On a Homoclinic Origin of Hénon-like Maps

Transcription

1 ISSN , Regular and Chaotic Dynamics, 2010, Vol. 15, Nos. 4 5, pp c Pleiades Publishing, Ltd., Special Issue: Valery Vasilievich Kozlov 60 On a Homoclinic Origin of Hénon-like Maps S. V. Gonchenko *, V.S. Gonchenko **, and L. P. Shilnikov *** Research Institute of Applied Mathematics and Cybernetics Nizhny Novgorod State University, Russia Received December 17, 2009; accepted January 15, 2010 Abstract We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds. MSC2000 numbers: 37C29, 37G25, 37D45 DOI: /S Key words: homoclinic tangency, Henon-like maps, saddle-focus fixed point, wild-hyperbolic attractor The classical Hénon map INTRODUCTION x = y, ȳ =1 bx + ay 2, (0.1) where (x, y) R 2 and a and B are real parameters (B is the Jacobian of (0.1), was introduced in [1] as the simplest nonlinear map demonstrating chaotic dynamics. Since 1976, the time of publishing [1], the Hénon map became quickly one of the most popular chaotic model map, and there appeared many papers devoted to the study of its dynamics. The main reason for such an interest is caused by the fact that the Hénon map can be considered as the simplest nonlinear mathematical model possessing important properties (regularities) of multidimensional chaotic systems, and these properties can be established rigorously (see [2, 3] etc). Besides, the Hénon map is one of those several artificial models whose use allows one to establish many fundamental phenomena in chaotic dynamics. For example, the famous Newhouse phenomenon was established in [4] mainly due to the scrupulous study of fine hyperbolic properties of the Hénon map. On the other hand, the Hénon map cannot be regarded as a purely artificial one. It may appear in the applied dynamics too: in particular, the systems with homoclinic tangencies are important sources for occurrence of Hénon-like maps [5 7]. The reason is that the first return map defined near a point of quadratic homoclinic tangency, can be often represented, for some rescaled coordinates, as maps close to certain standard quadratic maps. By now, the list of such homoclinic Hénon-like maps is not quite poor and includes, in particular, the following ones: 1. The parabola map ȳ = M y 2 ; (0.2) 2. The standard Hénon map (that is equivalent to (0.1) if a 0) x = y, ȳ = M bx y 2 ; (0.3) * gosv100@uic.nnov.ru ** gonchenko@gmail.com *** lpshilnikov@mail.ru 462

2 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS Generalized Hénon maps (GHM for abbreviation) of two main following forms and x = y, ȳ = M bx y 2 + ν 1 xy (0.4) x = y, ȳ = M bx y 2 + ν 1 xy + ν 2 y 3, (0.5) which are labelled, respectively, as the quadratic and cubic extensions of the Hénon map: the new coefficients ν 1 and ν 2 are small actually (they tend to 0 as the return time tends to infinity); 4. The Mirá map 5. Three-dimensional Hénon maps of two main following forms: x = y, ȳ = M 1 + M 2 y x 2 ; (0.6) and x = y, ȳ = z, z = M 1 + Bx + M 2 z y 2 (0.7) x = y, ȳ = z, z = M 1 + Bx + M 2 y z 2. (0.8) The parabola map is known as a homoclinic map from [5, 6]. The standard Hénon map, Mirá map and 3D Hénon map (0.7) (as well as the parabola map) were derived in [7, 8] as normal forms of rescaled first return maps for various cases of multidimensional systems with simple quadratic homoclinic tangencies (called also as quasi-transversal homoclinic intersections). GHMs and 3D Hénon map (0.8) have been discovered as homoclinic maps quite recently, in [9, 10] and in [11], respectively, for two- and three-dimensional diffeomorphisms with quadratic homoclinic tangencies to saddle fixed points with unit Jacobians. We reproduce the latter results in Sections 1.3 and 2.1 and give principal moments of proofs. Maps (0.2) (0.8) were found, again as normal forms of rescaled return maps, also in many other bifurcation problems: see papers [12] [16] and book [22] for more details. Note that the list above is far from completeness: one can add related results for conservative systems [17 20], for multidimensional systems having heteroclinic connections of special types [21] etc. Besides, many related problems are still in progress. However, it is evident that the study of the dynamics of these model maps can give a lot for homoclinic dynamics as a whole. See [15, 22] for more discussions of this. Note that, among maps from the list, the parabola map and the Mirá map are not diffeomorphisms. The point is that these model maps represent only the main (finite scale) part of rescaled return maps. For example, when the parabola map appears, the corresponding rescaled first return map T k is written in the following form (for the general multidimensional setting, see [7]) X = ε 1 k, Ȳ = M Y 2 + ε 2 k,v= ε3 k, (0.9) where ε k ε k (X, Y, V, M), (X, Y, V ) are coordinates (Y is one-dimensional, X and V can be multidimensional), M is the vector of parameters including M; norms of functions εk and their derivatives (up to certain order) tend to 0 uniformly in some region Q = (X, Y, V, M ) L as k. Note that map (0.9) is written in the so-called cross-form with respect to the strong unstable coordinate V. IfT is a diffeomorphism (near homoclinic orbit), the rescaled map (0.9) is adiffeomorphismtoo(onq), but its dynamics is mainly defined by the single coordinate Y, i.e. by the parabola map. An analogous situation takes place in other cases. However, the parabola map is a universal homoclinic map, since it can be found near any quadratic homoclinic tangency [15, 23] (excluding cases of systems with special structures, such as conservative, reversible etc, where the conservative Hénon-like maps are universal, [24, 25]). Therefore, the one-dimensional parabola map plays

3 464 GONCHENKO et al. a fundamental role both in homoclinic and chaotic multidimensional dynamics. Moreover, the parabola map and corresponding dissipative Hénon map (0.3) with small Jacobian b are considered as an important auxiliary tool for construction of the Newhouse theory of wild hyperbolicity and persistence of homoclinic tangencies [4, 26]. Mainly due to the universality of the parabola map, this two-dimensional theory was extended to the general multidimensional case [23, 27, 28], i.e. the existence of Newhouse regions (in which systems with homoclinic tangencies are dense) near any system with homoclinic tangency was proven. Note also that the Mirá map (0.6) itself has no Hénon-like form, but it appears in homoclinic problems as a limit map for the 3-dim Hénon map (0.7) with the Jacobian B = b k 0ask. Thus, in this sense, the Mirá map can be assigned to the Hénon-like maps. Remark 1. Consider GHM in form (0.4). By making the following linear change of coordinates x = x new ν 1,y= y new ν 1x new ν 1, then map (0.4) is recast as x = y ν 1x + O(ν 2 1), ȳ = M bx y 2 + O(ν 2 1), where M = M ν 1 4 (1 + b). Hence, when ν 1 is sufficiently small, we can assume that GHM (0.4) and the following one x = y ν 1x, ȳ = M bx y 2 are equivalent. Note also that there is one more equivalent (up to O(ν1 2 )) form of GHM given as x = y 1 2 ν 1y 2, ȳ = M bx y 2. Remark 2. Maps (0.7) and (0.8) can be considered as some partial representatives of the following general family of the three-dimensional Hénon maps 1) x = y, ȳ = z, z = M 1 + M (1) 2 y + M (2) 2 z + Bx + q 1y 2 + q 2 yz + q 3 z 2, (0.10) It was shown in [11, 29] that form (0.10) is universal for three-dimensional quadratic maps whose Jacobian is constant and whose inverse map is quadratic again. More exactly, any such map can be represented, up to affine coordinate transformation, either in form (0.10) or in one of two degenerate forms where the coordinates are decoupled (see [11, 29] for more details). In this connection, it seems to be very interesting to study the problem on homoclinic origin of map (0.10) with arbitrary coefficients. In the paper we consider several classes of systems with homoclinic tangencies in which Hénonlike maps of various types appear in natural way. We consider GHMs in more details in sections 1.3, 1.4, 1.5, and 3-dim Hénon maps in Section 2. The main part of the paper is based on quite recent results from [10, 11, 14] and we give proofs of some of them (assuming certain simplification conditions, for the reader s sake). We also recall classical results from [5, 6], Sections 1.1 and 1.2, giving elements of the proofs. 1. HOMOCLINIC TANGENCIES IN TWO-DIMENSIONAL CASE AND HÉNON-LIKE MAPS In this section we give a review of main results (incidentally sketching proofs) related to a homoclinic origin of the parabola map and generalized Hénon maps. We consider here only twodimensional maps with homoclinic tangencies, except for section 1.5 describing very briefly other cases where GHMs appear. 1) The family (0.10) contains 7 parameters, but this number can always be diminished to 5: by making an appropriate coordinate scaling we can assume, for example, that q 1 + q 2 + q 3 = 1; and, by a coordinate shift, one can make one of the coefficients M (1) 2 or M (2) 2 vanish identically.

4 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS Statement of the Problem Let f be a C r -smooth (r 3) two-dimensional map having a saddle fixed point O with multipliers λ and γ, where0< λ < 1 < γ. Suppose also that the stable and unstable invariant manifolds of O have a quadratic tangency at the points of some homoclinic orbit Γ (Fig. 1a). Fig. 1. Let U be a small neighborhood of O Γ. It is the union of a small disc U 0 containing O and a number of small disks surrounding those points of Γ which are located outside U 0 (Fig. 1b). Our main goal is to study bifurcations of single-round periodic orbits, i.e. those which pass around U only once. Every such orbit can be associated with a fixed point of the corresponding first return map defined near some homoclinic point. Let M + Wloc s and M Wloc u be two points of Γ, and Π + U 0 and Π U 0 be their sufficiently small neighborhoods. Denote the map f U0 as T 0,itis called the local map.letq be such an integer that f q (M )=M +.Wedenotethemapf q :Π Π + as T 1, it is called the global map. An initial point from Π + can reach Π under iterations of the local map T 0, a number of the iterations can be very large (the more the point closer to Wloc s ). Then, under T 1 the point from Π can come back into Π +. Thus, we can represent every such first return map as T k T 1 T0 k :Π+ Π +,wherek can be arbitrary large (k = k, k +1,...). For an analytical expression of T k we need to have local coordinates on U 0. Evidently, these coordinates must be both sufficiently smooth and such that iterations T0 k of the local map T 0 allow a simple representation for all large k. Remark 3. Such good coordinates (C r -smooth, to the point) exist always [30 32]. In these coordinates, map T 0 has the form x = λx + g(x, y, ε)x 2 y, ȳ = γy + h(x, y, ε)xy 2 (called the main normal form); the map T0 k can be written as x k = λ k x 0 (1 + ξ k (x 0,y k,ε)),y 0 = γ k y k (1 + η k (x 0,y k,ε)), where (x 0,y 0 )and(x k,y k )arepointsofu 0 such that T0 k(x 0,y 0 )=(x k,y k ). Functions ξ k and η k are asymptotically small along with all their derivatives up to order (r 1) and, besides, norm their derivatives up to order (r 2) can be estimate as follows: ξ k C r 2 L λ k, η k C r 2 L γ k where L is some positive constant independent of k. However, in this paper, we will not use the main normal form coordinates observed in Remark 3. Instead of this, we suppose, more for the sake of simplicity and ease of readers, that the local map T 0 has a linear form, i.e. it can be written as x = λx, ȳ = γy. (1.1) In these coordinates, O =(0, 0), the manifolds Wloc s and W loc u have equations y =0 and x =0, respectively; M + =(x +, 0), M =(0,y ) and, without loss of generality, suppose that x + > 0, y > 0. The global map T 1 canbewrittenas x x + = F (x, y y ), ȳ = G(x, y y ), (1.2)

5 466 GONCHENKO et al. Fig. 2. where F (0, 0) = G(0, 0) = 0, G(0, 0) y =0, 2d = 2 G(0, 0) y 2 0. These relations follow from the facts that T 1 (M )=M + and the curve T 1 (W u quadratic tangency with Wloc s = {y =0}. Then, T 1 can be rewritten as loc = {x =0}) hasa x x + = ax + b(y y )+..., ȳ = cx + d(y y ) , (1.3) where a, b, c, d are some coefficients, d 0 and the dots stand for high order terms. Next we will consider parameter families (unfoldings) of maps close to f. As an important control parameter we consider a parameter µ 1 of the splitting of the manifolds W u (O) andw s (O) near the homoclinic point M +. In this case, both maps T 0 and T 1 depend on parameters too, and T 1 can be written as x x + = ax + b(y y )+..., ȳ = µ 1 + cx + d(y y ) , (1.4) where the dependence on µ 1 is explicitly expressed: thus, µ 1 = 0 corresponds to the initial quadratic homoclinic tangency; at small µ 1 such that dµ 1 > 0 any diffeomorphism f µ1 has no (single-round) homoclinic orbits close to Γ; and for dµ 1 > 0 any diffeomorphism f µ1 has exactly two (single-round) homoclinic orbits close to Γ. See Fig.2. Finally, let points (x 0,y 0 )and(x k,y k )besuchpointsfromu 0 that (x k,y k )=T0 k(x 0,y 0 ), then, by (1.1), we can write x k = λ k x 0,y 0 = γ k y k. (1.5) 1.2. The Parabola Map and Two-dimensional Dissipative Diffeomorphisms with Quadratic Homoclinic Tangencies These results relate to the case where f is a dissipative diffeomorphism. It means that λγ < 1 and the Jacobian of T 1 is nonzero, i.e., by (1.3), bc 0. Theorem 1 (Rescaling lemma for two-dimensional dissipative diffeomorphisms). Let f µ1 be a one-parameter family of two-dimensional dissipative diffeomorphisms that unfolds generally the quadratic homoclinic tangency. Then, for any L>0 in any segment [ ɛ 0,ɛ 0 ] of values of µ 1 with ɛ 0 > 0, there exists an infinite sequence of intervals I k, k = k 0 (µ 0 ),k 0 +1,..., accumulating on µ 1 =0as k such that for µ 1 I k the first return map T k canbewrittenin some rescaled coordinates as X = Y + O(γ k ), Ȳ = M Y 2 bcλ k γ k X + O(γ k ), (1.6) where ( ) M = dγ 2k µ 1 + cλ k x + γ k y (1 +...), (1.7) and the ball (X, Y, M) L is the domain of definition of map (1.6).

6 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 467 Proof. By virtue of (1.5) and (1.4), we can write map T k (µ 1 ) in the following form where x x + = aλ k x + b(y y )+P 1 (x, y y,µ 1 ), γ k ȳ = µ 1 + cλ k x + d(y y ) 2 + P 2 (x, y y,µ 1 ), P 1 (x, y y,µ 1 ) O ( λ 2k x 2 + λ k x y y +(y y ) 2), P 2 (x, y y,µ 1 ) O ( λ 2k x 2 + λ k x y y + y y 3). (1.8) A coordinate shift x new = x x + + α 1 k,y new = y y + α 2 k, where αj k = O(λk ),j =1, 2, brings (1.8) to the form x = aλ k x + by + P 1, ȳ = γ k ( µ 1 + cλ k x + γ k y (1 +...) ) + cλ k γ k x + dγ k y 2 + P (1.9) 2, where the first and second equations do not contain, respectively, constant terms and linear in y terms. Map (1.9) can be easily reduced to a universal map close to the parabola map. It can be done using the rescaling method which was first applied to the homoclinic situation in [6]. Note that map (1.9) is well prepared for rescaling and its Hénon-like form is obtained almost immediately. Indeed, introduce new coordinates x = b d γ k X, y = 1 d γ k Y. Then, system (1.9) is recast as (1.6) and formula (1.7) is valid for M. Note that since the rescaled factors are asymptotically small, new coordinates X and Y can take arbitrary finite values for large k. The same is true for the parameter M which can take arbitrary finite values when µ 1 varies near (γ k y (1 +...) cλ k x + ). Remark 4. Form (1.9) was derived in the paper [5] of Gavrilov and Shilnikov. In fact, they found that if (x, y) are small, the first return map T k is close to a specific Hénon map x = y, ȳ = M + b k x + d k y 2, with small b k (b k 0ask ), big d k (d k as k ) and arbitrary M. Map (1.6) is close to the one-dimensional parabola map (0.2), whose bifurcations have been well studied, so that it is possible to recover the bifurcation picture for the initial map T k.forthe parabola map, the bifurcation set is contained in the interval [ 1 4, 2] of values of M: atm = 1 4 a fixed point with multiplier +1 appears, it falls into two fixed points one of which is attractive at M ( 1 4, 3 4 ) and undergoes a period-doubling bifurcation at M = 3 4 ; the cascade of period-doubling bifurcations lead to chaotic dynamics which alternates with stability windows and the bifurcations stop at M = 2 (when the restriction of the map onto the non-wandering set becomes conjugate to the one-side Bernoulli shift of two symbols) and no longer bifurcations as M increases. Similar bifurcations take place for the map T k. In particular, the map has an attractive fixed point O k at µ 1 (µ +1 k stability at µ = µ 1 k,µ 1 k ) which arises under the saddle-node bifurcation at µ = µ+1 k and loses under the period-doubling bifurcation. Here µ +1 k = γ k y cλ k x d γ 2k +..., µ 1 k = γ k y cλ k x + 3 4d γ 2k +... It gives immediately the classical result from [5], the so-called theorem on cascade of periodic sinks, on the existence of an infinite sequence of non-intersecting intervals of µ 1 corresponding to the existence of stable periodic orbits (periodic sinks). This theorem was extended to the multidimensional case in [33] (and, under certain linearization assumptions, in [27, 34]). General

7 468 GONCHENKO et al. criteria of birth of periodic attractors under bifurcations of homoclinic tangencies were deduced in [15] (announced in [7]). The corresponding results can be regarded as fundamental ones in chaotic dynamics, since they prevent many chaotic systems from having genuine strange attractors (see discussions in [35]) GHMs and Two-dimensional Diffeomorphisms with Homoclinic Tangencies to a Neutral Saddle We consider here the case λγ = 1. The corresponding fixed point O is called neutral saddle. Although, a general scheme of the proof follows closely the dissipative case, the problem of calculation of first return maps becomes more complicated. First, we must now consider two-parameter unfoldings with parameters µ 1 and µ 2,wherethe parameter µ 2 is taken to control the saddle value of O (for ex., it can be considered as µ 2 = λγ 1). Second, the resonant relations, such as λ = λ 2 γ and γ = λγ 2, present here and, thus, a sufficiently smooth linearization is impossible, in general. In [9, 10] the finitely smooth normal form (from [30, 31]) was used for T 0. Since the results obtained do not depend on nonlinear terms of this form, 2) we can assume (knowing the answer and for ease of the reader) that T 0 is linear now and, thus, we can use formulas from Section 1.1. Third (and this is the most important item), the direct rescaling results, as for the dissipative case, are not satisfactory. The point is that, following closely to the above consideration, we can deduce that the first return map T k (µ 1,µ 2 ) can be written in some rescaled coordinates in the following form X = Y + O(γ k ), Ȳ = M Y 2 + bc(1 + µ 2 ) k X + O(γ k ). (1.10) where formula (1.7) holds for M. Sincebc 0andµ 2 can take both positive and negative values, map (1.10), for sufficiently large k, is asymptotically close to the standard Hénon map (0.3) with the sign-defined Jacobian B = bc(1 + µ 2 ) k that can take arbitrarily values (depending on µ 2 and k). However, the Hénon map itself is degenerate with respect to bifurcations of periodic points with multipliers e ±iϕ (occurring at B = 1), whereas it is not the case for map (1.10) in general. Thus, we need to calculate more terms in (1.10). The latter was done in [9, 10] where the following result was proven. Theorem 2 (Rescaling lemma for 2D diffeomorphisms with a neutral saddle). Let f µ1,µ 2 be a two parameter family of two-dimensional diffeomorphisms that unfolds generally the quadratic homoclinic tangency with λγ =1. Then, for any L>0 in any neighborhood of the origin in the (µ 1,µ 2 )-parameter plane, there exists an infinite sequence of domains k, accumulating on µ 1 =0 as k such that for (µ 1,µ 2 ) k the first return map T k canbewritteninsomerescaled coordinates as X = Y + O(λ 2k ), Ȳ = M Y 2 BX + Rλ k XY + Qγ k Y 3 + O(λ 2k ), (1.11) where formula (1.7) holds for M, R 2a b d f 11 2 c d e 02, Q f 03 d 2 (1.12) and e 02,f 11 and f 03 are the following coefficients of nonlinear terms of the functions F and G from (1.2): e 02 = 1 2 F (0, 0) 2 y 2, f 11 = 2 G(0, 0) xy, f 03 = 1 3 G(0, 0) 6 y 3. 2) Certainly, the resonant terms can affect dynamics, especially for conservative and near-conservative one. For global bifurcations, this phenomenon was observed first in the classical paper of E.A.Leontovich [36] (for planar bifurcations of homoclinic loops of saddles). We can also refer to [18, 37] where it was shown that the conditions for the existence of infinitely many elliptic periodic points in area-preserving maps with nontransversal heteroclinic cycles include explicitly the Birkhoff coefficients (Taylor coefficients of the normal form in the map near a saddle)

8 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 469 Besides, the rescaled coordinates X and Y can take arbitrary values from the ball (X, Y ) L, when (µ 1,µ 2 ) vary in k the rescaled parameters (M,B) run arbitrary finite values from the ball (M,B) L, whereb takes only values of defined sign (B is positive if bcλ k γ k < 0 and B is negative if bcλ k γ k > 0). In fact, map (1.11) is GHM of form (0.5) with ν 1 = Rλ k,ν 2 = Qγ k. Fig. 3. Elements of bifurcation diagrams, (a) for the Henon map (0.3) and (b) for GHM (0.4) (with ν 1 < 0), on the half-plane B>1/2 ofthe(b,m)-parameter plane. The curves L +, L and L ϕ correspond to the existence of a fixed point with multipliers +1, 1 ande ±iϕ (0 <ϕ<π), respectively. Points B ++ and B correspond to the existence of fixed points with double multipliers (+1, +1) and ( 1, 1), respectively. Main bifurcations of GHMs were studied in [9, 10, 38], moreover, in [38] maps (0.4) and (0.5) with arbitrary ν 1 and ν 2 were also studied. One of the remarkable properties of GHMs is that they demonstrate nondegenerate Andronov Hopf bifurcations of fixed points, in contrast to the standard Hénon map. Note that if R 0, the strong resonances (ϕ 0,π,2π/3,π/2) are also nondegenerate in (1.11), except for the case ϕ = π/2, where a one more condition R 2Q is required [38]. In Figure 3 some elements of the bifurcation diagrams are shown: for the standard Hénon map (0.3), Fig. 3a), and for GHM (0.4) with a small coefficient ν 1. One can see, in particular, the sharp difference in the character of Andronov Hopf bifurcations (i.e. bifurcations occurring during transition through the curve L ϕ ). Note that map (1.11) with R 0 appears also in the conservative case [19, 39]; the term RXY is not conservative principally: it is not hard to check that Rd = J(T 1) (1.13) y x=0,y=y where J is the Jacobian of the global map T GHMs and Two-dimensional Endomorphisms with Critical Homoclinic Orbits Now we consider the case where f is not invertible and, moreover, its Jacobian vanishes exactly at some point of the orbit Γ. But we assume again that O is a saddle fixed point with multipliers λ and γ and, moreover, the case λγ > 1 is under consideration now. The main results of this section were obtained in [14] and we reproduce them in a shortened form.

9 470 GONCHENKO et al. Let us rewrite the global map T 1 (see (1.2) (1.3)) in the following form x x + = ax + b(y y )+e 02 (y y ) ȳ = cx + d(y y ) 2 + f 11 x(y y )+... where we write out explicitly two new (in contrast to (1.3)) quadratic terms e 02 (y y ) 2 and f 11 x(y y ) that are most important, whereas other quadratic, cubic and higher-order terms are not essential and enter to the dots. We consider the case where the map T 1 is not invertible in M. It means that bc = 0. Thus, we have a certain case of a critical homoclinic orbit [40]. 3) It is the natural transference of quadratic homoclinic tangencies to the case of noninvertible maps. Only, except for the following general conditions (i) either a) b =0andc 0;orb) c =0andb 0; (ii) d 0; we need to assume that the Jacobian J = J(x, y y )ofmapt 1 has the non-degenerate linear part [40]. We suppose that (see (1.12) and (1.13)) (iii) Rd 2ad bf 11 2ce Note that R =2a 2ce 02 /d in case a) and R =2a bf 11 /d in case b). For studying bifurcations of single-round periodic orbits we consider two-parameter families f µ1 µ 2 of maps close to f. As the parameters we take, first, the parameter µ 1 of splitting of manifolds W u and W s ; second, the parameter µ 2 is taken as µ 2 = b in the case a) and as µ 2 = c inthecaseb). Then, map T 1 (µ 1,µ 2 )iswritteneitherintheform inthecasea)b =0,c 0 or in the form x x + = ax + µ 2 (y y )+e 02 (y y ) ȳ = µ 1 + cx + d(y y ) 2 + f 11 x(y y )+... (1.14) x x + = ax + b(y y )+e 02 (y y ) (1.15) ȳ = µ 1 + µ 2 x + d(y y ) 2 + f 11 x(y y )+... inthecaseb)c =0,b 0. Again we consider the first return maps T k T 1 T0 k for various sufficiently large k, k = k, k + 1,... The case λγ < 1 is not principally different from the case of diffeomorphisms since all areas (in U) are contracted here. However, another situation appears in the case λγ > 1. Here, although the areas are expanded under iterations of f µ inside U 0, they can be contracted under the iterations near the global piece of Γ since f is noninvertible. Thus, we can expect in this case the existence of two-dimensional dynamics. Theorem 3 (The rescaling lemma for endomorphisms). Let f µ1 µ 2 be the two-parameter family under consideration and λγ > 1. Then, in the parameter plane (µ 1,µ 2 ), there exist infinitely many regions k accumulating at the origin as k such that the first return map T k at (µ 1,µ 2 ) k can be written in one of the following forms, up to small terms of order O( γ k + λ 2k ). 1) In the case b =0,c 0the form is where X = BY Rλk X, Ȳ = M 1 X Y 2 (1.16) M 1 = dγ 2k (µ 1 + cλ k x + γ k y + O(λ 2k )), B = c(µ 2 + O(λ k ))(λγ) k. (1.17) 3) We assume that the map f is not invertible in some point of Γ which lies out U 0. Evidently, this assumption implies that the global map T 1 is noninvertible in M since T 1 = f q.

10 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 471 2) In the case c =0,b 0the form is X = Y, Ȳ = M 1 BX Y 2 + Rλ k XY, (1.18) where M 1 = dγ 2k (µ 1 + λ k x + µ 2 γ k y + O(λ 2k )), B = b(µ 2 + O(λ k ))(λγ) k. (1.19) Proof. 1) In the case b =0,c 0, by (1.5) and (1.14), map T k canbewrittenintheform x x + = aλ k x + µ 2 (y y )+e 02 (y y ) ȳ = γ k µ 1 + cλ k γ k x + dγ k (y y ) 2 + f 11 λ k γ k x(y y )+... Shifting the coordinate x x x + + O(λ k ),y y y + O(λ k )weobtain x = aλ k x + µ 2 y + e 02 y 2 + O( λ k x y + y 3 + λ 2k x 2 ) ȳ = M + cλ k γ k x + dγ k y 2 + f 11 λ k γ k xy+ +O ( γ k y 3 + λ k γ k x ( λ k x + y 2 ) ) (1.20) where M = γ k (µ 1 γ k y + cλ k x + + O(λ 2k )). Now we rescale coordinates x = γ k cd(λγ) k X, y = 1 d γ k Y. Since λγ > 1and γ > 1, the rescaling factors tend to zero as k and, thus, the rescaled coordinates X and Y can take any (finite) values at large k. In the new coordinates map (1.20) is recast as X = aλ k X + BY + e 02c d λk Y 2 + O(λ k γ k ), Ȳ = M 1 X Y 2 + O( γ k + λ 2k ). where M 1 satisfies (1.17). By means of the change X new = X + d 1 ce 02 Y, Y new = Y and an additional coordinate shift to O(λ k ) we bring this map to the sought form (1.16). 2) In the case c =0,b 0, by (1.5) and (1.15), map T k canbewrittenintheform x x + = aλ k x + b(y y )+e 02 (y y ) ȳ = γ k µ 1 + µ 2 λ k γ k x + dγ k (y y ) 2 + f 11 λ k γ k x(y y )+... Shifting the coordinates x x x + + O(λ k ),y y y + O(λ k )weobtain x = aλ k x + by + e 02 y 2 + O( λ k x y + y 3 + λ 2k x 2 ) ȳ = M + λ k γ k x(µ 2 + ρ k )+dγ k y 2 + f 11 λ k γ k xy +O ( γ k y 3 + λ k γ k x ( λ k x + y 2 ) ) (1.21) where ρ k = O(λ k )andm = γ k (µ 1 γ k y + µ 2 λ k x + + O(λ 2k )). Now rescale coordinates Then map (1.21) is recast as x = b d γ k X, y = 1 d γ k Y. X = aλ k X + Y + O(γ k ), Ȳ = M 1 BX Y 2 bf 11 d λk XY + O( γ k + λ 2k ). (1.22)

11 472 GONCHENKO et al. where M 1 satisfies (1.19). After the change X new = X, Y new = Y + aλ k X, map (1.22) is recast as X = Y + O(γ k ), Ȳ = M 1 BX Y 2 + Rλ k XY + aλ k Y + O( γ k + λ 2k ). Finally, by a coordinate shift to O(λ k ), we bring this map to the final form (1.18) A Review of Other Cases Where GHMs Appear GHMs appear as rescaled first return maps in many other multidimensional systems with homoclinic and heteroclinic tangencies. We refer to the following ones. 1) Three-dimensional diffeomorphisms with a quadratic homoclinic tangencies to a saddle-focus fixed point with multipliers λe ±iϕ,γ such that 0 <λ<1 < γ, λ 2 γ < 1and λγ > 1[12].The corresponding multidimensional case was considered in [15]. 2) Three-dimensional diffeomorphisms with a non-simple quadratic homoclinic tangencies to a saddle fixed point with multipliers λ 1,λ 2,γ such that 0 < λ 2 < λ 1 < 1 < γ, λ 1 λ 2 γ < 1and λ 1 γ > 1 [14, 42]. See also [43] where bifurcations of such tangency were considered for the first time. 3) Three-dimensional diffeomorphisms with a quadratic homoclinic tangencies to a neutral saddle fixed point with multipliers λ 1,λ 2,γ such that 0 < λ 2 < λ 1 < 1 < γ, λ 1 λ 2 γ < 1and λ 1 γ =1[44]. 4) Two-dimensional diffeomorphisms with a non-transversal heteroclinic cycle having two saddle fixed points O 1 and O 2 such that (i) the Jacobian is less than 1 at one point and it is greater than 1 for the other point; (ii) one pair of the stable and unstable manifolds intersects transversally and the other one has a quadratic tangency at the points of some heteroclinic orbit [13]. 5) Three-dimensional flows close to a flow having a homoclinic loop of a saddle-focus equilibrium of conservative type, i.e. with eigenvalues λ ± iω, γ, whereγ 2λ =0[45]. 6) The conservative GHMs (maps (0.5) with ν 1 0) were derived (i) in [17] for three-dimensional divergence free flows close to a flow with a homoclinic loop of a saddle-focus equilibrium; (ii) in [19, 39, 46] for area-preserving maps close to a map with a quadratic homoclinic tangency. The above list is not complete, certainly, and, moreover, some related results are in progress yet. But it shows that GHMs are quite applicable maps and studying their dynamics can be important for the chaotic dynamics as a whole. 2. ON THE HOMOCLINIC ORIGIN OF THREE-DIMENSIONAL HÉNON MAPS Three-dimensional Hénon maps as well as two-dimensional ones are, in fact, artificial nonlinear (quadratic) maps demonstrating very nontrivial dynamics. However, these maps, and their generalizations, can be found in applied dynamics as rescaled return maps near homoclinic and heteroclinic tangencies. Many questions of the dynamics of two-dimensional Hénon-like maps have been studied in a very detailed way. It is not the case for three-dimensional Hénon maps. Only partial recent results have been obtained here, see, for ex., [11, 29, 35]. However, the dynamical phenomena appearing here are very interesting and fundamental. As an example, we refer only to one of such type results the existence of wild Lorenz-like attractors in three-dimensional Hénon maps (0.8), [35]. In this connection, we note that the Hénon map was introduced in [1] as a simple nonlinear map demonstrating a chaotic dynamics similar, as though, to that which the Lorenz system has. However, as it became clearly very soon, the chaotic dynamics of the Hénon map has an absolutely different nature. On the other hand, phase portraits of strange attractors of certain three-dimensional Hénon maps can be very analogous to those in thelorenzsystem,seefigure4takingfrom[35].

12 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 473 Fig. 4. Plots of attractors of 3D Hénon map (0.8) observed numerically for M 1 =0,B =0.7 andm 2 =0.85 (left) or M 2 =0.815 (right). The projections on the (x, y) variables and slices near the planes of constant z are also displayed. Numerically obtained 3D pictures of iterations of a single point by (0.8) are shown. The resemblance to the traditional picture of the Lorenz attractor is astonishing (left). Recall that this is a discrete trajectory of just one point. Still it mimics very well a motion along a continuous trajectory of the classical Lorenz attractor. From the right, the attractor picture is very similar to the Lorenz attractor with a lacuna. Remark 5. It was shown in [47] that local bifurcations of periodic points having a triplet ( 1, 1, +1) of multipliers can in certain cases lead to the appearance of wild Lorenz attractors. It was checked in [11, 35] that such bifurcations take place in map (0.8) (which has a fixed point with multipliers 1, 1, +1 at (M 1,M 2,B)=( 1/4; 1; 1)). It means that map (0.8) has a Lorenz-like attractor in some domain of the parameters adjoining to the point ( 1/4; 1; 1). This attractor is a genuine strange attractor: it and all close attractors possess a pseudo-hyperbolic structure [48, 49], 4) and, thus, do not contain periodic sinks [49, 50]; it is a wild strange attractor [48], i.e. it allows homoclinic tangencies and, hence, belongs to a Newhouse region); etc. Remark 6. Note that maps (0.7) and (0.8) are inverse to each other. However, they demonstrate different types of stable dynamics (when B < 1). For example, as we said, global Lorenz wild attractors exist in (0.8), whereas, wild strange attractors of such kind are not observed in (0.7), although certainly, a Lorenz wild repeller will be present dim Hénon map (0.7) and Simple Homoclinic Tangencies to Saddle-focus Fixed Points The first result of this topic goes back to [7] where map (0.7) was represented as a homoclinic map for the following bifurcation problem. Let a four-dimensional 5) diffeomorphism F have a fixed point O with multipliers ν 1,2 = λe ±iϕ and ν 3,4 = γe ±iψ,where0<λ<1 <γ, ϕ, ψ 0,π. Such a point is called a saddle-focus of type (2,2). We assume that λγ < 1. Suppose that the stable W s (O) and unstable W u (O) invariant manifolds of O have a quadratic tangency at the points of some homoclinic orbit Γ 0. Consider a three-parameter general unfolding F µ,whereµ =(µ 1,µ 2,µ 3 ). As the parameters, we consider the following ones: µ 1 is the splitting parameter with respect to some homoclinic point; µ 2 = ψ ψ 0 and µ 3 = ϕ ϕ 0.Hereψ 0 and ϕ 0 are the corresponding values of the complex arguments of the multipliers of the diffeomorphism F. 4) It means that the tangent space for the corresponding invariant set can be split into a direct sum E ss E cu, depending continuously on the point, such that along E ss there is a strong contraction, while the possible contraction in E cu is much weaker and, moreover, (two-dimensional) volumes in E cu are expanded. 5) Note that in [7] the general multidimensional case was announced and the proofs were given in [12] for the four-dimensional case and in [15] for the multidimensional one.

13 474 GONCHENKO et al. Theorem 4 ([7, 12]). There is an infinite sequence of domains k of the parameters such that k accumulate on the origin as k and the first return maps T k, after some appropriate rescaling of the coordinates and the parameters, can be brought to a map asymptotically close to the threedimensional Hénon map of form (0.8). Inthecaseλγ 2 > 1 (but λγ < 1 as before), all rescaled parameters M 1,M 2 and the Jacobian B can take arbitrary finite values (when µ vary in k ). In the case λγ 2 < 1 the Jacobian B = O(λ k γ 2k ) 0 as k (the limit map is the Mirá map in this case). Note that the following relations M 1 γ 4k (µ 1 ν k ),M 2 γ k cos(kψ + α), B (λγ 2 ) k cos(kϕ + β), take place between the rescaled and initial parameters (here α and β are some constants (invariants of F ), ν k = O(γ k ) is some small coefficient. We see that the formulae for the rescaled parameters contain asymptotically big in k factors (except for the factor (λγ 2 ) k when λγ 2 < 1). But varying the parameters µ 1,ψ and ϕ (when λγ 2 > 1) can result in the rescaled parameters M 1,M 2,B taking arbitrary finite values (for µ k ). Remark 7. Note that, as it was shown in [30, 41], diffeomorphisms with homoclinic tangencies possess continuous (topological) conjugacy invariants on the set of non-wandering orbits, so-called Ω-moduli. In the cases with a saddle-focus fixed point, the principal Ω-moduli are the angular arguments of complex multipliers of the saddle-focus, i.e. ϕ and ψ in the case under consideration. By definition, Ω-moduli are natural governing parameters. Therefore, we include them into the set of governing parameters dim Hénon Maps and Three-dimensional Diffeomorphisms with Quadratic Homoclinic Tangencies to a Saddle Focus of Conservative Type We consider two more cases of homoclinic tangency where three-dimensional Hénon maps appear. Both cases relate to three-dimensional diffeomorphisms close to those having quadratic homoclinic tangencies to saddle focus fixed points with the unit Jacobian. Our consideration follows closely [11]. Consider a three-dimensional diffeomorphism g 0 C r (R 3 ), r 5, that satisfies the following conditions (see Figure 5): (A) g 0 has a type (2, 1) saddle-focus fixed point O with multipliers (λ 0 e iϕ 0,λ 0 e iϕ 0,γ 0 )where 0 <λ 0 < 1 < γ 0 (w.l.o.g. 0 <ϕ 0 <π); (B) P 0 J 0 = λ 2 0 γ 0 =1; (C) the stable, W s, and unstable, W u, invariant manifolds of the fixed point O have a quadratic homoclinic tangency at the points of a homoclinic orbit Γ 0. As in Section 2.1, we consider a three-parameter general unfolding g ε,whereε =(ε 1,ε 2,ε 3 ). It is evident that the set of such parameters must include both a parameter µ = ε 1 of the splitting of the initial quadratic homoclinic tangency and a parameter ε 2, similar to ε 2 = P ε 1, that varies the Jacobian of the saddle-focus fixed point. As the third parameter, we consider the Ω-modulus ϕ, i.e. ε 3 = ϕ ϕ 0. Theorem 5. [11] Let g ε be the three parameter family defined above. Then, in any neighborhood of the origin, ε =0, of the space of parameters, there exist infinitely many open domains, k,such that the first return map gε k for ε k takes the form X = Y + o(1) k +, Ȳ = Z + o(1) k +, Z = M 1 + BX + M 2 Y Z 2 + o(1) k +, (2.1) where (X, Y, Z) are rescaled coordinates and the domains of definition of the new coordinates and the parameters (M 1,M 2,B) are arbitrarily large and cover, as k, all finite values. Furthermore, the Jacobian, B, is positive or negative depending on the orientability of the map gε k. The terms denoted o(1) are functions (of the coordinates and parameters) that tend to zero as k + along with all derivatives up to order r 2.

14 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 475 Fig. 5. Geometry of the homoclinic orbit to a type (2, 1) saddle-focus for the map g 0 (A) (C). under assumptions Theorem 5 shows that the 3D Hénon map (0.8) is a normal form for this homoclinic bifurcation, since it appears when we omit the o(1)-terms in (2.1). Besides, the new parameters M 1,M 2 and B can be given as functions of ε =(µ, ϕ ϕ 0,P ε 1) by, [11], M 1 γ 3k (µ + O(λ k )),M 2 (λγ) k cos(kϕ + β),b (λ 2 γ) k where β is some coefficient (an invariant of g 0 )and(λ(0),γ(0)) = (λ 0,γ 0 ). The exact relations will be given in formulas (2.12), (2.15) and (2.17) below. Solving the bifurcation problem for the family g ε allows us to solve almost automatically an analogous problem for a diffeomorphism g 0 C r (r 5) with a type (1, 2) saddle focus. Specifically, assume that g 0 satisfies (see Figure 6): (A ) g 0 has a type (1, 2) fixed point Õ with multipliers (λ 0,γ 0 e iψ 0,γ 0 e iψ 0 )where0< λ 0 < 1 <γ 0 (w.l.o.g. 0 <ψ 0 <π); (B ) P 0 (= λ 0 γ 2 0 =1;and (C ) the stable and unstable invariant manifolds of Õ have a quadratic homoclinic tangency at the points of some homoclinic orbit Γ 0. Fig. 6. Geometry of the homoclinic orbit to a type (1, 2) saddle-focus for the map g 0 (A )-(C ). under assumptions

15 476 GONCHENKO et al. As in the case of the type (2, 1) saddle-focus, we consider a three-parameter family g ε where ε =( µ, P ε 1,ψ ψ 0 ). We suppose here that µ is a splitting parameter, P ε = λ( ε)γ( ε) 2 and λ(0) = λ 0,γ(0) = γ 0. Theorem 6 ([11]). Let g ε be the three-parameter general family defined above. Then, in any neighborhood of ε =0 there exist infinitely many open domains, k, such that the first return map g k ε takes the form X = Y + o(1) k +, Ȳ = Z + o(1) k +, Z = M 1 + BX + M 2 Z Y 2 + o(1) k +, (2.2) where (X, Y, Z) are rescaled coordinates and the domains of definition of the new coordinates and the parameters (M 1,M 2,B) are arbitrarily large and cover, as k, all finite values. Furthermore, the Jacobian, B, is positive or negative depending on the orientability of the map g k ε. The terms denoted o(1) are functions (of the coordinates and parameters) that tend to zero as k + along with all derivatives up to order r 2. The proof follows immediately from Theorem 5. Indeed, the inverse g 0 1 is, evidently, a diffeomorphism satisfying the original conditions (A) (C). Thus, the map (2.1) is, in this case, the rescaled map for the inverse first return map g k ε. The latter can be be easily written in form (2.2), as we show explicitly below. The inverse of (0.7) is of interest in any case. Since (0.7) has the constant Jacobian B it is invertible whenever B 0.Thisinverseis ȳ = x, z = y, x = 1 B ( M 1 + z M 2 y + y 2 ). This map can be rewritten in the standard form of [29], if we define new variables (x new,y new,z new )=B 1 (z,y,x) and new parameters M1 new = M 1 B 2,Mnew 2 = M 2 B,Bnew = 1 B, then the resulting map has precisely the form (0.7) The proof of Theorem 5 For the sake of completeness, we give here the main elements of the proof of Theorem 5. We start with constructing the local and global maps T 0 and T 1. By [15, 31], one can introduce C r coordinates (x, y) =(x 1,x 2,y) in a neighborhood U of the fixed point O such that the local map T 0 (ε)( g ε U0 can be written, for small enough ε, intheform T 0 : x = λr ϕx + ξ(x, y) ȳ = γy + η(x, y) (2.3) where R ϕ = cos ϕ sin ϕ, ξ = O( x 2 y), η = O( x y 2 ). (2.4) sin ϕ cos ϕ When ε = 0, choose a homoclinic point M =(0, 0,y ) Wloc u on the homoclinic orbit Γ 0 and a corresponding point M + =(x + 1,x+ 2, 0) W loc s that is an image of M : M + = g n 0 0 (M )forsome n 0 > 0, recall figure 5. Let Π + and Π be some small neighborhoods of the points M + and M.

16 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 477 The global map T 1 (ε) g n 0 ε Π :Π Π + is well defined for small enough ε, and can be written in the coordinates of (2.3) as T 1 : x x+ = Ax + b(y y )+O( x 2 + y y 2 ), ȳ = y + (ε)+c T x + d(y y ) 2 + O( x 2 + x y y + y y 3 ), (2.5) where the 2 2matrixA, the vectors x +, b, andc, the coefficients y + and d, and the higher-order terms all depend smoothly on ε. We note especially the dependence of the coefficient y + (ε) that will become the splitting parameter for the manifolds W s and W u near the homoclinic point M +. We therefore denote it by µ = y + (ε) and consider as the first of the governing parameters. The assumption (C) that the tangency between W u and W s is quadratic implies that d 0.In addition, since g 0 is assumed to be a diffeomorphism, and since the Jacobian DT 1 (M )= A b (2.6) c T 0 then we must have b 0. Indeed, if b 2 (ε) is not identically zero, then one can use the linear coordinate change, (x new,y new )=(R ϑ x, y), where R ϑ is a rotation about an angle ϑ(ε), to transform the map (2.5) to a new map of the same form such that the vector b becomes b T =(b 1, 0). (2.7) Here the new matrix A, pointx +, and vector c are modified from the corresponding coefficients in the original (2.5). Now the fact that the determinant of the Jacobian (2.6) does not vanish implies that J 1 det DT 1 (M )=b 1 (a 21 c 2 a 22 c 1 ) 0. (2.8) One of the merits of the main normal form (2.3) is that the iterations T k 0 : U 0 U 0 for any k can be calculated quite easily. Namely, it was shown in [11] that the map T k 0 (ε) : (x 0,y 0 ) (x k,y k ) canbewritteninthecross-form x k = λ k R kϕ x 0 + λ 2k ξ k (x 0,y k,ε), (2.9) y 0 = γ k y k + λ k γ k η k (x 0,y k,ε), where the vector ξ k and the function η k are uniformly bounded along with all derivatives up to order (r 2). Lemma 1. For any sufficiently large k such that A 21 (kϕ) a 21 cos kϕ + a 22 sin kϕ 0 (2.10) the first return map T k can be brought, by means of an affine transformation of coordinates, to the following form Ȳ = Z + O(λ k ), X = Y + O(λ k + λγ k ), Z = M 1 Z 2 + b 1 λγ k (c 1 cos kϕ + c 2 sin kϕ + νk 1)Y (2.11) where ν 1,2 k +b 1 A 21 λ 2k γ k (c 2 cos kϕ c 1 sin kϕ + ν 2 k )X + O(λk ), = O(λ k ) are some coefficients, M 1 = dγ 2k [ µ + λ k ( cos kϕ(c 1 x c 2x + 2 )+sinkϕ(c 2x + 1 c 2x + 2 )+...)]. (2.12) Also, in (2.11) the O-terms have the indicated asymptotic behavior for all derivatives up to order (r 2).

17 478 GONCHENKO et al. Proof. Using formulas (2.5) with µ = y +, (2.9) and (2.7) we can write the first return map T k T 1 T0 k in the form x 1 = x λk (e 11 x 1 + e 12 x 2 )+b 1 (y y )+α 1, x 2 = x λk (e 21 x 1 + e 22 x 2 )+α 2, γ k ȳ + λ k γ k η k ( x, ȳ,ε) =µ + λ k (c 1 cos kϕ + c 2 sin kϕ)x 1 +λ k (c 2 cos kϕ c 1 sin kϕ)x 2 + d(y y ) 2 + α 3, (2.13) where the (2 2)-matrix E = AR kϕ and α 1,2 = O(λ 2k x + λ k x y y +(y y ) 2 ), α 3 = O(λ 2k x + λ k x y y + y y 3 ). Introduce new coordinates x new = x x + + β 1 k,y new = y y + β 2 k, where β k = O(λ k ), in order to eliminate the affine terms in the first two equations and the term linear in y in the third equation of (2.13). Then, the map (2.13) is rewritten as x 1 = e 11 λ k x 1 + e 12 λ k x 2 + by + O(λ 2k x + λ k x y + y 2 ), x 2 = e 21 λ k x 1 + e 22 λ k x 2 + O(λ 2k x + λ k x y + y 2 ), ȳ + λ k O( x + ȳ ) =M + dγ k y 2 + λ k γ k (c 1 cos kϕ (2.14) where d new = d + O(λ k )and +c 2 sin kϕ + O(λ k ))x 1 + λ k γ k (c 2 cos kϕ c 1 sin kϕ + O(λ k ))x 2 +γ k O(λ 2k x 2 + λ k x y + y 3 ), M = γ k [µ γ k y + λ k ( cos kϕ(c 1 x + 1 c 2x + 2 )+sinkϕ(c 2x + 1 c 2x + 2 )) + O(λ 2k )]. Next, we rescale the map (2.14) introducing new coordinates (X,Y,Z) defined through x 1 = b 1 d γ k Y, x 2 = b 1A 21 λ k γ k X, y = 1 d d γ k Z, where we suppose that values of ϕ are such that (2.10) holds. Then, map (2.14) is rewritten in the new coordinates in form (2.11). Now Theorem 5 easily follows from this lemma. Indeed, we have only to indicate those domains k of the initial parameters where the coefficients in the third equation of (2.11) are finite. Denote the coefficient of the third equation of (2.11) in front of Y as M 2, i.e. M 2 = b 1 λ k γ k (c 1 cos kϕ + c 2 sin kϕ + ν 1 k ). (2.15) Since γ > 1and λγ > 1, the coefficients M 1 and M 2 can take arbitrary finite values when varying parameters µ (for M 1 )andϕ (for M 2 ), see formulas (2.12) and (2.15). Moreover, the finiteness of M 2 means that the corresponding values of the trigonometrical term c 1 cos kϕ + c 2 sin kϕ are asymptotically small and, in any case, do not exceed λγ k in order. For such values of ϕ, we obtain c 2 cos kϕ c 1 sin kϕ = ± c c2 2 (1 +...), A 21 = a 21 cos kϕ + a 22 sin kϕ = ± a 21c 2 a 22 c 1 (1 +...) c c 2 2 (2.16)

18 ON A HOMOCLINIC ORIGIN OF HÉNON-LIKE MAPS 479 Thus, by (2.8), A 21 0 in this case. Denote as B the coefficient in front of X 2 from the third equation of (2.11). By virtue of (2.16) and (2.8) we have B = b 1 (a 21 c 2 a 22 c 1 )λ 2k γ k (1 +...)=J 1 λ 2k γ k (1 +...). (2.17) Since λ 2 γ =1 at ε = 0, it follows from (2.17) that B can take (when k is large) any arbitrary finite value (positive or negative depending on the sign of J 1 λ 2k γ k ) as the value of the Jacobian λ 2 γ of the local map T 0 varies. This completes the proof of Theorem ACKNOWLEDGMENTS The authors thank D.V. Turaev and J.D. Meiss for very fruitful discussions. The work was supported in part by grants of RFBR No , No , No , the regional grant RFBR No r-povolj e as well as the Royal Society grant Global Bifurcations. REFERENCES 1. Hénon, M., A Two-dimensional Mapping with a Strange Attractor, Comm. Math. Phys., 1976, vol. 50, pp Benedidicks, M. and Carleson, L., The Dynamics of the Hénon Map, Ann. Math., 1991, vol. 133, pp Lai, Y.-C., Grebogi, C., Yorke, J. and Kan, I., How Often are Chaotic Saddles Nonhyperbolic? Nonlinearity, 1993, vol. 6, pp Newhouse, S.E., The Abundance of Wild Hyperbolic Sets and Non-smooth Stable Sets for Diffeomorphisms, Publ. Math. IHES, 1979, vol. 50, pp Gavrilov, N.K. and Shilnikov, L.P., On Three-dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: Part I, Mat. Sb., 1972, vol. 88(130), no. 4(8), pp [Mathematics of the USSR-Sbornik, 1972, vol. 17, no. 4, pp ]; Part II, Mat. Sb., 1973, vol. 90(132), no. 1, pp [Mathematics of the USSR-Sbornik, 1973, vol. 19, no. 1, pp ]. 6. Tedeschini-Lalli, L. and Yorke, J.A., How Often do Simple Dynamical Processes Have Infinitely Many Coexisting Sinks? Commun. Math. Phys., 1986, vol. 106, pp Gonchenko, S.V., Shilnikov, L.P. and Turaev, D.V., Dynamical Phenomena in Systems with Structurally Unstable Poincaré Homoclinic Orbits, Dokl. Akad. Nauk, 1993, vol. 330, no. 2, pp [Acad. Sci. Dokl. Math., vol. 47, no. 3, pp ]. 8. Gonchenko, S.V., Shilnikov, L.P. and Turaev, D.V., Dynamical Phenomena in Systems with Structurally Unstable Poincaré Homoclinic Orbits, Chaos, 1996, vol. 6, no. 1, pp Gonchenko, S.V. and Gonchenko, V. S., On Andronov Hopf Bifurcations of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies, Preprint of Weierstrass Inst. for Applied Analysis and Stochastics, Berlin, 2000, no. 556; Gonchenko, S.V. and Gonchenko, V.S., On Bifurcations of the Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies, Tr. Mat. Inst. Steklova, 2004, vol. 244, pp [Proc. Steklov Inst. Math., 2004, vol. 244, no. 1, pp ]. 11. Gonchenko, S.V., Meiss, J.D., and Ovsyannikov, I.I., Chaotic Dynamics of Three-Dimensional Hénon Maps That Originate from a Homoclinic Bifurcation, Regul. Chaotic Dyn., 2006, vol. 11, no. 2, pp Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V., On the Dynamical Properties of Diffeomorphisms with Homoclinic Tangencies, Contemporary Mathematics and its Applications, 2003, vol. 7, pp (Russian) [English version: J. Math. Sci. (N.Y.), 2005, vol. 126, pp ]. 13. Gonchenko, S.V., Sten kin, O.V., and Shilnikov, L.P., On the Existence of Infinitely Stable and Unstable Invariant Tori for Systems from Newhouse Regions with Heteroclinic Tangencies, Rus. J. Nonlin. Dyn., 2006, vol. 2, no. 1, pp (Russian). 14. Gonchenko, S.V., Gonchenko, V.S., and Tatjer, J.C., Bifurcations of Three-dimensional Diffeomorphisms with Non-simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps, Regul. Chaotic Dyn., 2007, vol. 12, no. 3, pp Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V., On Dynamical Properties of Multidimensional Diffeomorphisms from Newhouse Regions: P. 1, Nonlinearity, 2008, vol. 21, no. 5, pp Gonchenko, S.V., Shilnikov, L.P., and Turaev, D.V., On Global Bifurcations in Three-Dimensional Diffeomorphisms Leading to Wild Lorenz-Like Attractors, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp