Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps V

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To cite this article: V. Gelfreich & V. Naudot (2009): Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps, Experimental Mathematics, 18:4, 409-427 To link to this article: http://dx.doi.org/10.1080/10586458.2009.10129058

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V. Gelfreich and V. Naudot

CONTENTS We study several families of planar quadratic diffeomorphisms 1. Introduction near a Bogdanov–Takens bifurcation. For each family, the skele- 2. Main Results ton of the associated bifurcation diagram can be deduced from 3. Computing the Width of the Homoclinic Zone the interpolating ﬂow. However, a zone of chaos conﬁned be- 4. Validation of the Numerical Method tween two lines of homoclinic bifurcation that are exponentially Acknowledgments close to one another is observed. The goal of this paper is to test References numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.

1. INTRODUCTION In this paper we study homoclinic bifurcations in the unfolding of a diffeomorphism near a fixed point of Bogdanov–Takens type. To begin with, we consider a planar diffeomorphism F : R2 → R2 with the origin as a fixed point and where

dF (0, 0) = Id+N,

where N ≡ 0 is nilpotent. The origin is said to be a fixed point of Bogdanov–Takens type. This latter terminology is better known for a singularity of a vector field X with linear part having double zero eigenvalues and a nonvanishing nilpotent part. Since this singularity is of Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 codimension 2, i.e., is twice degenerate, a generic unfolding will depend on two parameters, say (μ, ν). In the case of a vector field, such unfolding has been studied in [Arnol’d 83, Takens 74], and for maps it has been studied in [Broer et al. 96, Broer et al. 92]. For completeness, the corresponding bifurcation diagram is revisited in Figure 1 (left): a curve of homoclinic bifurcation emanates from the origin, below a curve of Hopf bifurcation; see [Broer et al. 91] for the terminology and more details. For parameters located between these two curves, the 2000 AMS Subject Classification: Primary 37D45, 37E30, 37G10 corresponding dynamics possess a stable limit cycle. Fi- nally, for the parameter {μ =0} on the ordinate, a saddle Keywords: Bogdanov map, Bogdanov-Takens bifurcation, Hénon map, separatic splitting, splitting function node occurs; see also [Broer et al. 91] for more details.

c A K Peters, Ltd. 1058-6458/2009 $ 0.50 per page Experimental Mathematics 18:4, page 409 410 Experimental Mathematics, Vol. 18 (2009), No. 4

+ ν (μ) Hopf Hopf − ν ν ν (μ ) Homoclinic bifurcation Homoclinic zone

μ μ

FIGURE 1. The Bogdanov–Takens bifurcation for a ﬂow (left) and for a diﬀeomorphism (right).

The Bogdanov–Takens bifurcation plays an impor- planar vector field. In the real analytic theory, this differ- tant role in dynamical systems, for instance from the ence is exponentially small [Broer and Roussarie 01, Gelf- bifurcation-theoretic point of view. Given any dynami- reich 03]. cal system depending on a parameter, the structure of the As stated above, for diffeomorphisms, the bifurcation bifurcation set can often be understood by the presence diagram (Figure 1, right) is essentially the same. How- of several high-codimension points that act as organiz- ever, there is no reason to expect a single homoclinic ing centers. Starting from the presence of (degenerate or curve, since a homoclinic orbit may be transverse and nondegenerate) Bogdanov–Takens points, we can initiate therefore persists. We observe a separatrices splitting, the search for subordinate bifurcation sets such as Hopf and instead of a single homoclinic curve, one observes bifurcation sets or homoclinic bifurcation sets. In this two curves ν+(μ)andν−(μ) respectively corresponding paper, we consider a nondegenerate Bogdanov–Takens to the first and last homoclinic tangencies. If a parame- point. ter (μ, ν) is (strictly) located in the region between those For the map F , an unfolding theory is developed in two curves, then the map Fμ,ν possesses transverse homo- [Broer et al. 96, Broer et al. 92]. It is very similar to the clinic trajectories. On the lower and upper boundaries, case of a flow. To be more precise, any unfolding the homoclinic connection becomes nontransverse. Un-

2 2 derstanding the width of this region is the main goal of Fμ,ν : R → R , (x, y) → (x1,y1), this paper. 2 of the map F (where (μ, ν) ∈ R and F = F0,0)can Before going any further, we set the following prelim- be embedded in a nonautonomous and periodic family inaries. Without loss of generality and up to an analytic Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 of vector fields Xμ,ν . The diffeomorphism coincides with change of coordinates, one has the time-1 map of that vector field; see also [Takens 74]. x1 = x + y, y1 = y + fμ,ν (x, y), (1–1) Using an averaging theorem [Neishtadt 84], the dependence on time is removed to exponentially small terms. where F Moreover, one can show that μ,ν is formally interpo- ∂f ∂f X˜ f 0,0 , 0,0 , . lated by an autonomous vector field μ,ν ; see [Gelfreich 0,0(0) = 0 = ∂x (0 0) = ∂y (0 0) 03]. This latter can be used to study the bifurcations of fixed points of Fμ,ν . Both approaches move the differ- We shall assume that ence between these two types of bifurcations beyond all ∂2f 0,0 , . algebraic order. ∂x2 (0 0) =0 (1–2) Although all the Taylor coefficients of X˜μ,ν can be x μ, ν written, there is no reason to expect convergence of the By the implicit function theorem, there exists ˜( ) such that corresponding series, since the dynamics for a planar dif- ∂f μ,ν x μ, ν , ≡ . feomorphism can be much richer than the dynamics of a ∂x (˜( ) 0) 0 Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 411

Applying a conjugacy of the form x =¯x +˜xμ,ν ,y=¯y depends on a perturbation parameter ε ≥ 0andahyper- (and after removing the bars) amounts to writing bolicity parameter h>0 and admits four separatrices, ε> 2 which break up when 0. fμ,ν (x, y)=−b (μ, ν)+b (μ, ν)x + b (μ, ν)y 00 20 01 In this special case, the author proposed an asymptotic b μ, ν xy . . . x, y , + 11( ) +ho t ( ) (1–3) expansion for the area of the main lobes of the resulting h.o.t.(x, y) stands for the higher-order terms in x and y. turnstile that takes the form of a power series (with even terms) in ε. See [Gelfreich and Simó 09, Delshams and From (1–2), we have b20(0, 0) =0. By a linear rescaling in the variables (x, y), we can fix Ramirez-Ros 99, Levallois and Tabanov 93] for more references on the computation of separatrices splitting. b20(μ, ν) ≡ 1. Furthermore, we put b11(0, 0) = γ and assume that the map In this paper our approach is experimental. We study examples and present strong numerical evidence for the (μ, ν) → (−fμ,ν (0, 0), following expansion of the width of the homoclinic zone: ∂fμ,ν − (0, 0)) = (−b (μ, ν),b (μ, ν)), log ν+(μ) − ν (μ) log K(μ, γ − 2) (1–8) ∂y 00 01 k/4 k/2 + mkμ +logμ nkμ , is a local diffeomorphism near (0, 0). From now on, we k≥1 k≥1 shall consider (b00,b01) as our parameters and rename where K is given by (1–7). them (again) by (μ, ν), i.e., we shall write (b00,b01)= (μ, ν). Therefore, our study concerns the map Remark 1.1. One easily checks that the following formula 2 fμ,ν (x, y)=−μ + x + νy + γxy +h.o.t.(x, y), (1–4) can be derived from (1–8): + − k/4 j and we restrict our study to the case in which μ>0. ν (μ) − ν (μ) K(μ, γ − 2) c˜k,jμ log μ, k≥ k In [Broer et al. 96] it was shown that 0 0≤j≤[ 2 ] (1–9) ± 5 √ 3 ν (μ)= (γ − 2) μ + O(μ 4 ). (1–5) 7 k k/ c where [ 2 ] stands for the integer part of 2andthe˜k,j’s In [Gelfreich 03] the following formula is proposed: depend on the mk’s and the nk’s. Observe that (1–9) is a double series with logarith- ν+ μ − ν− μ K μ, γ − Θ O μ1/4 μ , ( ) ( )= ( 2) γ + ( log ) mic terms, and numerically, for such an expansion, we (1–6) do not know any efficient techniques to compute the corresponding coefficients with high precision. However, the where √ √ asymptotic series (1–8) does not involve a double sum- 5 − 2π2/ 4 μ −6π2γ/˜ 7 K(μ, γ˜)= √ 5 · e e (1–7) mation and therefore the corresponding coefficients can μ 4 6 2 be computed with a much higher precision. is referred to as the leading part of the width and Θγ is an analytic invariant of the map F0,0 called a splitting Remark 1.2. Logarithmic terms may vanish. This occurs, constant; see [Gelfreich and Naudot 06]. for instance, in the case of the Hénon map; see the next Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 The goal of this paper is to establish, numerically, a section for more details. more accurate formula for the width of the homoclinic zone ν+(μ) − ν−(μ). The existence of asymptotic expan- Remark 1.3. From the numerical data, we are able to sions for the width of the homoclinic zone is currently an guess a simple analytic expression for the first logarithmic γ− 2 open question. Furthermore, if it does exist, it is very term in (1–8). More precisely, we have n = − 6( √ 2) , 1 7 2 hard to compute analytically. The difficulty here comes which is valid for all families studied in this article. from the fact that the normal form of the map coincides with that of the time-1 map of a vector field. Therefore Remark 1.4. Our computations require very high pre- the difference between the flow and the map is pushed cision, of several thousand digits. For this we use the beyond any algebraic order. package Mathematica, which allows us to set the preci- In the nearly integrable context, a polynomial asymp- sion of our computation as high as required. totic expansion for the splitting of the separatrices is proposed in [Ramirez-Ros 05]: the author considers the per- The paper is organized as follows. We shall consider turbation of a Hamiltonian (elliptic) billiard. The system three different families that satisfy (up to appropriate 412 Experimental Mathematics, Vol. 18 (2009), No. 4

smooth changes of coordinates) the setting above with 2.1 Asymptotic Sequences and Expansions diﬀerent nonlinear terms. As a result of our experiments, Let ε0 > 0 be given and let for each family we shall state the asymptotics for the

width of the homoclinic zone, confirming formula (1–8). S˜ = {f0,f1,...,fn,...}, Looking for the width of the homoclinic zone amounts f ≡ i> f ,ε → R to fixing one parameter, say μ, in the unfolding (1–3), where 0 1 and for each integer 0, i :(0 0) and finding the values of the second parameter, say ν, is a smooth positive function such that for which the system admits a first and a last homoclinic fi+1(x) tangency. We say that μ is the main parameter and ν is lim =0, x→0+ fi(x) the slave parameter. In Section 3 we briefly present the strategy that we follow. or in other words, fi+1(x)=o(fi(x)). Such a family S˜ is The rest of the section is devoted to the computation called an asymptotic sequence. of the invariant (stable and unstable) manifolds at the In this paper we shall consider the following asymp- saddle point. The splitting function, which is a key in- totic sequences: gredient of our techniques, is presented. Indeed, primary homoclinic orbits are in one-to-one P˜ = {1,x,x2 ...,xn,...}, (2–1) correspondence with zeros of the splitting function. i Therefore, the first and last homoclinic tangencies will that is, fi(x)=x and the Dulac asymptotic sequence correspond to double zeros of the splitting function. [Mardˇesic 94] Moreover, the splitting function is periodic, with expo- D˜ = 1,x,x2 log x, x2,x3,x4 log x, x4,...,x2n log(x), nentially decreasing harmonics, and is well approximated 2n 2n+1 by the splitting determinant. With good precision, com- x ,x ,... , puting the width of the zone amounts to the computation that is, for all integer n ≥ 0, of the first two harmonics of the splitting function and their dependence with respect to the slave parameter (the 2n 2n+1 f3n(x)=x ,f3n+1(x)=x , main parameter being fixed). f x x2n+2 x . For each family, we compute the width of the homo- 3n+2( )= log( ) (2–2) clinic zone for several hundred values of the main pa- Let φ :(0,ε0) → R be a smooth function. We say that rameter μ and collect the results in a set of renormalized data. φ(x) αnfn(x) (2–3) In the next step, the coefficients in (1–8) (considered n∈N as an ansatz) are extracted by interpolation techniques. φ {f } The remaining part of the paper is devoted to the is an asymptotic expansion of at 0 (where n n∈N is α verification of the validity of our results. More precisely, an asymptotic sequence and all n’s are real) if for all n we test the ansatz (1–8) and determine how precise our integer , data for the width of the homoclinic zone should be in {n}

Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 φ(x) − φ (x)=O(fn (x)), order to produce reliable results for the coeﬃcients of the +1

asymptotic expansion. φ{n} x n α f x where ( )= i=0 i i( ). With respect to expan- Finally, the constant coeﬃcient of the expansion sions of the form (2–3), no convergence is implied and should coincide with the splitting constant [Gelfreich 03]: often the αi’s are Gevrey-1, i.e., following the procedure developed in [Gelfreich and Nau- dot 06], we compare these constants with the constant there exist M>0, r>0, such that for all k ≥ 0, k coeﬃcients of the expansions. |αk|≤Mk!/r .

2.2 Quadratic Family Our ﬁrst example is the quadratic map 2. MAIN RESULTS 2 2 Q = Qμ,ν,γ : R → R , (2–4) Before presenting our main results, we ﬁrst introduce the x, y → x y,y x2 − μ γxy νy . following notions. ( ) ( + + + + ) Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 413

Observe that Q mimics the unfolding (1–1), i.e., it takes Arnold tongues, and chaos [Arrowsmith 93]; see also [Ar- the form of Fμ,ν and ignores the higher-order terms. We rowsmith and Place 90] for more details. normalize the width of the homoclinic zone associated to For this map the saddle point is located at the origin. the quadratic family by deﬁning This map can be transformed to the form (1–3). Indeed, let ν+(μ) − ν−(μ) Sγ (μ)= . a K(μ, γ − 2) u = x − , 2 Within the precision of our computations we observe a 2 a a v = y + x − +˜γ x − y + a x − + by. that 2 2 2 k/4 k/2 We retrieve the map (1–2) and higher-order terms (1–3) log Sγ (μ) Mk(γ)μ +logμ Nk(γ)μ , k≥0 k≥1 by putting (2–5) a ν = a + b − (˜γ +2) ,γ=˜γ +2,μ= a2/4, 2 where Mk(γ)andNk(γ) are real coeﬃcients that depend on the parameter γ. and 2 2 Comparing with (1–6), we see that fμ,ν =(x + y) − μ + γy . The parameter a is chosen to be the main parameter and exp(M (γ)) ≡ Θγ 0 b the slave parameter. From (1–5), the Bogdanov map

is the splitting constant associated with Q0,0,γ.More- admits a homoclinic zone near the line over, as we announced in the previous section, we have 6 b±(a)= aγ˜ + O(a3/2). 7 6(γ − 2) 2 N1(γ) ≡− √ . 7 2 The normalized width takes the form b+(a) − b−(a) For each value of γ,theMk’s and Nk’s can be computed S˜γ(a)= . ˜ K a2/ , γ with very high precision; see Table 1 for an illustration. ( 4 ˜) Formula (2–5) has been verified for the first 76 coef- Similarly to the quadratic family, our experiments M k ,..., N ,..., ficients: k, =0 50, and , =1 25. Al- showed that log S˜γ˜ satisfies the following asymptotics: though the precision decreases almost linearly as k and k/2 k increase, the first 76 coefficients can be computed with log S˜γ˜(a) Ak(˜γ)a +loga Bk(˜γ)a , (2–6) 60 correct digits. To compute these first coefficients, we k≥0 k≥1 need to compute the width of the homoclinic zone with at where Ak(˜γ)andBk(˜γ) are real coefficients that depend least 200 correct digits; see Section 4.2 for more details. on the parameter γ. Comparing with (1–6), we see that Even if we can propose an analytic expression for N γ 1( ), we have not been able to guess analytic expres- exp(A0(˜γ)) ≡ Θγ˜ sions for the other coefficients Nk and Mk. Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 is the splitting constant associated with B0,0,γ˜.More- 2 2.3 Bogdanov Family over, we observe numerically that B1(˜γ) ≡−(6˜γ/7) . Our second example is the Bogdanov map [Arrowsmith In Table 1, we provide typical results for our compu- 93, Arrowsmith and Place 90, Bogdanov 75] tation for the quadratic and Bogdanov maps. Although thefirst20coefficientsdonotshowatendencytogrow 2 2 B = Ba,b,γ˜ : R → R rapidly, we conjecture that the series (2–5) and (2–6) di- (x, y) → (x + y + x2 +˜γxy + ax + by, verge and belong to the Gevrey-1 class defined at the end of Section 2.1; compare with [Gelfreich and Simó 09]. y + x2 +˜γxy + ax + by).

The Bogdanov map, see for example [Arnol’d 72, Bog- 2.4 Hénon Map danov 81], is the Euler map of a two-dimensional system The last example to be considered in this paper is deﬁned of ordinary diﬀerential equations. by Arrowsmith studied the bifurcations and basins of H H R2 → R2, u, v → u ,v , attraction and showed the existence of mode locking, = a,˜ ˜b : ( ) ( 1 1) 414 Experimental Mathematics, Vol. 18 (2009), No. 4

Coeﬃcient Scale Value Coeﬃcient Scale Value A0 1 61.26721889 M0 1 −13.35083105 1/2 1/4 A1 a −29.82701974 M1 μ −35.34533603 1/2 B1 a log a −6.612244898 N1 μ log μ −9.183673469 1/2 A2 a 5.824479250 M2 μ −25.71572403 3/2 3/4 A3 a 17.41183781 M3 μ 60.69366755 2 B2 a log a 5.649967276 N2 μ log μ −41.92449575 2 A4 a −0.2874798361 M4 μ −215.4221683 5/2 5/4 A5 a −22.04012159 M5 μ −45.92851439 3 3/2 B3 a log a −6.966574583 N3 μ log μ −242.5333437 3 3/2 A6 a −6.250578833 M6 μ −960.8699623 7/2 7/4 A7 a 39.27382902 M7 μ 755.3601690 4 2 B4 a log a 10.92891913 N4 μ log μ −1587.303140 4 2 A8 a 19.31687979 M8 μ −3308.441120 9/2 9/4 A9 a −82.17477248 M9 μ 1090.837521 5 5/2 B5 a log a −20.01663759 N5 μ log μ −11017.80445 5 5/2 A10 a −50.35178499 M10 μ −134120.3771 11/2 11/4 A11 a 186.9039750 M11 μ 22519.75418 6 3 B6 a log a 40.63376347 N6 μ log μ −79363.78673 6 3 A12 a 128.7996196 M12 μ 904656.6104 13/2 13/4 A13 a −444.7385574 M13 μ 87833.05069

TABLE 1. The ﬁrst 20 coeﬃcients of the asymptotic expansion for the Bogdanov map (left,γ ˜ = 3) and the quadratic map (right, γ = −3). All the given digits are correct.

where where ν 2 2 ˜ u1 = v, v1 =ãv − bu +1. μ = 1+ − a,˜ ν = ˜b − 1. 2 See [Kirchgraber and Stoffer 06] for recent results con- The Hénon map admits a homoclinic zone near the line cerning this family. The Hénon map has a fixed point of ˜b± a ≡ , a ≥ . Bogdanov–Takens type ata ˜ = ˜b =1. (˜) 1 ˜ 1 a ˜b We chose ˜ as the main parameter and as the slave In the case of the Hénon map we define the normalized parameter. We note that the Hénon map is conjugate to width of the zone by the Bogdanov family in the special case ofγ ˜ =0.The ˜b+(ã) − ˜b−(ã) conjugacy is given by the following change of coordinates S˜(ã)= . and parameters: K(1 − a,˜ 0) Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 S˜ u = x, v = x + y + x2 + ax + by, Our numerical experiments show that has the following asymptotic expansion: ˜b = b +1, a˜ =(1+b/2)2 − a2/4. k/4 S˜(ã)= A˜k(1 − a˜) . (2–7) We also observe that the Hénon map can be transformed k≥0 to the form (1–1) with nonlinear term of the form (1–3) In contrast to the case of the Bogdanov map with by putting γ= 2 (i.e.,γ ˜= 0), the asymptotic expansion does not

1 ˜b +1 1 ˜b +1 1 contain logarithmic terms. We expect this property to u = x + ,v= x + + y. be closely related to the fact that the Hénon map con- a˜ 2 a˜ 2 a˜ tains a one-parameter subfamily of area-preserving maps: In the new system of coordinates, the Hénon map takes in [Brännström and Gelfreich 08], the authors study the form (1–3) with the exponentially small splitting of separatrices for area- preserving maps near a Hamiltonian saddle center bifur- 2 fμ,ν =(x + y) − μ + νy, cation. This study covers the case of the area-preserving Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 415

H´enon map. The authors show that the corresponding where

series does not contain logarithmic terms. In general, + − b (ai) − b (ai) γ S˜ a < c

3. COMPUTING THE WIDTH OF THE Remark 3.2. For the Hénon family, the set of data for the HOMOCLINIC ZONE normalized width is denoted by In this section, our approach concerns the quadratic fam- Z˜ = (|1 − a˜i|, S˜(ãi)), c<˜ |1 − a˜i| < d,˜ i =1,...,n˜ , ily Qμ,ν,γ. The other families (Bogdanov and Hénon) are (3–3) treated in a similar way. From now on, we do not mention the (μ, ν, γ) dependencies when it is not necessary, but where + − we may emphasize such dependence when it is needed. ˜b (ãi) − ˜b (ãi) S˜(ãi)= and 0 < c<˜ d.˜ K((1 − a˜i), 0) 3.1 Strategy (i) We assume an ansatz and in particular the one given in formula (1–8); 3.2 Invariant Manifolds We now compute the stable and unstable manifolds at the (ii) Computen ˜ (several hundred) values of the width for saddle point. In what follows, our description concerns values of μ1/4 ∈ [c, d], where 0 0 sufficiently small it is clear that λ1 < 1 <λ2. Then we compute the coefficients Mk, k =0,...,, At the saddle Sμ, the Taylor expansion of the local and Nk, k =1,...,/2, of the truncated expansion W s stable manifold loc and that of the local unstable man- W u /2 ifold loc are computed as follows. Denote by {3/2} k/4 k/2

Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 G μ M γ μ μ N γ μ ( )= k( ) +log k( ) 2 Φs :(R, 0) → (R , Sμ), k=0 k=1 √ ∞ ∞ H i k k to interpolate the set , i.e., for all integer = z → Φs(z)= μ + ϕkz , ψkz , 1,...,3/2+1,wehave k=1 k=1 /2 and k/4 k/2 log Sγ(μi)= Mk(γ)μi +logμi Nk(γ)μi . 2 k=0 k=1 Φu :(R, 0) → (C , Sμ), ∞ ∞ See Section 3.10 for more details. √ k k z → Φu(z)= μ + fkz , pkz , k=1 k=1 Remark 3.1. For the Bogdanov family, the set of data for the normalized width is denoted by the parameterizations that respectively satisfy 1/2 H˜ = (ai , log(S˜γ (ai))), c

u for all z near 0. Substituting the series into (3–4) and Let P0 =Φu(z0) ∈ W and choose m0 such that collecting terms of the same order in z,weget,fork ≥ 1, z λ−m0 z ≤ δ . 1 = 2 0 u k ϕk + ψk = λ ϕk, (3–5) k k 1 Then for any fixed m0,wehave ϕ ϕ γ ϕ ψ νψ λkψ , j k−j + j k−j + k = 1 k P Qm0 ◦ z , j=0 j=0 0 = lim Φu,Nmax ( 1) Nmax→∞ and and if z1 1, the convergence is fast. k pk + fk = λ pk, (3–6) Therefore, by putting k k 2 f f γ f p νp λkp . W u ≈ W u {Qm ◦ λ−mz , ≤ z ≤ z }, j k−j + j k−j + k = 2 k Nmax,m = Φu,Nmax ( 2 ) 0 0 j=0 j=0 m ≥ m0, we get an accurate estimate of the global un- λ > Q Since 2 1and is entire, from (3–4) we easily stable manifold. deduce that the radius of convergence of the series defined in (3–6) is infinite. Denote by the radius of convergence 3.3 Jacobian and Wronskian Functions N ∈ N of the series defined in (3–5). We fix max .Sincewe Before introducing the splitting function that will play a are after a single branch of the stable manifold, we write key role in the paper, we need to introduce two additional W s ≈ W s { z , ≤ z ≤ δ }, functions. loc Nmax = Φs,Nmax ( ) 0 s We first define where J : D→C,z→ det dQ(Φs(z)), Nmax Nmax √ k k Φs,N (z)= μ + ϕkz , ψkz max as the Jacobian of the map Q along the stable manifold k=1 k=1 z z , z . and where 0 <δs < . Φs( )=(Φs,x( ) Φs,y( )) We proceed in the same way for the local unstable A straightforward computation gives manifold, i.e., J z ν γ − z − γ z . W u ≈ W u { z , ≤ z ≤ δ }, ( )=1+ +( 2)Φs,x( ) Φs,y( ) (3–9) loc Nmax = Φu,Nmax ( ) 0 u In terms of series, from (3–9) we get where ∞ N N √ max max k k k J(z)= Jkz , (3–10) Φu,Nmax (z)= μ + fkz , pkz , (3–7) k k k =0 =1 =1 √ J ν γ − μ k> J and where 0 <δu 1. where 0 =1+ +( 2) and for all 0, k = The local invariant manifolds are computed with the (γ −2)φk −γψk. The Wronskian function (along the local following precision: stable manifold) Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 Nmax D→R,z→ z , Φs,N (z) − Φs(z) = O z , Ω: Ω( ) max z − z O zNmax . Φu,Nmax ( ) Φu( ) = satisfies

In particular, we have Ω(λ z)=J(z)Ω(z). (3–11) 1 λ z − Q ◦ z O zNmax . Φu( 2 ) Φu,Nmax ( ) = (3–8) We put Ω0 = 1 and look for a solution of (3–11) of the Since we need to study the map when homoclinic or- form

bits are present, we need a good estimate of the global ∞ z zlog J0/log λ1 zk . unstable manifold. Ω( )= 1+ Ωk (3–12) k=1 Recall that Φu is entire and therefore both components deﬁned in (3–7) converge for all z as Nmax →∞.How- With (3–11), (3–10), and (3–12), it follows that ever, for large z, the computation of the unstable man- n−1 ifold requires too many coeﬃcients, and therefore (3–7) 1 J J . Ωn = λ − J n + Ωj n−1−j is not very convenient. We then proceed as follows. 1 0 j=0 Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 417

Both series (3–10) and (3–12) are convergent. The Now we state the following (ﬂow box) lemma [Gelf- functions J and Ω will be approximated by reich 96].

Nmax k Lemma 3.3. There exist E0 > 0 and an analytic diﬀeo- JNmax (z)= Jkz k=0 morphism 2 and Ψ:(−E0,E0) × I0 → R ,

Nmax (E,t) → Ψ(E,t)=(X(E,t),Y(E,t)), log J0/log λ1 k ΩNmax (z)=z 1+ Ωkz k=1 such that the following hold: respectively. In this way, we have (i) Ψ(E,t +1)=Q ◦ Ψ(E,t); s | λ z − J z z | O |z|Nmax . (ii) Ψ(0, 0) = q0, Ψ(0,t) ∈ W for t ∈ I0; ΩNmax ( 1 ) Nmax ( )ΩNmax ( ) = ( ) loc (3–13) (iii) the Jacobian matrix

∂X/∂E ∂X/∂t 3.4 Splitting Function and Flow Box Theorem d E,t Ψ( )= ∂Y /∂E ∂Y /∂t (3–14) In this section, we introduce the key part of our techniques. Recall that in our investigation for the width of is such that the second column of dΨ(0,t) is the homoclinic zone, we fix the value of the main param- Γ˙ s = dΓs(t)/dt; eter and look for values ν+ and ν− of the slave parameter that correspond, respectively, to the first and the last ho- (iv) the map Ω(ˆ E,t)=detdΨ(E,t) satisfies Ω(0ˆ ,t)= z · λt moclinic tangencies. In order to find a homoclinic point Ω( s 1). we need to adjust the slave parameter in such a way that two curves on the plane have an intersection. Finding a The splitting function, denoted by Θμ,ν (t), is the first homoclinic tangency requires additional adjustments to component of make this intersection degenerate. This problem is much −1 −1 easier in the discrete flow box coordinates, in which the Ψ ◦ Γu(t) − Ψ ◦ Γs(t). stable curve coincides with the horizontal axis and the Applying Taylor’s theorem to the stable manifold, we get unstable one is a graph of a periodic function. A further −1 −1 simplification will be achieved by observing that this pe- Ψ ◦ Γu(t) − Ψ ◦ Γs(t) riodic function is very close to a trigonometric polynomial −1 = dΨ (Ψ(0,t)) · Γu(t) − Γs(t) (3–15) of the first order. 2 + O Γu(t) − Γs(t) . The splitting function Θ = Θμ,ν we shall introduce now is such that the first and the last tangencies corre- The following properties hold: spond to double zeros of Θμ,ν+ and Θμ,ν− respectively. Our investigation amounts then to finding values ν+ and • Let 0 < δ<π˜ .ThemapΘμ,ν has an analytic − Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 ν such that Θμ,ν+ and Θμ,ν− possess double zeros. continuation onto the rectangle In this section, we present the splitting function Θμ,ν B={t ∈ C | t = t + it ,t∈ I , |t |≤ , (3–16) for the quadratic map. In the case of the Bogdanov map, 0 | | < (π − δ˜)/| log λ |}. the splitting function is denoted by Θa,b. In what follows, 1 we assume that the parameter (μ, ν) is such that the The function Θμ,ν is periodic, so we can expand it Q map possesses a homoclinic orbit, i.e., the unstable into a Fourier series: manifold intersects the local stable manifold at a point ∞ u zu q s zs U 2iπt Φ ( )= 0 =Φ ( ). Then we fix a neighborhood of Θμ,ν (t)= Pj(μ, ν)e . q W s q the point 0. We parameterize loc near 0 by j=−∞ Γ I → z · λt , s : 0 Φs( s 1) As usual, the Fourier coefficients are defined by an integral: u where I0 =(−1, 1) and W near q0 by 1 −2ikπt t Pk(μ, ν)= Θμ,ν (t)e dt, for each k ∈ N. Γu I → u zu · λ . : 0 Φ ( 2) 0 418 Experimental Mathematics, Vol. 18 (2009), No. 4

Let 0 < <(π − δ˜)/| log(λ1)|. Since the integral where t e−2ikπt of Θμ,ν ( ) over the boundary of the rectangle 1 d ˜ t Γ t , Γ t − Γ t {(t + it ) | 0 ≤ t ≤ 1, 0 ≤ t ≤ } vanishes, we Θμ,ν ( )= z · λt det dt s( ) u( ) s( ) Ω( s 1) conclude that (3–24) 1 −2ikπt Θμ,ν (t)e dt (3–17) is the splitting determinant and 0 1 2 −2kπ −2ikπt |h˜μ,ν t | O | ˜ μ,ν t | . = e Θμ,ν (t + i )e dt. ( ) = sup Θ ( ) (3–25) t∈I0 0 Consequently, Note that even if the invariant manifolds and the Wronskian are computed with a very high precision, |P μ, ν |≤ | t i |·e−2|k|π, k( ) sup Θμ,ν ( + ) (3–18) μ,ν t t∈I the function Θ ( ) is evaluated only with a relative 0 O | | error of order (supt∈I0 Θμ,ν ). i.e., the harmonics of Θμ,ν decrease exponentially. The function Θμ,ν can be well approximated by the 3.5 Approaching a Primary Homoclinic Orbit sum of the zero- and first-order harmonics: In order to compute the width of the homoclinic zone, −2iπt we first find a value ν =¯ν where the map possesses a Θμ,ν (t)=P− (μ, ν)e + P (μ, ν) (3–19) 1 0 ν ν 2iπt primary homoclinic orbit. Near =¯, Lemma 3.3 will + P1(μ, ν)e + O2(t), then be applied, and the splitting determinant Θ˜ μ,ν will

double zeros) of Θμ,ν are in one-to-one correspondence Concretely, we write with primary homoclinic orbits (and homoclinic tangen- 1 1 cies) for the corresponding map; see [Gelfreich 96, Gelf- P (μ, ν) ≈ R (μ, ν)= Θ˜ μ,ν (0) + Θ˜ μ,ν 0 0 2 2 reich 03] for more details. We then replace the problem P− (μ, ν) ≈ R− (μ, ν) of finding homoclinic points and homoclinic tangencies to 1 1 that of finding double zeros of the splitting function Θμ,ν . 1 1 = Θ˜ μ,ν (0) − Θ˜ μ,ν (3–31) 4 2 3.6 First and Last Tangencies 1 −1 + i Θ˜ μ,ν −˜Θμ,ν The most natural way to compute the width of the 4 4 + + homoclinic zone is to estimate both ν = ν (μ)and P1(μ, ν) ≈ R1(μ, ν) − − ν = ν (μ). Write 1 1 = Θ˜ μ,ν (0) − Θ˜ μ,ν 4 2 t P μ, ν ˆ t . Θμ,ν ( )= 0( )+Θμ,ν ( ) (3–26) 1 −1 − i Θ˜ μ,ν − Θ˜ μ,ν . 4 4 At the first tangency (ν = ν−), the graph of the splitting function is located below the t-axis, and Θμ,ν− admits a From (3–19), thanks to (3–23) and (3–25), the approxi- − double zero. Therefore there exists t ∈ I0 such that mation here means that

− max{|R0(μ, ν) − P0(μ, ν)|, |R±1(μ, ν) − P±1(μ, ν)|} Θμ,ν− (t )=supΘμ,ν− (t) = 0 (3–27) t∈I 2 0 = O(sup |Θμ,ν (t)| ). (3–32) − t∈I0 = P0(μ, ν )+supΘˆ μ,ν− (t). t∈I0 Moreover, with (3–30) we have + At the last tangency (ν = ν ) the graph of the splitting 2 |R±1(μ, ν) − P±1(μ, ν)| = O(|R±1(μ, ν)| ). (3–33) function is located above the t-axis, and Θμ,ν+ admits a double zero. We then solve + Therefore there exists t ∈ I0 such that + + R0(μ, ν˜ ) − 2|R−1(μ, ν˜ )| =0, (3–34) + − − + t + t Θμ,ν ( )= infΘμ,ν ( ) = 0 (3–28) R0(μ, ν˜ )+2|R−1(μ, ν˜ )| =0. t∈I0 + = P0(μ, ν )+ inf Θˆ μ,ν+ (t). From (3–30), (3–32), (3–33), and (3–34), we have t∈I0 t− O R2 μ, ν− , Θμ,ν˜− ( )= ( −1( ˜ )) (3–35) If we neglect O2 in (3–20), then (3–27) and (3–28) are + 2 + + t O R μ, ν . equivalent to Θμ,ν˜ ( )= ( −1( ˜ )) Let t ∈ I . By the mean value theorem, we have P μ, ν+ − |P μ, ν+ | , 0 0 0( ) 2 −1( ) =0 (3–29) 2 − − R− (μ, ν¯) Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 P μ, ν |P μ, ν | . 0( )+2 −1( ) =0 |ν+ − ν˜+| = O 1 , (3–36) ∂Θμ,ν (t )/∂ν|ν ν 0 =¯ 2 In this way the problem of finding the first and the last R− (μ, ν¯) |ν− − ν˜−| = O 1 . tangencies is replaced by scalar equations in one variable ∂Θμ,ν (t0)/∂ν|ν=¯ν each. Therefore, instead of looking for intersections be- W u W u This approach gives a good estimate of the locus of tween loc and and their tangencies, we save consid- erable time by simply solving a scalar equation. Observe the homoclinic zone and therefore of the corresponding width, but requires the computation of both ν+ and ν− that for ν nearν ¯, for all t ∈ I0 we have to very high precision. To be more precise, assume that we want to compute the width of the homoclinic zone P0(μ, ν)=O(|P1(μ, ν)), (3–30) for a given value of the main parameter with N correct sup |Θμ,ν (t)| = O(|P1(μ, ν)). t∈I0 digits, while the width of the zone (roughly estimated with formula (1–6)) satisfies From (3–19) we need only four points per period to eval- 10−Nz−1 ≤ ν+ − ν− < 10−Nz , (3–37) uate P0 and P±1. 420 Experimental Mathematics, Vol. 18 (2009), No. 4

+ where Nz 1. Thus we need to compute both ν and We observe (numerically) that Θˆ μ,ν does not change − − ν with Nz +N correct digits. We observe (numerically) much with respect to ν. More precisely, for all ν ≤ ν3 ≤ + − + that ν , ν ≤ ν4 ≤ ν ,andforallt ∈ I0,

− |R−1|(μ, ν¯) |Θˆ μ,ν (t) − Θˆ μ,ν (t)| ν˜+ − ν˜ = O ; (3–38) 4 3 O |P | μ, ν . ∂ t /∂ν| = ( −1 ( ¯)) (3–43) Θμ,ν( 0) ν=¯ν ν4 − ν3

also compare with (3–42) below. Thus, with (3–20) and (3–30) we have Therefore with (3–36) and (3–38),ν ˜+ − ν˜− gives an sup Θμ,ν− (t) − inf Θμ,ν+ (t) (3–44) t∈I estimate of the width with a relative error of the same t∈I0 0 order. In particular, this means that we cannot choose N |P | μ, ν O P2 μ, ν . =4 −1 ( ¯)+ ( −1( ¯)) bigger than Nz. With this method, thanks to (3–33), the estimates of P0(μ, ν)andofP−1(μ, ν) are obtained with Furthermore, with (3–26) we have |P μ, ν | a relative error of the same order as −1( ¯) .Thisre- ∂P ∂Θμ,ν ∂Θˆ μ,ν quires the computation of the splitting determinant with 0 t t − t ∂ν ( )= ∂ν ( ) ∂ν ( ) (3–45) the same relative precision. When the main parameter ν=ν2 ν=ν2 ν=ν2 ∂ tends to 0, since the eigenvalues λ and λ tend to 1, the Θμ,ν 1 2 = (t)+O(P−1(μ, ν¯)). ∂ν ν=ν2 number of iterations (i.e., m0) and the number of terms N in (3–7) (i.e., max) required to compute the unstable With (3–22) and (3–23) we have manifold need to be chosen bigger and bigger. Moreover, in order to guarantee (3–33), we need to ∂Θμ,ν ∂Θ˜ μ,ν |ν ν (t)= (t)+O(Θ˜ μ,ν (t)). (3–46) ∂ν = 2 ∂ν ¯ have P0(μ, ν)=O(P−1(μ, ν)), i.e., (3–30), which re- ν=ν2 quires that the local stable and unstable manifolds be We then write close to one another, and more precisely, that ∂ ˜ ˜ t − ˜ t Θμ,ν t · ν − ν Θμ,ν4 ( ) Θμ,ν3 ( )= ( ) ( 4 3) Γu(t) − Γs(t) = O(K(μ, γ − 2)). (3–39) ∂ν ν=ν3 2 + O((ν4 − ν3) ), (3–47) As a conclusion, as the main parameter tends to 0, this approach becomes more and more delicate. and therefore In what follows, we propose another approach, one ˜ t − ˜ t ∂ ˜ Θμ,ν4 ( ) Θμ,ν3 ( ) Θμ,ν − that does not require the computation of P (μ, ν), yet = (t) + O((ν+ − ν )). 0 ν − ν ∂ν ν ν ν ν 4 3 = 2 still requires a ﬁrst value of =¯such that (3–30) and (3–48) gives an estimate of the width with the same precision. We observe (numerically) that the left-hand side of 3.7 The Real Approach (3–48) stays away from 0 as the main parameter tends v > From (3–27) and (3–28) we have to 0. More precisely, there exists 0 0 such that for all μ>0, ν3, ν4 nearν ¯ and for all t ∈ I0, Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 + − P0(μ, ν ) − P0(μ, ν )=− inf Θˆ μ,ν+ (t)+supΘˆ μ,ν− (t). t∈I ˜ ˜ 0 t∈I0 Θμ,ν4 (t) − Θμ,ν3 (t) >v0. (3–49) (3–40) ν4 − ν3

Furthermore, from the mean value theorem, there exists With (3–30), (3–45), and (3–48) and by choosing ν3 and ν− ≤ ν ≤ ν+ 2 such that ν4 suﬃciently close to one another, we have ∂P ∂P ˜ t − ˜ t + − 0 + − 0 Θμ,ν4 ( ) Θμ,ν3 ( ) P0(μ, ν ) − P0(μ, ν )= · (ν − ν ). | O |P | μ, ν ∂ν ν ν ν=ν2 = + ( −1 ( ¯)) = 2 ∂ν ν4 − ν3 (3–41) − + O(ν+ − ν ). (3–50)

Thus we get Therefore, with (3–42), (3–44), (3–49), and (3–50)) we can write ˆ − ˆ + supt∈I Θμ,ν (t) − inft∈I0 Θμ,ν (t) ν+ − ν− = 0 . (3–42) ∂P /∂ν ν+ − ν− Z μ O Z2 μ , 0 ν=ν2 = ( )+ ( ( )) (3–51) Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 421

where Since |P±1|(μ, ν0)=O(K(μ, γ − 2)), we then conclude 4|P− |(μ, ν¯)(ν − ν ) that P0(μ, ν0)=O(Δ0). Since Δ0 K(μ, γ − 2), Z(μ)= 1 3 4 ,t∈ I . (3–52) ˜ t − ˜ t 0 0 we are not able to compute precisely the ﬁrst harmonic Θμ,ν3 ( 0) Θμ,ν4 ( 0) P−1(μ, ν0) with the real approach. However, instead of Thanks to (3–31) we obtain the following estimate for considering t ∈ I0 as real, we now consider t in the com- the width of the homoclinic zone: plex interval [δi, δi + 1]. Recall that the Fourier coeﬃ- |R μ, ν | ν − ν + − 4 −1( ¯) ( 3 4) t ν − ν ≈Zr(μ)= . (3–53) cients of Θμ,ν0 ( )are Θ˜ μ,ν (t ) − Θ˜ μ,ν (t ) 3 0 4 0 1 P μ, ν t dt, P μ, ν With (3–33), (3–51), and (3–52) it follows that 0( 0)= Θμ,ν0 ( ) −1( 0) 0 + − 2 ν − ν −Zr μ O Z μ . 1 ( ) ( )= ( r ( )) (3–54) 2πit = e Θμ,ν0 (t)dt. This real approach gives a good estimate of the width 0

of the homoclinic zone with the same precision as before Since Θμ,ν0 is periodic and analytic in B defined in in (3–36). Moreover, it requires only the computation of (3–16), we have P− (μ, ν¯) and that of Θ˜ μ,ν (t) for two different values of 1 iδ+1 ν. However, we still need to find a value of ν =¯ν such P μ, ν e2πit t dt. −1( 0)= Θμ,ν0 ( ) (3–57) that (3–30) holds. iδ In what follows we present another way to compute With (3–19) we have the width: In the new approach, Γu does not need to e2πit t P μ, ν e2πitP μ, ν return near Γs as close as in (3–39). In this way, we will Θμ,ν0 ( )= −1( 0)+ 0( 0) (3–58) 4πit 2πit be able to compute the splitting determinant with less + e P1(μ, ν0)+e O2(t), precision. This alternative approach consists in looking where at the splitting function for complex values of t. O t O | 2 iδ t | . 2( )= sup Θμ,ν0 ( + ) (3–59) 3.8 The Complex Approach t ∈I0 Now we present another way to compute the first har- With (3–18) we have monic, with less precision than in the real approach, but −2π with less effort. P±1(μ, ν0)=O(e ). Recall that formulas (1–5) and (1–6) already give the P μ, ν O following estimate: Therefore, since 0( 0)= (Δ0), with (3–19) and (3–59), we have − ν+ − ν = O(K(μ, γ − 2)). (3–55) | | iδ t O e2πδ−2π O |P |e2πδ , sup Θμ,ν0 ( + )= = −1 Moreover, with (3–49) and (3–52), (3–55) gives us a t ∈I0 rough estimate of |P±1|, i.e., we have |P±1|(μ, ν)= (3–60) O(K(μ, γ − 2)). and further we have 2πδ Take 0 δ< and Δ0 = K(μ, γ − 2)e such that Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 4iπt −2π(+2δ) K(μ, γ − 2) Δ0. Assume that we have found a value P1(μ, ν0)e = O e , (3–61) ν ν of = 0 such that 2πit +2πit 2 4πδ e O2(t) = O |e |P−1| (μ, ν0)e | K μ, γ − t ≤ . −2π(2−δ) ( 2) sup Θμ,ν0 ( ) Δ0 (3–56) = O(e ). t∈I0

Observe that looking for such a value of ν = ν0 requires We distinguish two cases: less eﬀort than searching forν ¯ where supt∈I Θμ,ν¯ = δ> − δ< δ 0 Case 1: 3 .Inthiscase,2 +2 ,andfrom O(K(μ, γ − 2)). In particular, we need to compute (3–61) we have the splitting function with a relative error only of order 4iπt 2πit t |P1(μ, ν0)e ||e O2(t)|. supt∈I0 Θμ,ν0 ( ). With (3–19), there exists s ∈ I such that 0 Using (3–23), we write −2iπs sup Θμ,ν (t)=P (μ, ν )+P− (μ, ν )e 0 0 0 1 0 2πit 2πit ˜ 2πit˜ t∈I0 e Θμ,ν0 (t)=e Θμ,ν0 (t)+e hμ,ν0 (t) (3–62) 2iπs + P1(μ, ν0)e + O2(s). = A(t)+E1(t), 422 Experimental Mathematics, Vol. 18 (2009), No. 4

2πit where A(t)=P−1(μ, ν0)+e P0(μ, ν0), and with where (3–61), − π −δ E t O e 2 (2 ) . C−1(μ, ν0) 1( )= ( ) −2πδ 1 1 Observe that for t ∈ [iδ, iδ +1],wehave = e Θˆ μ,ν (0) − Θˆ μ,ν 4 0 0 2

2iπt˜ −2π(2−δ) − |e hμ,ν0 (t)| = O(e ). 1 1 − i Θˆ μ,ν − Θμ,ν , 0 4 0 4 In this case and iδ+1 A t dt 1 A iδ A iδ / . ( ) = ( )+ ( +1 2) (3–63) |r | O e−2π(2−δ) O C2 μ, ν e2πδ . iδ 2 ˜2 = ( )= ( −1( 0) ) But with (3–62), we have When δ> /3, the estimate given in (3–65) requires the iδ iδ ˜ +1 +1 computation of the splitting determinant Θμ,ν0 (t)atonly e2πit ˜ t dt A t dt O e−2π(2−δ) . Θμ,ν0 ( ) = ( ) + ( ) two different values of t. However, when δ ≤ /3, (3–66) iδ iδ t (3–64) requires four different values of . The computation in the first case is faster, but since δ is bigger, we lose some Finally, from (3–57), (3–62), (3–63), and (3–64) we get precision. From the complex approach, the width of the homo- P μ, ν C μ, ν r , −1( 0)= −1( 0)+˜1 (3–65) clinic zone is approximated by

where ν+ μ − ν− μ ≈Z μ , ( ) ( ) c( ) (3–67) 1 −2πδ 1 C− (μ, ν )= e Θ˜ μ,ν (iδ) − Θ˜ μ,ν iδ + , 1 0 2 0 0 2 where −2π(2−δ) |C μ, ν | ν − ν |r˜1| = O(e ). −1( 0) ( 3 4) Zc(μ)=4 , Θ˜ μ,ν (t) − Θ˜ μ,ν (t) 3 4 Case 2: δ ≤ . In this case, from (3–61) we have 3 ν ν ν C μ, ν with 3 and 4 chosen near 0.Since −1( 0)= 4iπt 2πit −2π |P1(μ, ν0)e |≥|e O2(t)|. O(e ) with (3–65) or (3–66), we have

P e4πit |Z μ − ν+ μ − ν− μ | O C2 μ, ν+ e2πδ . Therefore we cannot neglect the term 1 from the c( ) ( ( ) ( )) = ( −1( ) ) integration in (3–57). Thus we write (3–68)

e2iπt ˜ t P μ, ν e2πitP μ, ν Θμ,ν0 ( )= −1( 0)+ 0( 0) 3.9 Real versus Complex 4πit + P1(μ, ν0)e + E(t) The real approach provides a good estimate of the width = A˜(t)+E(t), of the homoclinic zone. More precisely, formula (3–53) gives an estimate of the width with a relative error of the Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 where with (3–61), we have same order; see (3–54).

2πit 4πit However, this approach requires that one compute the A˜(t)=P− (μ, ν )+e P (μ, ν )+P e , 1 0 0 0 1 splitting determinant with the same relative error. This −2π(2−δ) E(t)=O(e ). task becomes more and more delicate as the main parameter approaches zero. In this case, The complex approach requires less precision for the iδ+1 computation of the splitting determinant (and therefore A˜ t dt 1 A˜ iδ A˜ iδ 1 A˜ iδ 1 ( ) = ( )+ + + + can be computed much faster) as δ is chosen larger. How- iδ 4 2 4 3 ever, the estimate of the width is obtained with less pre- + A˜ iδ + , cision. 4 In the case of the Bogdanov map, we use similar nota- and we get tion: b is the slave parameter and a is the main parameter. The ﬁrst harmonic computed with (3–31) is denoted P−1(μ, ν0)=C−1(μ, ν0)+˜r2, (3–66) by R−1(a,¯b), where ¯b is the analogue ofν ¯ in the case of Gelfreich and Naudot: Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps 423

2πδ FIGURE 2. Left: (I) Graph of log10 |e | against the parameter a.Recallthatδ is chosen such that K(μ, γ − 2) μ,ν t ≤ 0 O 0 μ, ν0 | 1 a, b0 | a supt∈I0 Θ 0 ( ) Δ = (P ( )) (3–56). (II) Graph of log10 C ( ) against the parameter , which essentially coincides with the graph of log10 |R1(a,¯b)| against a. (III) The corresponding error, i.e., log10 |C1(a, b0)−R1(a,¯b)| against a. Right: (IV) Computation of the magnitude of the homoclinic zone with the real approach, i.e., log10(Zr(a)) against a.(V)Thegraphoflog10 |Zr(a) −Zc(a)| against a.

−300 the quadratic map. Similarly, C−1(a, b0) stands for the ative error or order 10 . Moreover, we just need to first harmonic computed with (3–65) or (3–66), where b0 find a first value of b = b0 such that ν is the analogue of 0 in the case of the quadratic map. Γ t − Γ t ≈− . log10 sup u( ) s( ) 300 For illustration, we compute the first harmonic and t∈I0 the width of the homoclinic zone using both approaches However, instead of having a relative error for the for the Bogdanov map (˜γ =3);seeFigure2. − width of order 10 1000 as with the real approach, we We easily verify that obtain an estimate of the width with a relative error of C a, b − R a,¯b −300 log10 −1( 0) −1( ) order 10 . ≈ |R| a,¯b e2πδ , Now that we can compute the width of the homoclinic 2log10( −1( )) + log10( ) zone, we do so forn ˜ (several hundred) values of μ1/4 and which follows from (3–54) and (3–68). Furthermore, we establish the set also verify that 1 4 + − H = (μ , log(δi)),δi = ν (μi) − ν (μi),c<μi

Coeff. Scale Hénon Map 6 4. VALIDATION OF THE NUMERICAL METHOD A˜0 1 2.4744255935532510538408 × 10 1/4 6 A˜1 |a˜ − 1| −2.878113364919828141704 × 10 To test the validity of our result, we propose three tests. 1/2 6 A˜2 |a˜ − 1| 1.8211174314566012763528 × 10 3/4 First, we test the validity of the ansatz. In what fol- A˜3 |a˜ − 1| −412552.07921345800366019 A˜4 |a˜ − 1| −309961.28583121907079391 lows, the experiments are presented in the cases of the 5/4 A˜5 |a˜ − 1| 257055.93487794037812901 Bogdanov map and the Hénon map, but the same test 3/2 A˜6 |a˜ − 1| −56830.201956139947433580 can be applied in the case of the quadratic map, thereby 7/4 A˜7 |a˜ − 1| −12386.990577003086404843 2 confirming formula (2–5). A˜8 |a˜ − 1| −11792.964908478734939516 9/4 A˜9 |a˜ − 1| 18742.189161591275288347 5/2 A˜10 |a˜ − 1| −4774.6727458595190485600 4.1 Extrapolability 11/4 A˜11 |a˜ − 1| −2822.9663193640187675835 3 We claim that the ansatz (1–8) is appropriate for an A˜12 |a˜ − 1| 3276.6438736125169964394 13/4 A˜13 |a˜ − 1| −1910.5466958542171966392 asymptotic expansion of the width if the following cri- 7/2 A˜14 |a˜ − 1| 7704.6605615546853854041 terion is satisfied. 15/4 A˜15 |a˜ − 1| −7827.0351891507566506398 G ,ε → R 4 Assume that a function :(0 0) possesses the A˜16 |a˜ − 1| 13919.102717097324631620 17/4 following asymptotics at 0: A˜17 |a˜ − 1| −11932.139780641352182621 A˜ |a − |9/2 . 18 ˜ 1 22120 721696311178434645 ∞ G(x) αifi(x), TABLE 2. The 19 first coefficients in (2–6). All the i given digits are correct. We also conjecture that the =0 series (2–6) belongs to the Gevrey-1 class. where {fi(x),i∈ N} is the asymptotic sequence defined in (2–2). that is, the coefficients αi have been constructed in such Define a way that the map 3k+3 {3k+3} {n˜} G (x)= αifi(x). (4–1) φ :(0,ε0) → R, (3–70) i=0 n˜ {n˜} x → φ (x)= αifi(x) We have i=0 {3k+3} 2k+3 |G(x) − G (x)| = x α k + ε (x) , (4–2) interpolates the set of data H. 3 +4 1 To illustrate our techniques, Table 1 indicates the first where ε1(x)=O(x). From (4–2) we get coefficients of the interpolation (ñ ≈ 100) in the case of γ { k } the Bogdanov map (left, ˜ = 3) and in the case of the log |G(x) − G 3 +3 (x)| quadratic map (right, γ = −3). In the case of the Hénon =log|α k | +(2k +3)logx +log 1+ε(x) (4–3) map, replacing the ansatz (1–8) by (2–7), we obtain the 3 +4 |α | k x ε x , coefficients indicated in Table 2. =log 3k+4 +(2 +3)log + 2( ) Redoing the above interpolation for different values of where ε2(x)=O(|x|). Downloaded by [Florida Atlantic University] at 13:05 21 March 2012 γ reveals that the first nonlinear term in the expansion This implies that the quantity log |G(x)− G{3k+3}(x)| satisfies x is approximatively linear in log . This must be satisfied 6(γ − 2) 2 for values of x outside the data set used for interpolation. N1(γ)=− √ (3–71) 7 2 Now we apply this criterion to the Bogdanov family. Recall that a is the slave parameter and b is the main in the case of the quadratic map, and parameter. Take an interval [c ,d ], where c

−40 −240 −50 −260 −60 −70 −280 −80 −300

−10.5 −10 −9.5 −8.5 −320 −100 −20 −19 −18 −17 FIGURE 3. Plot of the set Lc,d (4–4) for the Bogdanov −5 −3 . Plot of the set L˜c,d (4–6)fortheH´enon map in the caseγ ˜ =3,c =3.5 × 10 , d =9.4 × 10 , FIGURE 4 map, c =1.69 × 10−10, d =1.125 × 10−7, k˜ = 60, n˜ = 140, k = 36. n˜ = 140.

We plot the set √ The set {3k+3} + − Lc,d = (log(a), log |G ( a) − (b (a) − b (a))|), L˜c,d = log(˜a − 1), (4–6) c

γ˜ Θ(˜γ) − log10 |(Θ(˜γ) − exp(A0(˜γ)))/Θ(˜γ)| −2 0.28524883190581352 65.23 0 2.47442559355325105 × 106 90.01 3 4.05522622851113044 × 1026 62.04 6 2.70980378082897208 × 1047 60.03 7 3.09943158275750458 × 1054 59.6 9 5.18377311752952789 × 1068 55, 6

TABLE 3. The value of Θ(˜γ) for different values ofγ ˜. We clearly observe that the splitting constant coincides with the first term in (2–6) up to at leat the first 50 digits.

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V. Gelfreich, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ([email protected])

V. Naudot, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ([email protected])

Received June 28, 2008; accepted November 18, 2008.