A Subshift of Finite Type

in the Takens-Bogdanov Bifurcation

with D_{3} Symmetry

Karsten Matthies

Received: November 5, 1998 Revised: August 30, 1999

Communicated by Bernold Fiedler

Abstract. We study the versal unfolding of a vector field of codimension two, that has an algebraically double eigenvalue 0 in the linearisation of the origin and is equivariant under a representation of the symmetry group D_{3}. A subshift of finite type is encountered near a clover of homoclinic orbits. The subshift encodes the itinerary along the three different homoclinic orbits. In this subshift all those symbol sequences are realized for which consecutive symbols are different. In the parameter space we also locate a transcritical, three different Hopf and two global (homoclinic) bifurcations.

1991 Mathematics Subject Classification: 34C28, 58F14 Keywords and Phrases: Takens-Bogdanov, subshift of finite type, symmetry, homoclinic orbit

1 Introduction

A vector field has a Takens-Bogdanov point, if there is a Jordan block ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

in the linearisation of a steady state and if certain nondegeneracy conditions are fulfilled. This codimension two degeneracy with its unfolding is a key to understand several phenomena in dynamical systems (see [17, 3] and textbooks like [13]). Takens-Bogdanov points can also serve as a starting point for the path following in two-parameter flows of global Hopf bifurcation [6] and homoclinic orbits [7]. One parameter families of homoclinic orbits are created at Takens-Bogdanov points of two-parameter flows. Hence these bifurcation points play the same role for the creation of homoclinic orbits in twoparameter flows as Hopf bifurcation for periodic solutions in one-parameter flows. Near homoclinic orbits several bifurcations to other bounded solutions may occur. Thus Takens-Bogdanov points are important organizing centers for the bifurcation analysis of dynamical systems. Suppose the dynamical system ẋ = f(x) has constraints given by an equivariance under a symmetry group Γ, i.e. f(γx) = γf(x), for γ ∈ Γ. Then one often finds complicated bifurcation diagrams even at simple bifurcations, [11]. Similarly at the Takens-Bogdanov point with D_{3} symmetry the dynamics are much richer than in the non-symmetric case. Applications are given to systems of three coupled oscillators in a ring. The results could be also applied to mode interactions for pattern formation in convection problems, where solutions with D_{3} symmetry exist [12]. We will encounter a subshift of finite type, which is a novel dynamical feature in a bifurcation problem of dynamical systems defined by a vector field. Whereas in many bifurcations one can encounter Smale horseshoes giving rise to a full shift, the existence of a subshift of finite type is a rare phenomenon. A subshift σ(x_{n})_{n∈Z} = (x_{n+1})_{n∈Z} of finite type on three symbols {1, 2, 3} is defined on

X_{A} = {(x_{n})_{n∈Z}|x_{n} ∈ {1, 2, 3}, a_{x_{n},x_{n+1}} = 1}

where A = (a_{i,j})_{i,j∈{1,2,3}} is a 3×3 matrix with entries 0 and 1. The topology of X_{A} is defined as the product topology of the discrete set of symbols {1, 2, 3}. A subshift of finite type allows only those symbol sequences, for which consecutive symbols x_{n}, x_{n+1} are compatible with the transition matrix A. The symmetry group D_{3} will act on X_{A} in the following manner

‘flip’:  κ((x_{n})_{n∈Z}) = (κ̃x_{n})_{n∈Z} with κ̃1 = 1, κ̃2 = 3, κ̃3 = 2
‘rotation’: γ((x_{n})_{n∈Z}) = (x_{n} + 1 mod 3)_{n∈Z}.

(1)

The bifurcation analysis will be reduced in section 2 to the discussion of a vector field on R^{4} ≅ C^{2}, where D_{3} acts as

‘flip’:  κ(v, w) = (v̄, w̄)
‘rotation’: γ(v, w) = (exp(i\frac{2π}{3} )v, exp(i\frac{2π}{3} )w).

(2)

A vector field in normal form can be derived. Using additional parameters (μ_{1}, μ_{2}) to unfold the singularity the normal form is generically given - up to time reversal - by

v̇ = w  (3)
ẇ = μ_{1}v + μ_{2}w + v̄^{2} − v̄w̄ + [A|v|^{2} + B|w|^{2} + C(vw̄ + v̄w)]v + D|v|^{2}w.

Takens-Bogdanov Point with D_{3} Symmetry 465

A bifurcation diagram describing the complete plane of unfolding parameters is given in figure 1. The main result of this paper is formulated in theorem 1. It states the existence of a special form of a horseshoe for an open set of parameter Figure 1: Bifurcation diagram in parameter space: transcritical bifurcation of steady states (bold line), three kind of Hopf bifurcation (dashed lines), two different homoclinic bifurcations (dotted lines) and shift dynamics (shaded area).

values. Later we will rigorously define three Poincaré sections S_{1}^{in}, S_{2}^{in}, S_{3}^{in} as sections along three coexisting homoclinic orbits biasymptotic to the origin. P will be the return map on S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in}.

Theorem 1 For 0 < μ_{2} + \frac{6}{7}μ_{1} small, μ_{1} > 0, there exists an invariant hyperbolic Cantor set C ⊂ S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} such that the return map P : C → C induced by the flow of (3) is topological conjugate to the irreducible subshift of

finite type with transition matrix A = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ). Here means topological

conjugacy that there exists an homeomorphism τ : C → X_{A} such that τP = στ on C. Furthermore P and τ can be chosen to be D_{3}-equivariant, when using the representation (2) on C and (1) on X_{A}. C is D_{3}-invariant.

So in fact we have a D_{3}-subshift of finite type as defined in [8]. Neither the incomplete bifurcation diagram in figure 1 nor theorem 1 depend on the coefficients A, B, C, D and other higher order terms. In the bifurcation diagram the lines will be bended to curves by a near-identity diffeomorphism. This is peculiar to the case of D_{3} symmetry. When the system has some other symmetry group like O(2) [4] or D_{4} [1] many more parameters and several different bifurcation diagrams have to be discussed. Thus our analysis can only be a first step to a general analysis of D_{n} equivariant Takens-Bogdanov singularities, where one might hope to encounter the O(2) case as a limit. The rest of the paper is organized as follows. In section 2 we give the Taylor expansion near the origin of a generic vector field equivariant under (2) at the Takens-Bogdanov point, derive a normal form up to third order and unfold it. We discuss the basic dynamical behavior in section 3, i.e. we analyze steady states, Hopf bifurcations and the dynamics in invariant subspaces including homoclinic orbits. The existence of the subshift will be proved in section 4, where we use a definition of a general horseshoe. In the last section 5 we will discuss some further numerical studies and applications.

2 D_{3}-equivariant vector fields and normal forms

Before giving a Taylor expansion near the singularity we first use some representation theory to justify the representation like in (2). There are in general two possibilities that a representation space of a compact Lie group Γ admits a non-diagonalizable Γ-equivariant linearisation A at the origin. Similar to chapter XVI of [11] there must be a Γ-invariant subspace W , that is either of the form V ⊕V, where V is absolutely irreducible, or that is irreducible but not absolutely irreducible. The second case is not possible for the Takens-Bogdanov

singularity. The linearisation A contains the nilpotent matrix ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ). Sup-

pose W is irreducible but not absolutely irreducible, then A(W) = 0 or A(W) is isomorphic to W [11, Lemma XII.3.4]. But if A_{|W} contains the nilpotent Jordan block then A : W → A(W) cannot be an isomorphism. Hence A_{|W} = 0 and this is in contradiction that the Jordan block is non-zero. So we use a representation of the form V ⊕V. When choosing for V the standard representation of D_{3} on C ≅ R^{2} we get the representation (2).

Proposition 2 (i) The ring of all D_{3}-invariant germs acting on C ⊕ C as in (2) is generated by

s_{1} = vv̄, s_{2} = ww̄, s_{3} = vw̄ + v̄w and
t_{j} = v^{j}w^{3−j} + v̄^{j}w̄^{3−j} , j ∈ {0, …, 3}

(ii) The module of D_{3}-equivariant smooth mappings of C ⊕ C → C ⊕ C is generated by

g_{0} = ( \begin{matrix}v \\ 0 \end{matrix} ) , g_{1} = ( \begin{matrix}0 \\ v \end{matrix} ) , g_{2} = ( \begin{matrix}w \\ 0 \end{matrix} ) , g_{3} = ( \begin{matrix}0 \\ w \end{matrix} ) and

f_{j} = ( \begin{matrix}v̄^{j} & w̄^{2−j} \\ & 0 \end{matrix} ) , h_{j} = ( \begin{matrix}& 0 \\ v̄^{j} & w̄^{2−j} \end{matrix} ) , j ∈ \{0, 1, 2\},

i.e. all D_{3}-equivariant smooth germs of mappings h : R^{4} → R^{4} can be written in the form h(v, v̄, w, w̄) = p_{0}g_{0} + p_{1}g_{1} + p_{2}g_{2} + p_{3}g_{3} + q_{0}f_{0} + q_{1}f_{1} + q_{2}f_{2} + r_{0}h_{0} + r_{1}h_{1} + r_{2}h_{2} where p_{0}, …, p_{3}, q_{0}, q_{1}, q_{2}, r_{0}, r_{1}, r_{2} are smooth function germs of s_{1}, s_{2}, s_{3}, t_{0}, …, t_{3}.

Proof: For polynomials the completeness of the generators can be checked by lengthy by-hand calculations or by computer algebra, see [9]. These are then by Poénaru’s theorem also the generators of the module of germs of mappings.□ Then a general D_{3}-Takens-Bogdanov point has the following Taylor expansion up to third order with real coefficients a_{1}, b_{1}, …:

\begin{aligned}v̇ & = & w & + & a_{1}v̄^{2} + b_{1}v̄w̄ + c_{1}w̄^{2} \\ & & & + & v(d_{1}vv̄ + e_{1}ww̄ + f_{1}(vw̄ + v̄w)) \\ & & & + & w(g_{1}vv̄ + h_{1}ww̄ + i_{1}(vw̄ + v̄w)) \\ ẇ & = & & & a_{2}v̄^{2} + b_{2}v̄w̄ + c_{2}w̄^{2} \\ & & & + & v(d_{2}vv̄ + e_{2}ww̄ + f_{2}(vw̄ + v̄w)) \\ & & & + & w(g_{2}vv̄ + h_{2}ww̄ + i_{2}(vw̄ + v̄w)). \end{aligned}  (4)

First we try to remove as many second order terms as possible, therefore we choose a general near-identity D_{3}-equivariant coordinate change.

v = v′ + α_{1}v̄^{2} + β_{1}v̄w̄ + γ_{1}w̄^{2}
w = w′ + α_{2}v̄^{2} + β_{2}v̄w̄ + γ_{2}w̄^{2}

We rewrite (4) in the new coordinates and this yields to

v̇′ = w′ + (a_{1} + α_{2})v̄^{′2} + (b_{1} + β_{2} − 2α_{1})v̄′w̄′ + (c_{1} + γ_{2} − β_{1})w̄^{′2}
+ v′(d̃_{1}v′v̄′ + ẽ_{1}w′w̄′ + f̃_{1}(v′w̄′ + v̄′w′))
+ w′(g̃_{1}v′v̄′ + h̃_{1}w′w̄′ + ĩ_{1}(v′w̄′ + v̄′w′))
ẇ′ =  a_{2}v̄^{′2} + (b_{2} − 2α_{2})v̄′w̄′ + (c_{2} − β_{2})w̄^{′2}
+ v′(d̃_{2}v′v̄′ + ẽ_{2}w′w̄′ + f̃_{2}(v′w̄′ + v̄′w′))
+ w′(g̃_{2}v′v̄′ + h̃_{2}w′w̄′ + ĩ_{2}(v′w̄′ + v̄′w′))

(5)

where the˜ terms depend on the original term, a_{i}, b_{i}, c_{i} and α_{i}, β_{i}, γ_{i} for i = 1, 2. By choosing

α_{1} = \frac{1}{2}(b_{1} + c_{2}), α_{2} = −a_{1}, β_{1} = 0, β_{2} = c_{2}, γ_{1} = 0, γ_{2} = −c_{1}

we can remove all second order terms in the first component and w̄^{′2} in the second component in (5). All the third order terms are O(2)-equivariant. Thus we can use exactly the same coordinate change as Dangelmayr and Knobloch [4] (after removing the second order terms) without affecting the lower order terms to get the following simplified system:

v̇ = w
ẇ = Ev̄^{2} + F v̄w̄ + [A|v|^{2} + B|w|^{2} + C(vw̄ + v̄w)] v + D|v|^{2}w,

where E = a_{2}, F = b_{2} + 2a_{1}, A = d̃_{2}, B = d̃_{2} − g̃_{1} + f̃_{1} − 2ĩ_{2}, C = d̃_{1} + f̃_{2} and D = d̃_{1} + g̃_{2}. We use an unfolding to describe the behavior of generic families of vector fields near the singular point. Even if there is not a general method to unfold the whole vector field, the linear part can be unfolded by

( \begin{matrix}0 & 0 \\ μ_{1} & μ_{2} \end{matrix} ) such that all nearby linear parts can be reached up to conjugation

[2]. After scaling v, w, t and possibly a time-reversal we can set generically E = 1, F = −1, if the transformed second order terms are nonzero. Then we get the normal form as in equation (3):

v̇ = w
ẇ = μ_{1}v + μ_{2}w + v̄^{2} − v̄w̄ + [A|v|^{2} + B|w|^{2} + C(vw̄ + v̄w)]v + D|v|^{2}w.

3 Bifurcations

Standard computations show some symmetry breaking bifurcations. Here especially the behavior inside the flow-invariant fixed point space Fix(κ) = {(v, w)|κ(v, w) = (v, w)} = {(v, w)|v, w ∈ R} will be considered. The same dynamics can be encountered in the rotated spaces γFix(κ) and γ^{2}Fix(κ). The bifurcation inside these planes is a Takens-Bogdanov bifurcation, in which the origin remains a singular point. This was analyzed by Hirschberg and Knobloch [14]. In general cubic and quintic terms cannot be neglected in Hopf bifurcation with D_{3} symmetry. But in our situation the higher order terms are not important as long μ_{1}, μ_{2} are small enough. To see this we have to perform the normal form calculations including these terms. The terms involving e.g. A, B, C, D are all of higher order in μ_{1}, μ_{2} and hence can be neglected in a small neighborhood of 0 in the μ_{1}, μ_{2} plane. For illustration we consider a Hopf bifurcation of the origin at μ_{2} = 0, μ_{1} < 0 inside Fix(κ). After calculating a normal form for Hopf bifurcation like in [13] the direction of branching is determined by the sign of the term a = −\frac{1}{8|μ_{1}|} + \frac{2C+D}{8} . So the higher order terms can be neglected inside a neighborhood of (0, 0) in parameter space (μ_{1}, μ_{2}). Similar results hold for the other bifurcations too. We suppress therefore the dependence on these terms. They only bend some lines in the bifurcation diagram to curves by a near-identity diffeomorphism. See also figure 1.

• The only stable feature is the origin for μ_{1}, μ_{2} < 0.

• For μ_{1} = 0 there is a transcritical bifurcation of secondary steady state N_{1} = (−μ_{1}, 0) and the rotated points N_{2} = γN_{1}, N_{3} = γ^{2}N_{1} each with isotropy Z_{2}.

• For μ_{2} = 0, μ_{1} < 0 the spectrum of the origin is purely imaginary and the system undergoes a D_{3}-Hopf bifurcation [11], where three different types of periodic solutions appear (isotropy type Z̃_{3} for μ_{2} < 0; solutions of isotropy type Z̃_{2} and inside Fix(κ) of isotropy type Z_{2} both for μ_{2} > 0; all these solutions are of saddle type).

• For μ_{2} = μ_{1} < 0 N_{1}, N_{2}, N_{3} undergo Hopf bifurcations, where the imaginary eigenvalues have eigenvectors outside the invariant subspaces and the solutions have isotropy Z̃_{2}.

• For 0 > μ_{2} = −μ_{1} N_{1}, N_{2}, N_{3} undergo Hopf bifurcations inside the invariant subspaces, i.e. the periodic orbit have isotropy Z_{2}.

• For some curve with μ_{2} ≈ −\frac{6}{7}μ_{1}, μ_{1} > 0 there exists an orbit inside Fix(κ) homoclinic to 0, see [14].

• Similar there are orbits homoclinic to N_{1}, N_{2}, N_{3} for μ_{2} ≈ −\frac{1}{7}μ_{1}, μ_{1} < 0.

For the homoclinic orbits we have even nearly explicit expressions. Scaling the equation (3)

τ = ϵt, v = ϵ^{2}x, w = ϵ^{3}y, μ_{1} = ϵ^{2}ν_{1}, μ_{2} = ϵ^{2}ν_{2}.

and ˙ = \frac{d}{dτ} give

ẋ = y  (6)
ẏ = ν_{1}x + x̄^{2} + ϵ(ν_{2}y − x̄ȳ) + O(ϵ^{2}).

Letting ϵ = 0 the system has an explicit homoclinic orbit for ν_{1} > 0 inside Fix(κ):

q_{0}(t) = ( \begin{matrix}x_{q}(t) \\ y_{q}(t) \end{matrix} ) = ( \begin{matrix}& −\frac{3ν_{1}}{2} (1 − tanh^{2} (\frac{\sqrt{ν_{1}}}{2} t)) \\ \frac{3ν_{1}}{2} & \sqrt{ν_{1}} & sech & 2 & (\frac{\sqrt{ν_{1}}}{2} t) & tanh & (\frac{\sqrt{ν_{1}}}{2} t) \end{matrix} ) .

Using the Melnikov method, see e.g. Guckenheimer and Holmes [13], we can then compute parameter values for which the homoclinic orbit persists for ϵ > 0 to get the above results. By symmetry there are homoclinic orbits biasymptotic to the origin inside the other two invariant fixed point spaces γFix(κ) and γ^{2}Fix(κ) for the same parameter values too. So there exists a ‘clover’ like structure of homoclinic orbits, see figure 2.

4 General Horseshoes and Proof of Theorem 1

In this section we prove the existence of the subshift of finite type near the clover of homoclinic orbits. We will compute a Poincaré map near the homoclinic orbits with varying unfolding parameters μ_{1} and μ_{2}. For each of the three homoclinic orbits we define an ‘in’ and an ‘out’ section, called S_{i}^{in} and S_{i}^{out} (figure 3). The return map P : S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} → S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} is discussed by dividing it into local parts near the steady state, which can be described by Figure 2: A sketch of the clover of homoclinic orbits. The three orbits lie all in different planes, which intersect only in the origin.

Figure 3: The sections S_{1}^{out} and S_{1}^{in} at the homoclinic orbit projected to Fix(κ).

its linearisation (lemma 4) and global parts along the homoclinic orbit. This technique can also be used to analyze several other homoclinic bifurcations, see for example the textbook [10]. Before we give the technical details of the analysis of P, we describe the geometric idea: The sections S_{i}^{in} and S_{i}^{out} are cubes in R^{3}. We identify those regions in S_{i}^{out}, which have a preimage in some S_{i}^{in} under the local maps (see figure 4). Similarly we compute the regions in S_{i}^{in}, which are mapped by the local maps to some S_{i}^{out} (see figure 5). The global map P will map the cube in figure 4 to the cube in figure 5. For appropriately chosen parameters (μ_{1}, μ_{2}) the slabs marked ‘2’ and ‘3’ in figure 4 will intersect the slabs ‘2’ and ‘3’ in figure 5. We can then show that there is a Smale horseshoe in three dimensions in the upper half of the cube. But because of the symmetry we have three copies of these cubes and the possible itineraries inside the invariant set are more complicated. In the figures 4 and 5 the sections of the homoclinic orbit marked ‘1’ in figure 2 are shown. The trajectories of points in the regions ‘2’ and ‘3’ in figure 4 were in the sections S_{2}^{in} and S_{3}^{in} before. In the same way the slabs ‘2’ and ‘3’ in figure 5 are those regions, where the forward orbit will reach the section S_{2}^{out} and Figure 4: The section S_{1}^{out} with the images of S_{1}^{in}, S_{2}^{in} and S_{3}^{in}. The line in the middle is the section with Fix(κ).

S_{3}^{out} next. Hence the itineraries, described in the figures 4 and 5, have first a symbol ‘2’ or ‘3’ then the symbol ‘1’, because they are now at the homoclinic orbit with symbol ‘1’, and then proceed with ‘2’ or ‘3’. At the other sections there is the same behavior after following once along the homoclinic loop: The trajectories of points inside the invariant set will lead to another section and hence to another symbol. Therefore the subshift described in theorem 1 can be realized, but no other infinite symbol sequences. To rigorously prove the existence of the subshift, we describe briefly the notion of a general horseshoe in R^{3} following Katok and Hasselblatt [15]. First we explain the meaning of ‘full intersection’. Then using cone conditions we give precise meaning to ‘horizontal expansion’ and ‘vertical contraction’. We prove a technical lemma to justify the complete linearisation near the steady state before computing the local and global maps. We will consider a rectangle Δ = D_{1} × D_{2} ⊂ R ⊕ R^{2} = R^{3} where D_{1} and D_{2} are discs. The projections on the components are denoted by π_{1} (“horizontal”) and π_{2} (“vertical”). Let Δ ⊂ U ⊂ R^{3} be a rectangle and f : U → R^{3} be a diffeomorphism. Then we call a connected component S′ = fS ⊂ Δ ∩ fΔ full, if

1. π_{2}(S) = D_{2},

2. for all z ∈ S, π_{1}_{|f(S∩(D_{1}×π_{2}(z)))} is a bijection onto D_{1}.

The first condition implies that S reaches completely along the vertical direction and second one that the image of every horizontal fiber in S meets Δ and Figure 5: The section S_{1}^{in} where the preimages of S_{1}^{out}, S_{2}^{out} and S_{3}^{out} are the dotted slabs.

traverses it completely. Next we introduce cone conditions. A horizontal s-cone H_{x}^{s} is defined by H_{x}^{s} = {(u, v) ∈ T_{x}R^{3}|∥v∥ ≤ s∥u∥}, similarly a vertical s-cone V_{x}^{s} by V_{x}^{s} = {(u, v) ∈ T_{x}R^{3}|∥u∥ ≤ s∥v∥} at x ∈ R^{3} for some s. A map f preserves a family H_{x} of horizontal cones for x ∈ U ⊂ R^{3}, if Df_{x}(H_{x}) ⊂ int(H_{f(x)}) ∪ {0}. It is called expanding on a horizontal cone family H_{x}, if ∥Df_{x}ξ∥ ≥ μ∥ξ∥ for ξ ∈ H_{x} and some fixed μ > 1. We want to express a contraction property in the vertical direction, thus we consider f^{−1} on vertical cone families. It preserves the vertical cone family V_{x}, if Df_{x}^{−1}(V_{f(x)}) ⊂ int(V_{x}) ∪ {0} and f^{−1} expands them, if ∥Df_{x}^{−1}ξ∥ ≥ λ^{−1}∥ξ∥ for ξ ∈ V_{f(x)} and some uniform λ < 1. Then the appropriate generalization of a Smale horseshoe in higher space dimensions is given by

Definition 3 [15] Let Δ ⊂ U ⊂ R^{3} be a rectangle and f : U → R^{3} be a diffeomorphism. Δ ∩ f(Δ) is called a horseshoe if it contains at least two full components Δ_{1} and Δ_{2} such that for Δ′ = Δ_{1} ∪ Δ_{2} the following conditions hold:

1. π_{2}(Δ′) ⊂ int(D_{2}) and π_{1}(f^{−1}(Δ′)) ⊂ int(D_{1}),

2. D(f_{|f^{−1}(Δ′)}) preserves and expands a horizontal cone family on f^{−1}(Δ′),

3. D(f_{|Δ′}^{−1}) preserves and expands a vertical cone family on Δ′.

To compute the return map P we will first prove that we can completely linearize the local maps.

Lemma 4 Suppose that the distinct eigenvalues of the linearisation A at 0

λ_{1,2} = \frac{μ_{2}}{2} ± \sqrt{\frac{μ_{2}^{2}}{4} + μ_{1}} are not in resonance, i.e. λ_{i} − (kλ_{1} + lλ_{2}) ≠ 0 for k, l ∈ N, k + l > 1. Then there exists a D_{3}-equivariant smooth diffeomorphism H conjugating the flow Φ_{t} of (3) and exp(At) on some neighborhood U of the origin: HΦ_{t} = exp(At)H.

Proof: We consider first the time-one-map Φ_{1}, again the linear part is diagonal with eigenvalues e^{λ_{1}}, e^{λ_{2}}. For these the non-resonance conditions for maps exp λ_{i} ≠ exp(kλ_{1}) · exp(lλ_{2}) for k, l ∈ N, k + l > 1 hold. The non-resonance conditions imply that we can formally remove all terms of algebraic order by a near-identity coordinate change. This is possible even in a D_{3}-equivariant setting [11]. So we still have to remove flat terms and discuss convergence. To remove these flat terms we use a version of Sternberg’s theorem [15, theorem 6.6.7]. The assumptions are fulfilled: The linear part is diagonal and the normal form which can be achieved by the above coordinate change is a convergent power series, since it is only linear. The theorem then gives the existence of a smooth diffeomorphism conjugating Φ_{1} and its normal form. Thus there exists a smooth diffeomorphism H_{1} linearizing Φ_{1} in a neighborhood of the origin. Furthermore the construction in [15] can be chosen to preserve D_{3}-equivariance, when we use invariant cut-off functions. Then the D_{3}-equivariant diffeomorphism H = ∫_{0}^{1} exp(−At)H_{1}Φ_{t}dt is the needed conjugacy for the entire flow on some neighborhood U of 0. This can be seen when using exp(−A)H_{1}Φ_{1} = H_{1}

exp(−As)HΦ_{s} = ∫_{0}^{1} exp(−A(t + s))H_{1}Φ_{t+s}dt = ∫_{s}^{s+1} exp(−Au)H_{1}Φ_{u}du = H − ∫_{0}^{s} exp(−Au)H_{1}Φ_{u}du + ∫_{1}^{s+1} exp(−Au)H_{1}Φ_{u}du = H − ∫_{0}^{s} exp(−A(u + 1))H_{1}Φ_{u+1}du + ∫_{1}^{s+1} exp(−Au)H_{1}Φ_{u}du = H□

Now we can compute the map P. After the coordinate change of the lemma the local maps are given by a linear flow. Then the stable and unstable manifolds coincide with the stable and unstable eigenspaces. To carry out the analysis we use again the scaled coordinates x = x_{1} + ix_{2}, y = y_{1} + iy_{2} ∈ C for some ϵ > 0 small. We know the homoclinic orbits explicitly by section 3 up to perturbations of order O(ϵ). While neglecting terms of order O(ϵ^{2}) the system in C^{2} is given by equation (6). Local maps: To compute the local maps we use a basis of eigenvectors of the

linearized system: For the eigenvalue λ_{1} = \frac{ϵν_{2}}{2} + \sqrt{\frac{ϵ^{2}ν_{2}^{2}}{4} + ν_{1}} > 0 we choose

v_{1}, v_{2} and for the eigenvalue λ_{2} = \frac{ϵν_{2}}{2} − \sqrt{\frac{ϵ^{2}ν_{2}^{2}}{4} + ν_{1}} < 0 the vectors v_{3}, v_{4}. The original basis of R^{4} ≅ C^{2} is given by (x_{1}, y_{1}, x_{2}, y_{2}).

\begin{matrix}v_{1} = (1 + λ_{1}^{2})^{− \frac{1}{2}} (1, λ_{1}, 0, 0)^{T} , & v_{2} = (1 + λ_{1}^{2})^{− \frac{1}{2}} (0, 0, 1, λ_{1})^{T} , \\ v_{3} = (1 + λ_{2}^{2})^{− \frac{1}{2}} (1, λ_{2}, 0, 0)^{T} , & v_{4} = (1 + λ_{2}^{2})^{− \frac{1}{2}} (0, 0, 1, λ_{2})^{T} . \end{matrix}

A vector a ∈ R^{4} is then denoted as a = a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3} + a_{4}v_{4}. The eigenvectors v_{1} and v_{3} span Fix(κ). The section S_{1}^{out} is then defined by

a_{1} = −c and max{|a_{2}|, |a_{3}|, |a_{4}|} < δ

with c small and 0 < δ ≪ c such that the section is completely inside U, where the flow is linearized. S_{1}^{in} is given by

a_{3} = −c and max{|a_{1}|, |a_{2}|, |a_{4}|} < δ.

We will also use rotated coordinate systems with basis vectors v′_{l} = γv_{l} and v′′_{l} = γ^{2}v_{l} with coefficients a′_{l}, a′′_{l}. Thus we can define the sections of the rotated homoclinic orbits.

S_{2}^{out} : a′_{1} = −c, max{|a′_{2}|, |a′_{3}|, |a′_{4}|} < δ
S_{3}^{out} : a′′_{1} = −c, max{|a′′_{2}|, |a′′_{3}|, |a′′_{4}|} < δ
S_{2}^{in} : a′_{3} = −c, max{|a′_{1}|, |a′_{2}|, |a′_{4}|} < δ
S_{3}^{in} : a′′_{3} = −c, max{|a′′_{1}|, |a′′_{2}|, |a′′_{4}|} < δ

First we compute P_{l}^{loc}, l ∈ {1, 2, 3}. The flow of the linear system is given by

Φ_{t}(a) = a_{1}v_{1}e^{λ_{1}t} + a_{2}v_{2}e^{λ_{1}t} + a_{3}v_{3}e^{λ_{2}t} + a_{4}v_{4}e^{λ_{2}t},  (7)

similarly in the primed versions for the rotated coordinate systems. Starting at a vector a ∈ S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} with P_{l}^{loc}(a) ∈ S_{1}^{out} (i.e. especially a_{1} < 0), the time t = (ln \frac{c}{|a_{1}|})/λ_{1} is needed to reach the S_{1}^{out} section. Then P_{l}^{loc}(a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3} + a_{4}v_{4})

= (−cv_{1} + a_{2} |\frac{c}{a_{1}} | v_{2} + a_{3} |\frac{a_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} v_{3} + a_{4} |\frac{a_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} v_{4})  (8)

with \frac{|λ_{2}|}{λ_{1}} = 1 + \frac{18}{49}ϵ^{2}ν_{1} + \frac{6}{7}ϵ\sqrt{\frac{9}{49}ϵ^{2}ν_{1}^{2} + ν_{1}} + O(ϵ^{3}). To understand the geometry of the local maps we compute how the preimage of the ‘out’-sections S_{l}^{out}, l ∈ {1, 2, 3} intersects the ‘in’-sections S_{l}^{in}, l ∈ {1, 2, 3} and how the images of S_{l}^{in} intersect the ‘out’-sections S_{l}^{out}. We start with the preimage of S_{1}^{out} intersected with S_{1}^{in}

S_{1,1}^{in}

= S_{1}^{in} ∩ P_{1}^{loc}^{−1}(S_{1}^{out})
= {(a_{1}, a_{2}, a_{3}, a_{4})|a_{3} = −c, max{|a_{1}|, |a_{2}|, |a_{4}|} < δ}
∩ \{(a_{1}, a_{2}, a_{3}, a_{4})|a_{1} < 0, max\{|a_{2}| |\frac{c}{a_{1}} | , |a_{3}| |\frac{a_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} , |a_{4}| |\frac{a_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} \} < δ\}
= \{(a_{1}, a_{2}, a_{3}, a_{4})| − δ < a_{1} < 0, |a_{2}| < δ |\frac{a_{1}}{c} | , a_{3} = −c, |a_{4}| < δ\}.

This is the slab with label ‘1’ infigure 5. Then the image of S_{1}^{in} inside S_{1}^{out} is given by S_{1}^{out} ∩ P_{1}^{loc}(S_{1}^{in}) = P_{1}^{loc}(S_{1,1}^{in})

= \{(a_{1}, a_{2}, a_{3}, a_{4})|a_{1} = −c, |a_{2}| < δ, −δ^{\frac{|λ_{2}|}{λ_{1}}} < a_{3} < 0, |a_{4}| < \frac{δ}{c} |a_{3}|\}.

This set is the slab with label ‘1’ in figure 4. To determine the images P_{2}^{loc}(S_{2}^{in}) ∩ S_{1}^{out} and P_{3}^{loc}(S_{3}^{in}) ∩ S_{1}^{out} we have just to rotate a part of the coordinate system. Inside the stable eigenspace (v_{3}, v_{4}) is changed to (v′_{3}, v′_{4}) and (v′′_{3}, v′′_{4}) respectively. Equation (7) holds for each eigenspace independently. Thus the restrictions are essentially the same as for S_{1}^{out} ∩ P_{1}^{loc}(S_{1}^{in}) just with a′_{3}, a′_{4} and a′′_{3}, a′′_{4} instead of a_{3}, a_{4}. Hence the slab S_{1}^{out}∩P_{1}^{loc}(S_{1}^{in}) has just to be rotated by 2π/3 and 4π/3 inside the (v_{3}, v_{4}) plane to get S_{1}^{out} ∩ P_{2}^{loc}(S_{2}^{in}) and S_{1}^{out} ∩ P_{3}^{loc}(S_{3}^{in}). A sketch of section S_{1}^{out} with the images of S_{l}^{in}, l ∈ {1, 2, 3} is given in figure 4. Next we will compute the preimage of S_{2}^{out} and S_{3}^{out} under P_{1}^{loc} to get the structure of S_{1}^{in}. When we use a rotated coordinate system (v′_{1}, v′_{2}) instead of (v_{1}, v_{2}) inside the unstable eigenspace, the time t = (ln \frac{c}{|a′_{1}|})/λ_{1} is needed to reach S_{2}^{out}. This yields to

P_{1}^{loc}(a′_{1}v′_{1} + a′_{2}v′_{2} − cv_{3} + a_{4}v_{4})
= (−cv′_{1} + a′_{2} |\frac{c}{a′_{1}} | v′_{2} − c |\frac{a′_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} v_{3} + a_{4} |\frac{a′_{1}}{c} |^{\frac{|λ_{2}|}{λ_{1}}} v_{4}).

So the preimage of S_{2}^{out} under P_{1}^{loc} is just S_{1,1}^{in} rotated by 2π/3 inside the unstable eigenspace. And finally for the preimage of S_{3}^{out} the coordinate system has to be rotated by 4π/3 in the unstable eigenspace. The section S_{1}^{in} with the preimages of S_{l}^{out}, l ∈ {1, 2, 3} is drawn in figure 5. Global maps: Next we approximate P_{l}^{glo} : S_{l}^{out} → S_{l}^{in} by an Taylor expansion using the linearisation along the homoclinic orbit. This approximation is valid by a general perturbation argument for hyperbolic sets, when we choose the size of the cubes δ small enough. We get a constant term of the global map when considering the splitting of the homoclinic orbit. The point (−c, 0, 0, 0) ∈ S_{1}^{out} is inside Fix(κ), hence it will be mapped to S_{1}^{in} ∩ Fix(κ). Thus the constant term is the distance of the stable and unstable manifolds inside Fix(κ). Using [13, Eq.(4.5.11)] this distance is given by d(ν_{2}, ϵ) = \frac{ϵM(ν_{2})}{∥f(q)∥} + O(ϵ^{2}), with Melnikov functional M(ν_{2}) and vector field f on Fix(κ). For our system this is d(ν_{2}) = ϵ\frac{4}{5c} \sqrt{ν_{1}}(ν_{2} + \frac{6}{7}ν_{1}). In (x_{1}, y_{1}, x_{2}, y_{2}) coordinates the linearisation along the homoclinic solution for ϵ > 0 is given by B = D_{(x,y)}f_{|(x(t),y(t))} =

( \begin{matrix}0 & 1 & 0 & 0 \\ ν_{1} + 2x_{1}(t) − ϵy_{1}(t) & ϵ(ν_{2} − x_{1}(t)) & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & ν_{1} − 2x_{1}(t) + ϵy_{1}(t) & ϵ(ν_{2} + x_{1}(t)) \end{matrix} ) ,

(9)

where x_{1}(t), y_{1}(t) are the non-zero components of the homoclinic orbit. This means that we have to solve the non-autonomous linear differential equation ξ̇ = Bξ. We use the block diagonal structure of the matrix. The first block describes the behavior inside the invariant subspace Fix(κ) and the second block the orthogonal complement Fix(κ)^{⊥}.

In the first block we are interested in the initial values ξ_{0} = a_{3}v_{3} inside S_{1}^{out}. One solution of the variational equation inside Fix(κ) is given by q̇_{0}(t) for ϵ = 0. Letting q_{0}(0) ∈ S_{1}^{out} and q_{0}(T) ∈ S_{1}^{in}, then ∥q_{0}(0)∥ = ∥q_{0}(T)∥ by symmetry. The vectors ξ_{0}, q̇_{0}(0) restricted to Fix(κ) are a fundamental system. The Wronskian W of this system is constant by Liouville’s theorem: Ẇ = trace(B_{|Fix(κ)})W = 0 . Therefore, as q̇_{0}(0) = kv_{1} and q̇_{0}(T) = kv_{3}, the projection of ξ_{0}(T) onto v_{1} is a_{3}. By smooth dependence on parameters this yields to P_{1}^{glo}(a_{3}v_{3}) = (1 + O(ϵ))a_{3}v_{1}. In the second block we consider initial values ξ_{1} = a_{2}v_{2} and ξ_{2} = a_{4}v_{4}. First assume that ϵ = 0. As (ν_{1} − 2x_{1}(t)) > ν_{1} > 0 and ξ_{1}^{(1)}(0), ξ_{1}^{(2)}(0) > 0 hold, the two components ξ_{1}^{(1)}(t) and ξ_{1}^{(2)}(t) are increasing. The global map also expands this vector for ϵ > 0 by the smooth dependence on the parameter ϵ for finite time. Hence in linear approximation we get P_{1}^{glo}(a_{2}v_{2}) = a_{2}(α_{1}v_{2} + α_{2}v_{4}) with α_{1}^{2} + α_{2}^{2} ≥ 1. Furthermore

α_{1} ≥ 0.9|α_{2}|  (10)

holds because the coefficients of the solution are positive in the x_{2}, y_{2} coordinates. Applying Liouville’s theorem again for ϵ = 0, the second initial vector is mapped to P_{1}^{glo}(a_{4}v_{4}) = a_{4}(β_{1}v_{2} + β_{2}v_{4}) with α_{1}β_{2} − α_{2}β_{1} = 1. Again ϵ > 0 will give perturbations of type 1 + O(ϵ), which we will suppress by still using the same notation. Full Map: We now consider only those points which are mapped under the local maps from S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} to S_{1}^{out}. When we use (v_{1}, v_{2}, v_{3}, v_{4}) as a coordinate system for all three ‘in’-sections then the composed mapping is given by

P_{1}^{glo} ∘ P_{l}^{loc} : S_{l}^{in} → S_{1}^{in}, l ∈ {1, 2, 3}

( \begin{matrix}a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{matrix} ) ↦ (\begin{matrix}\frac{4ϵ}{5c} \sqrt{ν_{1}}(\frac{6}{7}ν_{1} + ν_{2}) + (1 + O(ϵ))a_{3} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \\ α_{1}a_{2}|\frac{c}{a_{1}} | + β_{1}a_{4} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \\ −c \\ α_{2}a_{2}|\frac{c}{a_{1}} | + β_{2}a_{4} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \end{matrix} )(11)

Now we can use this to determine the return map P : S_{1}^{in} ∪ S_{2}^{in} ∪ S_{3}^{in} → S_{1}^{in}∪S_{2}^{in}∪S_{3}^{in}, where it is defined. Because of the symmetry the maps P_{2}^{glo}∘P_{l}^{loc} and P_{3}^{glo} ∘ P_{l}^{loc} are related to (11) by simple rotations of whole R^{4}. When changing to the rotated coordinates, the maps P_{2}^{glo} ∘ P_{l}^{loc} and P_{3}^{glo} ∘ P_{l}^{loc} are given by equation (11) with a_{i} replaced by a′_{i} and a′′_{i} . Therefore it is enough to consider a reduced map P̃ just as a map from one section S^{in} to itself. We just have to change the original labels ‘1’, ‘2’ and ‘3’ in the S_{2}^{in} and S_{3}^{in} sections. We will use a labeling relative to our position and call our position ‘1’, the next homoclinic orbit in the direction of the rotation is called ‘2’ and the other one ‘3’. Proof of theorem 1: The existence of a horseshoe for this reduced map P̃ will be shown. Analyzing the implications for the full map will prove the theorem. The conditions of definition 3 will be checked for the map:

P̃ : S^{in} → S^{in}

( \begin{matrix}a_{1} \\ a_{2} \\ a_{4} \end{matrix} ) ↦ ( \begin{matrix}\frac{4ϵ}{5c} \sqrt{ν_{1}}(\frac{6}{7}ν_{1} + ν_{2}) + (1 + O(ϵ))|a_{1}| |\frac{a_{1}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} \\ α_{1}a_{2}|\frac{c}{a_{1}} | + β_{1}a_{4} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \\ α_{2}a_{2}|\frac{c}{a_{1}} | + β_{2}a_{4} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \end{matrix} ) . (12)

The horizontal direction is α_{1}v_{2} + α_{2}v_{4} and the vertical directions are v_{1} and v_{4}. Using these as a new basis with coefficients ζ_{1}, ζ_{2} and ζ_{3} we define Δ as the product of discs with radii 2δ in ζ_{1} for D_{1} and δ in (ζ_{2}, ζ_{3}) for D_{2} with the further restriction \frac{δ}{6} < ζ_{2} < \frac{δ}{3}. This means we choose a coordinate system such that we can ignore any rotation of figure 4 under the global mapping to figure 5, even if |α_{2}| is not small. This can be done, because we estimated α_{1} ≥ 0.9|α_{2}| in (10). We just have to change the labels from a_{2} to ζ_{1}, a_{1} to ζ_{2} and a_{4} to ζ_{3}. As above we denote the rotated coordinates by ζ′_{i} and ζ′′_{i}. The rectangle is given in figure 6. We choose the distance of splitting d = \frac{δ}{6}. The two full components Δ_{1} and Δ_{2}, which have to be contained in Δ ∩ P̃(Δ), are the two top dotted slabs in figure 6. We consider the preimages of these two slabs under the original return map P, i.e. we are interested in the preimages of Δ_{1}, Δ_{2} ⊂ S_{1}^{in} under P_{1}^{glo} ∘ P_{2,3}^{loc}. Then we get (P_{1}^{glo} ∘ P_{2}^{loc})^{−1}(Δ_{1}) =

Γ̃_{1} = \{(ζ_{1}, ζ_{2}, ζ′_{3}) ∈ γ · Δ|0 < −ζ_{2} < 2δ, |ζ_{1}| ≤ δ |\frac{ζ_{2}}{cα_{1}} |\} ⊂ γΔ ⊂ S_{2}^{in}

and similarly (P_{1}^{glo} ∘ P_{3}^{loc})^{−1}(Δ_{2}) =

Γ̃_{2} = \{(ζ_{1}, ζ_{2}, ζ′′_{3}) ∈ γ^{2} · Δ|0 < −ζ_{2} < 2δ, |ζ_{1}| ≤ δ |\frac{ζ_{2}}{cα_{1}} |\} ⊂ γ^{2}Δ ⊂ S_{3}^{in}.

As we identified the three sections in this analysis of P̃, we deal with Γ_{1} and Γ_{2}, which are contained in the slabs with labels 2 and 3 in figure 6. The further restrictions are due to the possible additional expanding of the global map, i.e. the slabs are defined by

Γ_{1} = γ^{−1}Γ̃_{1} = \{(ζ′′_{1}, ζ′′_{2}, ζ_{3}) ∈ Δ|0 < −ζ′′_{2} < 2δ, |ζ′′_{1}| ≤ δ |\frac{ζ′′_{2}}{cα_{1}} |\}. (13)
Γ_{2} = γ^{−2}Γ̃_{2} = \{(ζ′_{1}, ζ′_{2}, ζ_{3}) ∈ Δ|0 < −ζ′_{2} < 2δ, |ζ′_{1}| ≤ δ |\frac{ζ′_{2}}{cα_{1}} |\}, (14)

After relabeling we have Δ_{1} = P̃(Γ_{1}) and Δ_{2} = P̃(Γ_{2}): The slab Γ_{1} is mapped by P_{1}^{loc} to S_{3}^{out} and then by P_{3}^{glo}, because of our relabeling it will be the slab coming from S_{2}^{out}, hence it is the dotted slab with label 2 and therefore Δ_{1} = P̃(Γ_{1}). In the same manner we get Δ_{2} = P̃(Γ_{2}).

Figure 6: The section S^{in} with rectangle Δ = D_{1}×D_{2} including Δ_{1,2} = P̃(Γ_{1,2})

So we can now check the conditions in the definition of the horseshoe. The two components Δ_{1} = P (Γ_{1}) and Δ_{2} = P (Γ_{2}) are full: π_{2}(Γ_{i}) = D_{2} for i = 1, 2, because we can choose ζ_{3} freely and ζ′_{2} < 0 (respective ζ′′_{2}) freely with |ζ′_{1}| (respective ζ′′_{1}) small inside Δ, i.e. we get all wanted ζ_{2} > 0 in the definition of Γ_{i}. For all ζ ∈ Γ_{i} the restriction π_{1}_{|f(Γ_{i}∩(D_{1}×π_{2}(ζ)))} is a bijection onto D_{1}. When we vary ζ_{1} for any given z = (a_{1}, α_{1}ζ_{1}, α_{2}ζ_{1} + a_{4}) ∈ Γ_{i} then P is affine linear (see (12)) and the projection π_{1} to the ζ_{1} component is injective, which is the a_{2} component in P. It is also surjective onto D_{1}, because the restrictions on a_{2} inside Γ_{i} ((14) and (13)) were given such that the maximal modulus of the a_{2} component is δ in the image. Next we check the first condition in definition 3. π_{2}(Δ′) ⊂ int(D_{2}) holds because of the contraction in the a_{1} = ζ_{2} and ζ_{3} component when choosing \frac{6}{7}ν_{1} + ν_{2} small enough. The ζ_{3} component is given by

|(α_{1}ζ_{3} − α_{2}ζ_{1}) |\frac{ζ_{2}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} | ≪ δ.

The other condition π_{1}(P^{−1}Δ′) ⊂ int(D_{1}) also holds, because |ζ_{1}| ≤ |ζ′′_{1}|/2 + \sqrt{3}|ζ′′_{2}|/2 and ζ′_{1}, ζ′_{2} and ζ′′_{1}, ζ′′_{2} are small enough by the definition of Γ_{1} and Γ_{2} (see (14),(13)). Finally we have to check the cone conditions, for which we need the linearisations of P̃ and P̃^{−1}. Suppressing all factors 1 + O(ϵ) these are given in the original (a_{1}, a_{2}, a_{4}) coordinates by DP̃_{x}(a_{1}, a_{2}, a_{4}) =

( \begin{matrix}− |\frac{a_{1}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & 0 & 0 \\ α_{1}a_{2} \frac{c}{a_{1}^{2}} − β_{1}a_{4} |\frac{a_{1}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & α_{1} |\frac{c}{a_{1}} | & β |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \\ α_{2}a_{2} \frac{c}{a_{1}^{2}} − β_{2}a_{4} |\frac{a_{1}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & α_{2} |\frac{c}{a_{1}} | & β_{2} |\frac{a_{1}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \end{matrix} )  (15)

and if (a_{1}, a_{2}, a_{4}) = P̃^{−1}(z) then DP̃_{x}^{−1}(z) = (DP̃_{x}(a_{1}, a_{2}, a_{4}))^{−1} is

( \begin{matrix}− |\frac{c}{a_{1}} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & 0 & 0 \\ −\frac{a_{2}}{a_{1}} |\frac{c}{a_{1}} |^{1− \frac{6}{7} ϵ\sqrt{ν_{1}}} a_{2} & β_{2} |\frac{a_{1}}{c} | & −β |\frac{a_{1}}{c} | \\ −a_{4} |\frac{c}{a_{1}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & −α_{2} |\frac{c}{a_{1}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & α |\frac{c}{a_{1}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \end{matrix} ) . (16)

Changing to the new ζ coordinates, we can easily check the cone conditions: The matrix DP̃(ζ_{2}, ζ_{1}, ζ_{3}) is given by

( \begin{matrix}− |\frac{ζ_{2}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & 0 & 0 \\ \frac{ζ_{1}}{α_{1}} _{\frac{c}{ζ^{2}_{2}}} −(α_{1}ζ_{3}−α_{2}ζ_{1}) \frac{β_{1}}{α_{1}} |\frac{ζ_{2}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & α_{1}|\frac{c}{ζ_{2}} |+ \frac{α_{2}}{α_{1}} |\frac{ζ_{2}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & \frac{β_{1}}{α_{1}} |\frac{ζ_{2}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \\ −(α_{1}ζ_{3}−α_{2}ζ_{1})|\frac{ζ_{2}}{c} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & α_{2}|\frac{ζ_{2}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & |\frac{ζ_{2}}{c} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} \end{matrix} )

(17)

and DP̃^{−1} by

( \begin{matrix}− |\frac{c}{ζ_{2}} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} & 0 & 0 \\ − |\frac{c}{ζ_{2}} |^{\frac{6}{7} ϵ\sqrt{ν_{1}}} \frac{ζ_{1}}{ζ_{2}α_{1}^{2}} & |\frac{ζ_{2}}{cα_{1}} | & −\frac{β_{1}}{α_{1}} |\frac{ζ_{2}}{c} | \\ ( \frac{α_{2}ζ_{1}}{c} −α_{1}(α_{1}ζ_{3}−α_{2}ζ_{1}))|\frac{c}{ζ_{2}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & −α_{2}|\frac{c}{ζ_{2}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} & α_{1}^{2}|\frac{c}{ζ_{2}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} +β_{1}α_{2}|\frac{ζ_{2}}{c} | \end{matrix} ) .

(18)

Now it is straightforward to see, that the term α_{1} |\frac{c}{ζ_{2}} | is the largest entry in the matrix (17). Then it preserves horizontal cones with constant e.g. s = 0.3 and expands them with expansion rate μ = α_{1} \frac{c}{2δ} > 1. Similar we see,

that α_{1}^{2} |\frac{c}{ζ_{2}} |^{1+ \frac{6}{7} ϵ\sqrt{ν_{1}}} is the leading term of the last two lines in (18). Hence it preserves vertical cones with constants s = 0.3 and expands them with constant λ^{−1} for λ = 2\frac{δ}{c} ^{\frac{6}{7} ϵ\sqrt{ν_{1}}} < 1. By Katok and Hasselblatt [15, p.274] we have the existence of an invariant hyperbolic Cantor set for the reduced map P̃, such that the dynamics are topological conjugate to the shift on two symbols for this reduced map. Then for the complete return map P there exists the shift of finite type with the transition matrix of the theorem: if an orbit is near the loop l in the present, then as the shift is on the symbols 2 and 3 the next loop in the itinerary has to be l + 1 mod 3 or l + 2 mod 3. Similarly the previous one was l + 1 mod 3 or l + 2 mod 3. Thus possible sequences (x_{n})_{n∈Z} have the form x_{n} ≠ x_{n+1}. The realization of all these sequences are guaranteed by the existence of the full shift on two symbols for P̃. Proposition 6.5.3 in [15] gives then even persistence under small C^{1} perturbations i.e. for an open set in parameter space. Hence we can include higher order terms. This also verifies the linear approximation of the global maps, for which all equivariant higher order terms can be neglected. It remains to check the symmetry properties of C, P and τ. The sections S_{k}^{in} are related by symmetry: S_{2}^{in} = γS_{1}^{in} and S_{3}^{in} = γ^{2}S_{1}^{in}. Furthermore S_{1}^{in} is κ-invariant and S_{3}^{in} = κS_{2}^{in}. Then P^{loc} is equivariant, because the linearizing diffeomorphism is D_{3}-equivariant. The global part is equivariant under rotation γ by construction. It is equivariant under κ by the following argument:

κ^{−1}P^{glo}κx = κ^{−1}Φ_{t(κx)}(κx) = Φ_{t(κx)}(x)
P^{glo}x = Φ_{t(x)}(x)

As the times t(κx) and t(x) are both close to the time needed of the homoclinic orbits from the ‘out’ section to the ‘in’ section, we get t(κx) ≈ t(x). As Φ_{t(x)}(x), Φ_{t(κx)}(x) ∈ S_{k}^{in} for the same k, we get t(x) = t(κx). Hence P^{glo} and P are equivariant. Then C = ∩_{n=−∞}^{∞}P^{n}(∪_{i=1,2,3}S_{i}^{in}) is D_{3} invariant, because P^{n} is equivariant and ∪_{i=1,2,3}S_{i}^{in} is invariant. If x ∈ C and x = P^{n}(a_{n}) with a_{n} ∈ S_{x_{n}}^{in}, then τ(x) = (x_{n})_{n∈Z} and the equivariance of τ can be easily checked using the representations (2) and (1). □

5 Discussion

In this section we give a more complete bifurcation diagram of the Takens-Bogdanov point with D_{3}-symmetry, using numerical studies of the normal form equations. Then we will describe an application to coupled oscillators. A major drawback in all further numerical studies is that there are not any stable dynamic features except the origin for some parameter values (μ_{1}, μ_{2} < 0). Therefore all direct simulations will not give much insight. Some conjectures about the periodic solutions created at Hopf bifurcations are possible using the path-following program AUTO [5]. The dynamics are fully understood in the invariant plane Fix(κ) by [14], see also [16]. There are two branches of periodic orbits starting from the D_{3}-Hopf bifurcation of 0 and the Hopf of N_{1} at μ_{2} = −μ_{1}, μ_{1} > 0. These branches do not undergo any folds and end at the homoclinic orbit. The global behavior of the other branches of periodic solutions are analyzed using AUTO. These branches of periodic solutions outside Fix(κ) seem to break down at the clover Figure 7: A periodic orbit with sequence 1213 of isotropy type Z̃_{2} of the branch coming from the D_{3}-Hopf. Parameter values are near the existence of the homoclinic clover. A projection on the v plane is shown, the crosses denote steady states. The trajectory of periodic orbit was approximated by integrating the differential equation starting at points, which described the periodic solution for AUTO.

structure of homoclinic orbits. Probably they are some of the periodic orbits of the subshift:

• The periodic solutions with isotropy type Z̃_{2} coming from the D_{3}-Hopf bifurcation have period 4 created by the sequences 1213, 2321 and 3132, see figure7.

• The solutions coming from the Hopf bifurcation of N_{1,2,3} at μ_{2} = μ_{1}, μ_{1} < 0 seem to have period 2, see figure 8.

• Even if the author could not pick up the Z̃_{3} periodic solutions starting at the D_{3}-Hopf bifurcation for path-following with AUTO. We might conjecture that this branch also ends at the homoclinic clover. They are probably of period 3 with sequences 123 and 132.

Figure 8: A periodic solution with sequence 23 with isotropy type Z̃_{2} on the branch coming from the Hopf bifurcation of N_{1}.

The entire horseshoe does persist for some parameter by a general perturbation argument for hyperbolic sets. It remains an open question how long for example the other periodic orbits created by the horseshoe persist. This will probably involve even more complicated bifurcations. We will consider an application to three coupled oscillators following Fiedler [6]. The system is given by

ẋ_{i} = f(x_{i}) + D(x_{i−1} + x_{i+1} − 2x_{i}) (mod 3), i = 1, 2, 3,  (19)

where x_{i} ∈ R^{k} and D = diag(d_{1}, …, d_{k}). This system is equivariant under permutations of x_{1}, x_{2} and x_{3}. The symmetry group is isomorphic to D_{3}. If we have a homogeneous solution, it will stay homogeneous under the evolution of time. We change to (x, y, z) coordinates where

x = x_{1} + x_{2} + x_{3}, y = x_{1} − x_{2}, z = x_{2} − x_{3}.

In the new coordinate system we then have:

ẋ = f(\frac{x + z + 2y}{3} ) + f(\frac{x + z − y}{3} ) + f(\frac{x − y − 2z}{3} )
ẏ = f(\frac{x + z + 2y}{3} ) − f(\frac{x + z − y}{3} ) − 3Dy
ż = f(\frac{x + z − y}{3} ) − f(\frac{x − y − 2z}{3} ) − 3Dz  (20)

We consider the homogeneous equilibrium (x_{0}, x_{0}, x_{0}) with linearisation f′(x_{0}) = A. In the new coordinates the equilibrium is (3x_{0}, 0, 0). Its Jacobian in the entire system is given by the block diagonal matrix diag(A, A − 3D, A − 3D). We choose k = 2 and for f the dynamics of the Brusselator as an easy example. It gives some insight into the possible behavior of chemical oscillator. So f = (f_{1}, f_{2}) is given by f_{1}(ξ_{1}, ξ_{2}) = a − (b + 1)ξ_{1} + ξ_{1}^{2}ξ_{2}, f_{2}(ξ_{1}, ξ_{2}) = bξ_{1} − ξ_{1}^{2}ξ_{2} with

a, b > 0, the equilibrium is x_{0} = (a, \frac{b}{a}) and A = ( \begin{matrix}b − 1 & a^{2} \\ −b & −a^{2} \end{matrix} ). We choose

D = \frac{1}{3} ( \begin{matrix}λ_{1} & 0 \\ 0 & λ_{2} \end{matrix} ) . Then A − 3D has a double eigenvalues 0 if

det(A − 3D) = λ_{1}λ_{2} + λ_{1}a^{2} − λ_{2}(b − 1) + a^{2} = 0
trace(A − 3D) = b − 1 − a^{2} − λ_{1} − λ_{2}  = 0

The solution is given by (λ_{1}, λ_{2}) = (b − 1 − a\sqrt{b}, −a^{2} + a\sqrt{b}), the diffusion constants λ_{1}, λ_{2} are positive and therefore somehow realistic for a < \frac{b−1}{\sqrt{b}}. So this D_{3}-equivariant system has a Takens-Bogdanov point since there is a double zero eigenvalue and A − 3D ≠ 0. We apply our bifurcation analysis for Takens-Bogdanov points with D_{3}-symmetry to this problem. It will be valid on a four-dimensional center manifold which is tangent to the subspace spanned by y and z. As trace(A) > trace(A − 3D) = 0 holds for λ_{1} + λ_{2} > 0, the matrix A has at least one eigenvalue with positive real part. Hence all dynamical features will be unstable if we consider the entire system. We could stabilize the system when using negative diffusion rates. But still all branching solutions have unstable directions due to the Takens-Bogdanov point making them inaccessible for direct numerical simulation. The origin will still correspond to the homogeneous solution even after the needed coordinate changes. Then an interpretation of a D_{3}-Hopf bifurcation in a ring of three coupled oscillators is given in [11, XVII.4]. The three different types of periodic solutions give different waveforms, phase shifts and resonances for the three cells. We furthermore expect near the bifurcation point the existence of inhomogeneous steady state solutions with two cells being in the same state. The periodic solutions coming from the Hopf bifurcations of these fixed points oscillate around these inhomogeneous steady states. In the first type two cells are in phase and in the other type two cells have a phase shift of π. The periodic solutions collapse at the homoclinic orbits, since by moving in parameter space parts of the periodic orbits reach a state very close to the homogeneous equilibrium. For these parameter values the system is already ‘chaotic’ because of the existence of shift dynamics. When the solution follows one of the loops of the ‘clover’ structure it has nearly a Z_{2} symmetry, i.e. two cells have nearly the same state. Hence within the shift dynamics we have arbitrary changes of two out of three cells being nearly in phase. The structure of our subshift forces the system to change to another pair of cells being in phase after some time. Because of the unstable directions of the hyperbolic structure this behavior is only observable as a transient motion to infinity or to some stable solutions far away from the Takens-Bogdanov point.

Acknowledgments

I would like to thank Peter Ashwin, Bernold Fiedler, Mark Roberts, Arnd Scheel and Ian Stewart for discussions.

References

[1] Armbruster,D., Guckenheimer,J. and Kim,S. [1989]. Chaotic dynamics in systems with square symmetry, Physics Letters A 140 no 7/8, p 416-420.

[2] Arnold,V.I.[1983].Geometrical methods in the theory of ordinary differential equations. Grundlehren der mathematischen Wissenschaften 250, Springer-Verlag,New York.

[3] Bogdanov,R.I.[1975]. Versal deformations of a singular point on the plane in the case of zero eigenvalues.Functional Anal. Appl.(2)9, p 144-145.

[4] Dangelmayr,G. and Knobloch,E.[1987].The Takens-Bogdanov bifurcation with O(2) symmetry.Phil. Trans. R. Soc. London A322, p 243-279.

[5] Doedel,E.J.,Wang,X.J. and Fairgrieve,T.[1994]. AUTO94 Software for continuation and bifurcation problems in ordinary differential equations. California Institute of Technology.

[6] Fiedler,B. [1986]. Global Hopf bifurcation of two-parameter flows, Arch. Rational Mech. Anal.94, p 59-81.

[7] Fiedler,B. [1996]. Global path following of homoclinic orbits in twoparameter flows, Pitman Math. Research Notes, No 352, p 79-146.

[8] Field,M.J.[1983]. Isotopy and stability of equivariant diffeomorphisms, Proc. London Math. Soc. 46, p 487-516.

[9] Gatermann, K. and Guyard, F. [1999] Gröbner bases, invariant theory and equivariant dynamics, Journal of Symbolic Calculation 27, p 1-28; program:URL: http://www.zib.de/gatermann /symmetry.html.

[10] Glendeninning,P.[1994] Stability, instability and Chaos. Cambridge University Press.

[11] Golubitsky,M., Stewart,I. and Schaeffer,D.G. [1988]. Singularities and groups in bifurcation theory, volume II.Appl.Math.Sci. 69,Springer-Verlag, New York.

[12] Golubitsky,M., Swift,J.W. and Knobloch,E. [1984]. Symmetries and pattern selection in Rayleigh-Bénard convection,Physica D 10,p 249-276.

[13] Guckenheimer,J. and Holmes,P. [1983].Nonlinear oscillation, dynamical systems, and bifurcations of vector fields. Appl.Math.Sci.42,Springer-Verlag, New York.

[14] Hirschberg,P. and Knobloch,E.[1991]. An unfolding of the Takens-Bogdanov singularity.Quart. Appl. Math. 49, p 281-287.

[15] Katok,A. and Hasselblatt,B.[1995]. Introduction to the modern theory of dynamical systems. Encyclopedia of mathematics and its applications v.54, Cambridge University Press.

[16] Lari-Lavassani,A., Langford,W.F., Huseyin,K. and Gatermann,K.[1999]. Steady-state mode interactions for D_{3} and D_{4}-symmetric systems. Dynamics of Continuous, Discrete and Impulsive Systems 6,p 169–209.

[17] Takens,F.[1974]. Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43, p 47-100.

Karsten Matthies

Institut für Mathematik I Freie Universität Berlin Arnimallee 2-6 D - 14195 Berlin Germany matthies@math.fu-berlin.de 486

Documenta Mathematica 4 (1999)