Bifurcation Analysis of Homoclinic Flips at Principal Eigenvalues Resonance

One orbit flip and two inclination flips bifurcation is considered with resonant principal eigenvalues. We introduce a local active coordinate system to establish bifurcation equation and obtain the conditions when the original homoclinic orbit is kept or broken. We also prove the existence and the existence regions of double 1-periodic orbit bifurcation. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately, and are well located.


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][6][7], and also resonant homoclinic flips in [8][9][10][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][16][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 2 -homoclinic orbit and their corresponding bifurcation surfaces.

Hypothesis
We assume that system (1.2) has a homoclinic orbit to an equilibrium 0 z  , which is hyperbolic and has two negative and two positive eigenvalues, denoted by 1 2  ) the stable (resp.strong stable) manifold and unstable (resp.strong unstable) manifold of the equilibrium u W uu W 0 z  , respectively.Now we further make three assumptions:
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 2 -homoclinic orbit.The last section gives a conclusion of the work.n

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 U (see [10,11,18]) with the assumption (H4) .
, 0, 0 0, 0, , 0 0 and   d  are parameters depending on  .Notice that we have straightened the corresponding invariant manifolds.So it is possible to choose some moment , such that , where  is small enough and Now we turn to consider the linear variational system and its adjoint system First we introduce a lemma, see [10,11] Lemma 2.1 There exists a fundamental solution matrix , and .
is bounded and tends to zero exponentially as t   due to where .We can well regard as a new local coordinate sys-tem along , and choose : , , , 2 as the cross sections of at t and  T  t T   respectively.Under the transformation of (2.4), system (1.1) becomes A simple integrating of both sides from to T of the above equation, we further achieve where are the Melnikov vectors (see [18]).Lemma 2.2 Actually a regular map is given by (2.5) as (see Figure 1(a))  But this map is established in the new coordinate system, so we should look for the relationship between two coordinate systems.Set , , , , , , , Next, we start to set up a singular map induced by the solutions of system (2.1) in the neighborhood , for example where  is the time going from 0 Denote the Silnikov time , then there is Similarly, there are With Equations (2.6)-(2.9),Equation (2.5) well defines the Poincaré map The above fact enables one to achieve the associated as follows: (2.10)

Main Results
To begin the bifurcation study, and    by one owing to (H1) for concision. Set 1) has a unique periodic orbit as or a unique homoclinic orbit as  , and they do not coexist.Furthermore,   , 0 F s   has explicitly a sufficiently small positive solution ) and an 1-homoclinic orbit exists as > 0 1 H   , but they do not coexist.
In the following stage, we try to look for bifurcations according to the case , .

. . . w M w M h o t w
with the normal vector The second equation permits a solution

. . . w M s h o t w
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 w M   increases (resp.decreases), the line must intersects the curve at two (resp.no) sufficiently small positive points.Now the proof is complete.

F s s w w w M w s M s h ot
has always two nonnegative solutions and if and only if 14 1 0 w M   .The proof is complete.
From the above proof, we see that if the line   , W P s  has a small positive section with the axis or small positive slope, then there exists a small positive . 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 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 : 0, ... 0, ... 0 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 1  and 2  be the time going from and , , , F q q F q q F q q F q q q 0      .
Then recall the process of the establishment of (2.10), similarly we may get the associated second returning successor function , , , ,  , , , , ,  G s s u u y y G G G G G G  Eliminating again and 3 from and , and assuming we obtain We know that a 2-homoclinic orbit corresponds to the solution 1 and 2 or 1 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 In the region defined by Then one may derive   which informs that 1 s increases (resp.decreases) as  moves along the direction 14 1 (resp.the opposite direction) such that a -periodic orbit bifurcates from the -homoclinic orbit  w w M   , so there does not exist any 2-periodic orbit when  is near 1 H . Therefore in the region bounded by the surfaces 2 H to 1 H , 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 .: 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   To well illustrate our results, a bifurcation diagram is drawn in Figure 3, where represents a -periodic orbit.

Conclusion
Homoclinic orbits generically occur as a codimensionone phenomenon, while if the genericity conditions are 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.

n
In the present context, we consider the following system

. 1 )
this is the bifurcation equation.Here we have omitted the parameter  in 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 31 33 1 0 w w M   (resp. To begin with we divide (3.1) into two parts:  .Notice that, when the tangency takes place, the line

1 )
has only one 1-periodic orbit near  as    and 14 1) has not any 1-periodic orbit or 1-homoclinic orbit as    and 14 w M   or has only a zero solution 1 0 s  for 44 42 4 0 w w M   .If    , there is, on the contrary, has not any 1-periodic orbit as 14 1 0 w M   and 42 44 4 0 w w M   .

2  as  leaves 2 H
for the side pointed by .

2 P
The above reasonings can repeat itself many times to find
to the variable s .
 , 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