Lorenz attractors in unfoldings of homoclinic flip bifurcations

Transcription

1 Lorenz attractors in nfoldings of homoclinic flip bifrcations A. Golmakani Department of Mathematics, Ferdowsi University of Mashhad A.J. Hombrg KdV Institte for Mathematics, University of Amsterdam Janary 8, 200 Abstract Lorenz like attractors are known to appear in nfoldings from certain codimension two homoclinic bifrcations for differential eqations in R 3 that possess a reflectional symmetry. This incldes homoclinic loops nder a resonance condition and the inclination flip homoclinic loops. We show that Lorenz like attractors also appear in the third possible codimension two homoclinic bifrcation (for homoclinic loops to eqilibria with real different eigenvales); the orbit flip homoclinic bifrcation. We moreover provide a bifrcation analysis, compting the bifrcation crves of bifrcations from periodic orbits and discssing the creation and destrction of the Lorenz like attractors. Known reslts for the inclination-flip are extended to inclde a bifrcation analysis. Introdction The Lorenz eqations, the system of ordinary differential eqations on R 3 given by ẋ = σx + σy, ẏ = ρx y xz, ż = βz + xy, was coined in the 960s by Lorenz [6] as a mch simplified model for Rayleigh-Bénard convection. Nmerical comptations done by Lorenz revealed the existence of strange attractors for parameter vales σ = 0, b = 8/3, ρ = 28. These nmerical observations of strange attractors were first explained in geometric Lorenz models [, 0, 35]. A fndamental property of the strange attractor is its robst occrrence nder variations of the parameters (or pertrbations of the system, details of the dynamics may change). The occrrence of a robst strange attractor in the actal Lorenz system was proven by Tcker [33, 34]. A general theory, for three dimensional flows, of robst partially hyperbolic attractors containing one or more eqilibria has been developed, see [9]. These attractors are called singlar hyperbolic attractors. We refer to [2, 4, 5, 7, 8] for frther reslts, mostly discssing ergodic properties, on Lorenz like attractors and other singlar hyperbolic attractors. We se the phrase Lorenz like attractor for a singlar hyperbolic attractor in three dimensional differential eqations with a geometry as in the Lorenz system (that is, containing a single eqilibrim). There has been sbstantial interest in the qestion how Lorenz like attractors may appear throgh bifrcations from ODEs with simpler dynamics. For instance miniatre Lorenz like attractors were shown to occr in nfoldings of certain nilpotent singlarities [9]. Shilnikov [32] sggested that Lorenz-like attractors may occr in bifrcations from ODEs with two homoclinic loops for three different codimension two (in the

2 context of Z 2 -eqivariant systems) bifrcations: resonant leading eigenvales, an inclination-flip condition or an orbit-flip condition. These are all codimension two homoclinic bifrcations involving a hyperbolic eqilibrim with real distinct eigenvales. The first possibility was worked ot by Robinson [26 28] (see also [20, 2]), the second by Rychlik [29]. Both Robinson and Rychlik provided cbic differential eqations containing Lorenz like attractors. We treat the third type of bifrcation, the orbit-flip bifrcation, and establish the existence of Lorenz like attractors in its nfolding. We present a more detailed bifrcation analysis going beyond the observation that Lorenz like attractors occr in the nfolding, ths giving more insight in the creation of the attractors. The word flip in both orbit flip and inclination flip indicates a change in geometry of the two dimensional stable manifold along the homoclinic orbit; both bifrcations trigger a change in orientation where the stable manifold near the homoclinic orbit is either an annls or a Möbis band. Both bifrcations show many similarities and in fact a bifrcation analysis for the inclination flip can be given along the lines of this paper. This is frther discssed in 5. The sbject of or stdy will be two parameter families = f(, µ) () of differential eqations with R 3 and µ = (µ, µ 2 ) R 2. The differential eqations we consider are symmetric with respect to a reflection against an axis and possess a symmetric hyperbolic eqilibrim at the origin. Symmetry is given by a linear involtion R (a linear map with R 2 = id) which maps orbits to orbits. More formally this means that Rf(, µ) = f(r, µ). (2) We assme that Df(0, 0) has three real eigenvales λ ss, λ s, λ with λ ss < λ s < 0 < λ. Write α = λ ss /λ, β = λ s /λ. Note that α, β are the stable eigenvales one obtains by a time rescaling that brings the nstable eigenvale to. The eigenspaces corresponding to λ ss, λ s are called the strong and leading stable directions. The eqilibrim at the origin possesses ths a one dimensional nstable manifold W (0) consisting of points whose negative orbit converges to the origin and a two dimensional stable manifold W s (0) consisting of points whose positive orbit converges to the origin. Within the stable manifold there is a one dimensional strong stable manifold W ss (0) consisting of points whose positive orbit converges at a rate e αt to the origin. Locally near the origin one can constrct center nstable manifolds W s, (0), tangent to the sm of the leading stable direction and the nstable direction at the origin. Sch manifolds are not niqe, bt are at least C [] and possess a niqe tangent bndle along the nstable manifold [2]. We assme that for µ = 0, a homoclinic orbit γ = { lim = 0} (3) t ± to the eqilibrim at the origin exists. We assme that Rγ is a second homoclinic orbit different from γ. We treat orbit flip bifrcations characterized by the following hypothesis. Hypothesis. (Orbit flip). The homoclinic orbit γ is a generically nfolding orbit flip homoclinic orbit:. Orbit flip: the homoclinic orbit γ is contained in the strong stable manifold W ss (0). 2. No inclination flip: along the homoclinic orbit γ, W s, (0) is transverse to W s (0). 3. Generic nfolding: the traces of the parameter dependent manifolds W ss (0) and W (0) in the prodct of state space and parameter space R 3 R 2, intersect transversally along (γ, 0). Let Σ be a smooth cross section transverse to γ. The generic nfolding condition can also be formlated as follows: the map assigning the difference vector in Σ between the two points W (0) Σ and W ss (0) Σ 2

3 γ Σ ss RΣ Σ Rγ RΣ ss Figre : A doble homoclinic loop at an orbit flip bifrcation. The dashed crves indicate a doble homoclinic loop in a btterfly configration bifrcating from the orbit flip. Cross sections as introdced in 2.2 are also indicated. (assming that Σ is contained in a linear sbspace) is locally injective at µ = 0. We will se the nfolding condition to reparameterize the parameter plane so that, given sitable coordinates on Σ, in new parameters µ = (µ, µ 2 ) we can write W ss (0) Σ = (0, 0), W (0) Σ = (µ 2, µ ) and a homoclinic orbit exists for {µ 2 = 0}. Depending on the dimension of the fixed point space of the involtion R (which is either 0 if R is a reflection against the origin or if R is a reflection against a line), there are two different nfoldings of the symmetric orbit flip homoclinic loop. We restrict or stdy to the case of one dimensional fixed point space Fix(R) = { R = }, this is the case where Lorenz like attractors occr. Hypothesis.2 (Symmetry). We assme eqivariance with respect to a linear involtion R for which dimfix(r) =. Note that γ and Rγ form a figre eight, meaning R lim t = lim t, R lim t = lim t. It is a conseqence of Hypothesis.2 that pertrbed homoclinic orbits that are no longer forming an orbit flip, are in a btterfly configration (see Figre ); still writing γ and Rγ for the pertrbed homoclinic orbits this means R lim t = lim t, R lim t = lim t. In contrast, if dimfix(r) = 0, then pertrbed homoclinic orbits are still forming a figre eight. We consider eigenvale conditions for which the nfolding of a single orbit flip homoclinic orbit gives rise to a homoclinic dobling bifrcation [4, 30]. Hypothesis.3 (Eigenvale conditions). Consider the following eigenvale conditions: α >, 2 < β <. The following reslt is or main theorem, discssing bifrcations from a codimension two orbit flip bifrcation of two symmetric homoclinic orbits and the occrrence of Lorenz like attractors. The following sections contain frther discssions, elaborating on the bifrcation reslt. 3

4 Het2Per SB Het2Per Hom Hom Hom2 Het2Per SB Het2Per SN Het2Per µ 2 PD Het2Per Hom2 µ Figre 2: Bifrcation diagram for Theorem.. Theorem.. Let ẋ = f(x, µ), µ R 2, be a smooth two parameter family of ODEs on R 3 for which Hypotheses.,.2,.3 are met. Up to a reparameterization of the parameter plane, the bifrcation diagram is as in the following description referring to Figre 2. The contination of the homoclinic orbits γ and Rγ exists along µ 2 = 0. Both in µ < 0 and µ > 0 there are crves Hom2 of homoclinic orbits branching from the codimension two bifrcation point: in µ < 0 with doble homoclinic loops following γ (or Rγ by symmetry) twice before closing, in µ > 0 with homoclinic loops following near γ Rγ before closing. Frther bifrcation crves are indicated by SN (Saddle-node bifrcations of periodic orbits), SB (Symmetry-breaking bifrcations of periodic orbits), PD (Period-dobling bifrcations of periodic orbits) and Het2Per (Heteroclinic connections from the origin to periodic orbits). The bifrcation crves are contained in a wedge for some C > 0. µ 2 C µ A Lorenz like attractor exists inside a wedge that is contained in the depicted filled-in region. We conjectre that Lorenz like attractors exist inside the entire region that is filled-in in the figre (i.e. between two heteroclinic bifrcation crves of heteroclinic connections from the eqilibrim to periodic orbits). Lacnae (see []) in the Lorenz like attractors are formed along the heteroclinic bifrcation crve Het2Per inside the filled-in region. Possibly other similar bifrcation crves exist, depending on the vale of β. This will be discssed in terms of redced interval maps below. By singlar rescalings, a first retrn map on a cross section contains small pertrbations from interval maps. The interval maps are of the form x sign(x)( µ 2 + q x β ), for varying µ 2 (related to µ 2 throgh a rescaling). The coefficient q has the same sign as µ, so that redctions to interval maps both with positive and negative slope occr. This paper, and in particlar the proof of Theorem., is organized as follows. Finding asymptotic expressions for a first retrn map on a cross section, is facilitated by a local normal form. The derivation of the local normal form is given in 2. This section also gives the asymptotic expressions for a first retrn map, see 2.2. In 3 we stdy renormalizations to small pertrbations of interval maps. The constrction of continosly differentiable stable foliations for the first retrn map provides a rigoros dimension redction to interval maps and enables an existence proof of Lorenz like attractors. Finally, we stdy bifrcations of β 4

5 periodic orbits to arrive at the bifrcation diagram. For this one cannot rely on the redction to interval maps sing a stable foliation, as the stable foliation is only continosly differentiable and higher order differentiability is reqired for the bifrcation stdy. One can however se a Lyapnov-Schmidt redction to obtain redced bifrcation eqations. The constrction and analysis of the redced bifrcation eqations is in 4. 2 Normal forms and retrn maps We arrive at the main theorems, yielding bifrcation crves and the existence of Lorenz like attractors in the nfolding of the symmetric orbit flip, throgh the derivation and analysis of a retrn map on a cross section. A crcial role lies in the derivation of asymptotic expansions for sch a retrn map. We remark that we do not assme conditions on eigenvales other than Hypothesis.3. In particlar we do not apply linearization reslts which hold only generically (see e.g. [5]). As a conseqence there is no need to check for nonresonance conditions on the eigenvales in order to apply or reslts. In the following section, 2., we constrct a local normal form, close enogh to a linear vector field to enable the derivation of sitable asymptotic expansions for a local transition map. This, as well as asymptotic expansions for a first retrn map, are considered in 2.2 below. 2. Local normal forms Take coordinates = (x, x s, x ss ) so that = f(, µ) is given by a set of ordinary differential eqations of the following form: ẋ = x + F (x, x s, x ss ; µ), ẋ s = βx s + F s (x, x s, x ss ; µ), (4) ẋ ss = αx ss + F ss (x, x s, x ss ; µ) where D i F j (0; µ) = 0 for i = 0, and j =, s, ss. We may assme R(x, x s, x ss ) = ( x, x s, x ss ). (5) A natral approach to stdying bifrcations of homoclinic orbits is throgh the constrction of Poincaré retrn maps, given as compositions of local transition maps (sing the flow near the origin) and global transition maps. The main technical obstacle lies in finding expansions for the flow near the eqilibrim at the origin. The comptation of a local transition map, throgh estimates of integrals appearing in variation of constants formlae, is facilitated by the change to a normal form. Proposition 2.. The system of differential eqations (4), eqivariant nder the action of (5), is smoothly eqivalent to a system of the same expression with F = 0, F s = O( x x ss, x s ), F ss = O( x x ss, x s ). The eqivalence is by mltiplication by a fnction and a conjgacy by a diffeomorphism depending smoothly on µ. The eqivalence varies smoothly with parameters and leaves the action of the symmetry naltered. Proof. By a smooth coordinate change near the origin we may assme that local stable and nstable manifolds are linear: W s loc (0) {x = 0} (6) and W loc(0) {x s, x ss = 0}. (7) 5

6 By (6), F = O(x ), so that mltiplying the vector field with the positive fnction brings the eqation for ẋ to the given form. x x + F (x, x s, x ss ; µ) Note that (7) ensres that F s and F ss are of order O( x s, x ss ). Consider the differential eqations restricted to the stable manifold {x = 0}. Choosing coordinates in which the strong stable foliation is affine, makes that the eqation for ẋ s depends only on x s. Since one-dimensional vector fields near a sink can always be smoothly linearized, a smooth coordinate change gives F s (0, x s, x ss ; µ) = 0. From eqivariance of the differential eqations, the obtained eqations are of the form ẋ s = βx s, ẋ ss = αx ss + x ss f(x s, x ss ; µ) We go on to remove the term x ss f(x s, x ss ; µ). Consider for this a smooth change of variables y s = x s, y ss = x ss + x ss q(x s, x ss ; µ). for a fnction q which satisfies q(x s, x ss ; µ) = q(x s, x ss ; µ). Calclating derivatives with respect to time t, we have ẏ s = βy s, ẏ ss = αy ss + ( + q) y ss (f + qf + q). In order to arrive at or aim, F ss (0, x s, x ss ; µ) = 0, consider the following eqations, treating q as a variable, ẏ s = βy s, ẏ ss = αy ss q = qf f. The eigenvales of the linearized eqations abot (y s, y ss, q) = (0, 0, 0), are β, α, 0. Ths q can be obtained from the two-dimensional stable manifold of the above system. The expressions for F s and F ss from the statement of the proposition follow. Remark 2.. Note that the normal form restricted to the local stable manifold is linear. Symmetry is essential for this normal form. Formal comptations show that the proposition is not valid in the following example when α = 2β: ẋ = x, ẋ s = βx s, ẋ ss = αx ss + x 2 s. This system of differential eqations does in particlar not possess a smooth weak stable manifold; any weak stable manifold is only C. Symmetry however forces the existence of a smooth weak stable manifold, namely the symmetry axis. 2.2 Transition maps Following a discssion of a local transition map for the flow near the origin, we give asymptotic expansions for a first retrn map in Theorem 2. below. For some small positive δ, let Σ = {(δ, x s, x ss ) x s, x ss < δ}, Σ ss = {(x, x s, δ) x, x s < δ} be cross sections which intersect the homoclinic orbit γ transversally. By a linear rescaling we may assme that δ =. The rescaling makes the higher order terms in the differential eqations of order δ: F ss (), F s () Cδ 2, (8) for some C > 0. Pt Σ in = Σ ss RΣ ss, Σ ot = Σ RΣ, see Figre for an illstration. Denote by Π loc : Σ in Σ ot the local transition map. The following proposition provides exponential expansions for Π loc in coordinate charts from the normal form given in Proposition 2.. Note that the leading terms of the expansions eqal the exact formlae for a locally linear vector field. 6

7 Proposition 2.2. For (x ot s, x ot ss ) = Π loc (x in, x in s ) and µ near 0, the following asymptotic formlae apply. If (x in, xin s ) Σ ss, then x ot s = x in β x in s + xin β+ω x in s ϕ s (xin, xin s x ot ss = x in α + x in α+ω ϕ ss (x in, xin s, µ), for some ω > 0. Here ϕ s, ϕ2 s, ϕ ss are smooth fnctions for x in RΣ ss given by symmetry., µ) + xin +ω ϕ 2 s (xin, xin s, µ), 0. Similar asymptotics hold for (xin, xin s ) Proof. Write (x (v), x s (v), x ss (v)) for the soltions to (4) (with higher order terms given by Proposition 2.). We have, for 0 v τ, x (v) = sign(x )e (τ v). With x ss (0) = ± and writing, for this proof, x s (0) = ξ s, the variation of constants formla gives, for 0 v τ, x s (v) = e βv ξ s + x ss (v) = e αv + v 0 v 0 e β(v ζ) F s (x (ζ), x s (ζ), x ss (ζ); µ)dζ, e α(v ζ) F ss (x (ζ), x s (ζ), x ss (ζ); µ)dζ. Write this as (x ss, x s ) = Γ(x ss, x s ). For ω > 0, consider a set B K of continos fnctions (x ss, x s ) : [0, τ] R 2 sch that x ss (v) e (β+ω)v K, x s (v) e βv Kξ s. Direct estimates, making se of (8), show that for ω < min(β, α β, ) and some K > 0, Γ maps B K into itself. This implies x s (v) = e βv ξ s + O(e (β+ω)v ), x ss (v) = e αv + O(e (β+ω)v ), Plg in τ = ln x to obtain the end points in Σ ot. One treats derivatives with respect to τ, ξ s and parameters µ by differentiating the integral formlas and stdying them as above. This leads to the following statement. For k 0, there are positive constants C k sch that, for 0 t τ and µ near µ 0, k (t, ξ s ; µ) k x s(t, τ, ξ s ; µ) C k e βt, k (t, ξ s ; µ) k x ss(t, τ, ξ s ; µ) C k e αt, k (t, ξ s ; µ) k τ x s(t, τ, ξ s ; µ) C k e βt+(t τ), k (t, ξ s ; µ) k τ x ss(t, τ, ξ s ; µ) C k e αt+(t τ). Similar estimates yield expansions in case ξ s = 0, proving the proposition. The global transition map Π far : Σ Σ ss is a local diffeomorphism by the flow box theorem. By the generic nfolding condition in Hypothesis., we may by applying a reparameterization assme that Π far (0, 0) = (µ 2, µ ). Write Π far (x s, x ss ; µ) = µ 2 + q (µ)x s + r (µ)x ss + f (x s, x ss ; µ), Π s far(x s, x ss ; µ) = µ + q 2 (µ)x s + r 2 (µ)x ss + f 2 (x s, x ss ; µ), where f and f 2 are qadratic and higher order terms. In the seqel we sppress the dependence of q, q 2, r and r 2 on µ. Also note that µ 2 = 0 if and only if there are homoclinic orbits to the origin and (µ, µ 2 ) = (0, 0) 7

8 if and only if there are orbit-flip homoclinic orbits to the origin. Symmetry indces a global transition map RΠ far defined on RΣ with vales in RΣ ss. The retrn map Π on Σ in is obtained by composing the global with the local transition maps. The following reslt is obtained, being deliberately nprecise in the notation by making no difference between Σ ss and RΣ ss. Theorem 2.. There are smooth coordinates (x, x s ) on Σ in, sch that with a smooth reparametrization of the parameter plane, the Poincaré retrn map has the following asymptotic expansions: ( ) sign(x )µ 2 + sign(x )q x β x s + O( x β+ω x s ) + O( x +ω ) Π(x, x s ) = µ + q 2 x β x s + O( x β+ω x s ) + O( x +ω, ) where ω is some positive nmber. 3 Lorenz like attractors A renormalization shows how Lorenz like retrn maps, small pertrbations of interval maps embedded in planar maps, arise in the stdy of the orbit-flip. As in [26, 29] the occrrence of Lorenz like attractors can be stdied by constrcting a stable foliation and examining the redced interval maps. Consider rescaled coordinates x = ( x s, x ) given by x = µ β x, x s = µ d x s + µ, (9) with < d < β β. Note that this reqires β > /2. This coordinate change on Σin becomes singlar for µ = 0. Define a rescaled parameter µ 2 by µ 2 = µ 2 µ Let Σ in Σ in be regions, depending on µ 2 and µ, on which x = ( x, x s ) and µ 2 are bonded. We denote the retrn map in rescaled coordinates (the renormalized retrn map) by ( x, x s ) Π( x, x s, µ 2, µ ). Write also Π = ( Π s, Π ). From Theorem 2. we obtain Π( x, x s, µ 2, µ ) = ( β. sign( x )( µ 2 + q x β ) + µ κ k ( x, x s, µ 2, µ ) µ κ k 2( x, x s, µ 2, µ ) where κ is a positive constant. We obtain convergence to a one-dimensional map when µ 0. ) (0) Proposition 3.. For µ 0, the Poincaré retrn map Π converges to ( sign( x )( µ 2 + q x Π β ) 0 ( x, x s, µ 2, µ ) = 0 ). Note that an additional rescaling of the x variable brings q to ±. The redction to an interval map is completed by the constrction of C invariant stable foliations for Π. Here, a C foliation is a foliation whose (in or case one dimensional) leaves can be mapped into straight lines by a C coordinate change. Existence of C stable foliations in related contexts is stdied in [2, 25, 26, 3]. We follow [3]. Proposition 3.2. The Poincaré retrn map Π admits a C foliation F s. Proof. This essentially follows from [3]. For applying [3] there are two isses to be resolved. In or setting there are two cross sections, instead of a single cross section. It is easily seen that this does not pose a problem; one can think of the two cross sections being connected to form a single cross section again. The main reslt in [3] assmes a form for the map in which the coefficients of the lowest order terms x β are constant, i.e. not depending on x s. In or case the coefficients do depend on x s (throgh terms in 8

9 the fnctions k, k 2 ). Additional coordinate changes remove this dependence (compare also [23]). Indeed, a coordinate change of the form x s x s + x a( x s ), x x for some fnction a removes terms with a factor x β from the eqation for Π s. A frther coordinate change of the form x s x s, x x b( x s ) for some fnction b removes terms with a factor x β x s from the eqation for Π. Identifying points on leaves of F s indces a one dimensional map π which is C close to f : [, ] \ {0} R, f( x) = sign( x)( µ 2 + q x β ) () (the projection along leaves of F s is C close to the coordinate projection ( x ss, x ) x if µ is small, compare [23]). The existence of an invariant interval I on which π > implies the existence of a Lorenz like attractor [3, 6, 7, 22]. It sffices to consider f as one can take µ as small as reqired. Sppose q > 0. A direct comptation shows that f > on the invariant interval [ µ 2, µ 2 ] for (q β) β < µ2 < ( q 2 ) β. Likewise, if q < 0, then for ( q 2 ) β < µ2 < ( q β) β we find f > on [ µ 2, µ 2 ]. This interval map f, as well as the related continos map { f( x), x < 0, f( x) = f( x), x > 0, occr in other bifrcation problems as well, see e.g. [23, 24]. Note that a periodic point of period k for f is a periodic point of period k or 2k for f, and vice versa. In [23] an open sbinterval of ( 2, ) of vales for β is given for which it is proved that a Lorenz like attractor exists for all parameter in the filled-in region in Theorem. (see also Figre 3) Figre 3: Nmerical comptation of the attractors inside [,,] for the interval map f as in () for varying parameter µ 2 on the horizontal axis. Here β = 0.7, details of the bifrcation diagram depend on β. 4 Bifrcation crves The analysis of bifrcations of periodic orbits sch as saddle-node bifrcations or symmetry-breaking bifrcations, reqires more differentiability than C and can ths not be done sing redced interval maps. Note frther that we have not inclded parameter dependence in the discssion of stable foliations. The asymptotic expansions for the (rescaled) retrn map allow one to set-p the bifrcation eqations and derive redced bifrcation eqations from a Lyapnov-Schmidt procedre. These are then easily stdied, following an analysis along the lines of e.g. [8, 3, 23, 30]. Here we indicate the constrction of redced bifrcation 9

10 eqations, in 4. below we consider the bifrcation crves from the bifrcation theorem. As a similar analysis can be fond in other papers, like the ones jst mentioned, we will not inclde fll details. For 0 j < N, let Ψ j = ( x in,j+ x in,j+ s ) Π(x in,j, x in,j s ; µ). The existence of N-periodic and N-homoclinic orbits is redced to solving a set of eqations where Ψ = (Ψ 0,..., Ψ N ), x = (x in,0 Ψ(x,x s ; µ) = 0, (2),..., x in,n modlo N. Note that for an N-periodic orbit x in,j 0 < j N yields an N-homoclinic orbit. It is easy to see that with x = (x in,0,...,x in,n ), x s = (x in,0 s 0 for all j. Also x in,n ), x s = (x in,0 s,...,x in,n s ), rank(l := D xs Ψ x =0) = N.,..., x in,n s ) and the indices are taken = x in,0 = 0 and x in,j 0 for Following the setp of of Lyapnov-Schmidt redction method, split (2) into an eqivalent pair of eqations PΨ(x,x s ; µ) = 0, (I P)Ψ(x,x s ; µ) = 0, (3) where P is the orthogonal projection onto the image Im D xs Ψ x =0; i.e. P : R 2N Im D xs Ψ x =0 R N (x,x s ) (0,x s ). Applying the Implicit Fnction Theorem to solve the first part of 3 for x s as a fnction of x and µ leads to the following lemma: Lemma 4.. The eqation PΨ = 0 can be solved for x s as a fnction of x and µ. Here x s (0, µ) = µ and the following estimate holds for k, l 0: k+l k µ l x s (x, µ) µ x C k+l x β l (4) Proof. The remarks preceding the lemma proves the existence, niqeness and smoothness of x s (x, µ). Estimate 4 follows from the asymptotic expansion for the rescaled retrn map in (0). We sbstitte x s (x, µ) into the second part of (3) to obtain the redced bifrcation eqation with Φ : kerl R 2 R N. This gives the following reslt. Φ(x, µ) = (I P)Ψ(x,x s (x, µ); µ) = 0, (5) Proposition 4.. With the above notations, the redced bifrcation eqations have the expansion x in,j+ = sign(x in,j )(µ 2 + q µ x in,j β ) + U j (x, µ), (6) for 0 j < N. The fnction U j is smooth for x in,j 0, 0 j < N and k+l k µ l U j (x, µ) x C k+l x υ l, for some υ > β 0

11 4. Bifrcation analysis We proceed with an analysis of bifrcation crves. Bifrcations of periodic orbits are treated sing the redced bifrcation eqation 6. We restrict to an analysis of bifrcation crves indicated in Theorem. and do for instance not prove that no additional bifrcations occr otside the wedge arond the µ axis indicated in Theorem.. Compare [3, Appendix]. Homoclinic bifrcations: Obviosly, at µ 2 = 0 the redced bifrcation eqation x = sign(x )(µ 2 + q µ x β ) + O( x υ ) has the soltion x = 0 for all µ, so along the line µ 2 = 0 a cople of symmetrical -homoclinic orbits exists. Sppose for simplicity that q < 0 and µ > 0. For 2-homoclinic orbits we solve the eqations with x in,0 and therefore = 0 and x in, x in, = sign(x in,0 x in,0 = sign(x in, )(µ 2 + q µ x in,0 )(µ 2 + q µ x in, 0. Then, for say x in, > 0, x in, = µ 2 β ) + U 0 (x in,0, x in, ; µ), β ) + U (x in,0, x in, ; µ), 0 = µ 2 + q µ µ 2 β + U (0, µ 2 ; µ), µ 2 = q µ β + o((q µ ) β ). Saddle node bifrcations: Sppose for simplicity that q < 0 and µ > 0. To compte the crve of parameter vales with a saddle-node bifrcation of a periodic orbit, consider the bifrcation eqations Solving these eqations, see [8], we obtain x = sign(x )(µ 2 + q µ x β ) + U 0 (x, µ), = sign(x )βq µ x β + U 0 (x, µ). µ 2 = q µ β (β β β β β ) + o((q µ ) β ). Symmetry breaking and period dobling bifrcations: Consider q > 0. A symmetry breaking bifrcation occrs if Solving these eqations, we get x = µ 2 q µ x β + O( x υ ), = βq µ x β + O( x υ ). µ 2 = (q µ ) β (β β + β β β ) + o((q µ ) β ). Note that this gives the bifrcation crve, and does not entail an analysis of the nfolding (compare [8, 3, 30]). In a similar way one treats the crve of periodic dobling bifrcations when q < 0. Heteroclinic connections to periodic orbits: For the analysis of bifrcation crves with heteroclinic connections between the origin and hyperbolic periodic orbits, we refer to [24]. Note that here estimates on first order derivatives sffice and knowledge on higher order derivatives is not reqired. One can stdy the connections for the limit map given in Proposition 3. and conclde their existence for small vales of µ.

12 5 Lorenz like attractors from inclination flip bifrcations Inclination-flip homoclinic orbits in Z 2 symmetric differential eqations can be stdied in mch the same way, sing the normal form and renormalization as in [3, Appendix] (where inclination-flips withot symmetry are considered). This gives an approach to Rychliks reslts inclding a bifrcation analysis. We merely state the reslts. Hypothesis 5. (Inclination flip). The homoclinic orbit γ is a generically nfolding inclination flip homoclinic orbit:. Inclination flip: Along the homoclinic orbit γ for µ = 0, W s, (0) is tangent to W s (0). 2. No orbit flip: for µ = 0, the homoclinic orbit γ is not contained in the strong stable manifold W ss (0). 3. Generic nfolding: the traces of the parameter dependent manifolds W s, (0) and W s (0) in the prodct of state space and parameter space R 3 R 2, intersect transversally along (γ, 0). Theorem 5.. Let ẋ = f(x, µ), µ R 2, be a smooth two parameter family of ODEs on R 3 for which Hypotheses 5.,.2,.3 are met. Up to a reparameterization of the parameter plane, the bifrcation diagram is as in the following description referring to Figre 2. The contination of the homoclinic orbits γ and Rγ exists along µ 2 = 0. Both in µ < 0 and µ > 0 there are crves Hom2 of homoclinic orbits branching from the codimension two bifrcation point: in µ < 0 with doble homoclinic loops following γ (or Rγ by symmetry) twice before closing, in µ > 0 with homoclinic loops following γ Rγ close before closing. Frther bifrcation crves are indicated by SN (Saddle-node bifrcations of periodic orbits), SB (Symmetry-breaking bifrcations of periodic orbits), PD (Period-dobling bifrcations of periodic orbits) and Het2Per (Heteroclinic connections from the origin to periodic orbits). The bifrcation crves are contained in a wedge for some C > 0. µ 2 C µ A Lorenz like attractor exists inside a wedge that is contained in the depicted filled-in region. β References [] V. S. Afraĭmovich, V. V. Bykov, and L. P. Shil nikov. On strctrally nstable attracting limit sets of Lorenz attractor type. Trans. Mosc. Math. Soc., 983(2):53 26, 983. [2] V. S. Afraĭmovich and Ya. B. Pesin. Dimension of Lorenz type attractors. In Mathematical physics reviews, Vol. 6, volme 6 of Soviet Sci. Rev. Sect. C Math. Phys. Rev., pages Harwood Academic Pbl., Chr, 987. [3] J. F. Alves, J. L. Fachada, and J. Sosa Ramos. A condition for transitivity of Lorenz maps. In Proceedings of the Eighth International Conference on Difference Eqations and Applications, pages 7 3. Chapman & Hall/CRC, Boca Raton, FL, [4] V. Araújo, M. J. Pacífico, E. R. Pjals, and M. Viana. Singlar-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc., 36(5): , [5] S. Batista and C. A. Morales. Existence of periodic orbits for singlar-hyperbolic sets. Mosc. Math. J., 6(2): , [6] Y. Choi. Attractors from one dimensional Lorenz-like maps. Discrete Contin. Dyn. Syst., (2-3):75 730,

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