermore,
has explicitly a sufficiently small positive solution
if
. On the other hand, it has a solution
when
, so we have Theorem 3.1 Suppose that
and
hold, then system (1.1) has at most one 1-periodic orbit or one 1-homoclinic orbit in the neighborhood of
. Moreover an 1-periodic orbit exists (resp. does not exist) as
in the region defined by
(resp.
) and an 1-homoclinic orbit exists as
, but they do not coexist.
In the following stage, we try to look for bifurcations according to the case
for
.
To begin with we divide (3.1) into two parts:
![](https://www.scirp.org/html/1-7401314\4792a6e1-0644-41bd-9e10-b3c6a192c911.jpg)
Therefore
, where
is a line and
is a curve with
according to the variable
.
Theorem 3.2 Suppose that
and
, system (1.1) then has a unique double 1-periodic orbit near
, and two (resp. not any) 1-periodic orbits near
when
lies on the side of
which points to the direction
(resp. in the opposite direction of
). The corresponding double 1-periodic orbit bifurcation surface
is
![](https://www.scirp.org/html/1-7401314\b020a1e9-81a9-446e-a0b3-4a43c0926c74.jpg)
with the normal vector
at
.
Proof Consider equations ![](https://www.scirp.org/html/1-7401314\4d602178-708d-4dfc-a02f-bd084a303427.jpg)
and
, that is,
(3.2)
The second equation permits a solution
![](https://www.scirp.org/html/1-7401314\7ab3dd18-0dab-40d5-9a17-efead4270029.jpg)
as
.
Substituting it into the first equation of (3.2), we obtain the tangency condition, which corresponds to the existence of the double periodic orbit bifurcation surface
situated in the region
and
. Notice that, when the tangency takes place, the line
lies under the curve
. So if
increases (resp. decreases), the line must intersects the curve at two (resp. no) sufficiently small positive points. Now the proof is complete.
Theorem 3.3 Suppose that
and
are true, then there exists two codimension-one hypersurfaces
![](https://www.scirp.org/html/1-7401314\b62ba722-27ec-4dd8-8f87-5fb76fee1d37.jpg)
and
such that System (1.1) has only one 1-homoclinic orbit near
as
and
;
System (1.1) has only one 1-periodic orbit near
as
and
;
System (1.1) has exactly one 1-homoclinic orbit and one 1-periodic orbit near
as
and
;
System (1.1) has not any 1-periodic orbit or 1-homoclinic orbit as
and
.
Proof When
, we have at once
![](https://www.scirp.org/html/1-7401314\57a8af3c-76d8-4fe6-8083-f5fdade3d142.jpg)
and
, therefore
![](https://www.scirp.org/html/1-7401314\20fddb6b-350c-4ecf-baad-bf8259e95b44.jpg)
has always two nonnegative solutions
and
![](https://www.scirp.org/html/1-7401314\bc43f4e0-939f-4845-8428-b216ec7acec6.jpg)
for
or has only a zero solution ![](https://www.scirp.org/html/1-7401314\3894cd62-e0f2-453e-a5ed-815459605dc5.jpg)
for
. If
, there is, on the contrary,
but
, apparently the line
is horizontal. So
has a solution
![](https://www.scirp.org/html/1-7401314\5916e50f-5794-438f-86ba-f2a06c953da0.jpg)
if and only if
. The proof is complete.
From the above proof, we see that if the line
has a small positive section with the
- axis or small positive slope, then there exists a small positive
such that
. Thus the following corollary is valid, which is a complement of Theorem 3.2.
Corollary 3.4 Assume that the hypotheses of Theorem 3.2 are valid, system (1.1) then has a unique 1-periodic orbit near
as
is situated in the region defined by
and
or
,
and
; has not any 1-periodic orbit as
and
.
Notice that in Theorem 3.3, system (1.1) has a codimension-1 1-homoclinic orbit, see Figure 2(a), that is the existing homoclinic orbit has no longer orbit flip. But an orbit flip homoclinic orbit could still exist if
see Figure 2(b).
Corollary 3.5 Suppose that
and
hold, system (1.1) has a codimension-2 orbit-flip homoclinic orbit as
.
Now we turn to study the homoclinic doubling bifurcations. To begin with we look for the 2-homoclinic orbit and 2-periodic orbit bifurcation surfaces. Reset
and
be the time going from
to
and from
to
respectively,
and
.
Then recall the process of the establishment of (2.10), similarly we may get the associated second returning successor function
expressed as
:
(a)
(b)
Figure 2. 1-homoclinic orbit (1-H) and (1-OH). (a) μ ∈ Σ1; (b) F(0, μ) = 0, y0 = M4μ + h.o.t. = 0.
![](https://www.scirp.org/html/1-7401314\a395feae-dca3-4d29-a012-45a936004b42.jpg)
Eliminating again
and
from
and
, and assuming
![](https://www.scirp.org/html/1-7401314\ab536b62-4b14-405e-baa1-9a22a47e073f.jpg)
we obtain
(3.3)
(3.4)
We know that a 2-homoclinic orbit
corresponds to the solution
and
or
and
of (3.3) and (3.4), that means an orbit returns once nearby the singular point in limit time and twice in limitless time. So it is sufficient to seek the small solutions of
and
by symmetry of
. Therefore
(3.5)
(3.6)
Clearly (3.5) yields
![](https://www.scirp.org/html/1-7401314\ef75f017-e5eb-489e-9588-cedb8bc07cf0.jpg)
for
, and
sufficiently small. With this, Equation (3.6) determines a 2-homoclinic orbit bifurcation surface
![](https://www.scirp.org/html/1-7401314\6dcc3675-1f66-4dd1-868e-c9c066ebc2a2.jpg)
for
and
, which has a normal vector
at
.
Continually, differentiating both sides of (3.3) and (3.4) with respect to
and for
, we obtain
![](https://www.scirp.org/html/1-7401314\99e79059-2446-4098-80a2-ca9e149fe102.jpg)
In the region defined by
and
the 2-homoclinic orbit bifurcation surface
is simplified to be
.
Accordingly
.
Then one may derive
![](https://www.scirp.org/html/1-7401314\f467fafa-ceda-45e1-a1e2-89eef744c323.jpg)
which informs that
increases (resp. decreases) as
moves along the direction
(resp. the opposite direction) such that a
-periodic orbit bifurcates from the
-homoclinic orbit
as
leaves
for the side pointed by
.
Notice that confined on the surface
, (3.3) and (3.4) has a unique positive solution, meanwhile Theorem 3.3 indicates exactly the existence of one 1-periodic orbit when
for
, so there does not exist any 2-periodic orbit when
is near
. Therefore in the region bounded by the surfaces
to
, there must exist another bifurcation surface which merges the 1-periodic orbit and the 2-periodic orbit into a new 1-periodic orbit with the different stability from the original one. We call this surface the period-doubling bifurcation surface and denote it by
.
The above reasonings can repeat itself many times to find the
-homoclinic orbit bifurcation surface
![](https://www.scirp.org/html/1-7401314\06d73135-fe37-49c8-a838-dbdcebb3c127.jpg)
in the same region of
space and simultaneously the presence of period-doubling bifurcation surface
of
-periodic orbit.
In short, we conclude that:
Theorem 3.6 Suppose that
![](https://www.scirp.org/html/1-7401314\a703f8a7-471f-48ba-b500-0ae67e7e842f.jpg)
and
hold, then for
![](https://www.scirp.org/html/1-7401314\b7b8e8fb-e263-4af1-b1c7-fa6590787037.jpg)
and
, there exists a
-homoclinic orbit bifurcation surface
with the normal vector
at
and the period-doubling bifurcation surface
of
-periodic orbit in the small neighborhood of the origin of
space. Moreover system (1.1) has exactly a
-homoclinic orbit as
and a
-periodic orbit as
moves away to the side of
pointing to the direction
and none on the other side.
To well illustrate our results, a bifurcation diagram is drawn in Figure 3, where
represents a
-periodic orbit.
4. Conclusion
Homoclinic orbits generically occur as a codimensionone phenomenon, while if the genericity conditions are
![](https://www.scirp.org/html/1-7401314\48f3ca2d-ba2b-4b91-82e2-a67168fc952d.jpg)
Figure 3. Location of bifurcation surfaces for rank (M1, M3, M4) = 3, w33 = 0, 2λ1 > λ2 > ρ2.
broken, some high codimension instance including the resonant and flips cases, concomitant usually with chaotic behavior, may take place. Homburg and Oldeman studied two kinds of resonant homoclinic flips in [8,9] with unfolding techniques and numerical methods respectively. Zhang in [10,11] continued to research on these problems and gave some theoretical proofs of the existence of
-periodic orbit and
-homoclinic orbit and also their existence regions via the method initially established in [18]. Besides these the flip heterodimensional cycles have also attracted attentions nowadays, see [16]. In this paper, we extend the method to fit a higher codimension case of 3 flips with resonant. With the delicate analysis, the existence of 1-periodic orbit, 1-homoclinic orbit, and double periodic orbit are proven and also the
-homoclinic orbit and their corresponding bifurcation surfaces. With the work, we find the extensive existence of the double periodic orbit bifurcation and the homoclinic-doubling bifurcation, which efficiently advance the development of the flips homoclinic study.
NOTES