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One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior
Chaos, Solitons and Fractals 29 (2006) 36–47 www.elsevier.com/locate/chaos
One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior S. Cerbelli, M. Giona
*
Centro Interuniversitario sui Sistemi Disordinati, e sui Frattali nell’Ingegneria Chimica, Dipartimento di Ingegneria Chimica, Universita` di Roma ‘‘La Sapienza’’, Via Eudossiana, 18-00154 Roma, Italy Accepted 8 August 2005
Abstract Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a ‘‘massive’’ hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The theoretical understanding and interpretation of chaos in low-dimensional conservative dynamics is still largely based on the paradigm of uniformly hyperbolic systems (also referred to as Anosov systems [1]) that was laid out in the 60s by Anosov [2] and Smale [3]. Shortly, a uniformly hyperbolic transformation f of a smooth manifold M. is a measure-preserving map for which there exists, at any point x of M, a splitting of the tangent space T x ¼ Eux Esx , invariant under the differential Df of transformation, that is, Dfjx ðEax Þ ¼ EafðxÞ , where a = s, u. The splitting is uniquely identified by the property that the norm of vectors v (w) belonging to the subspace Eux ðEsx Þ shrinks exponentially fast to zero for *
Corresponding author. Tel.: +39 6 445 85 892; fax: +39 6 445 85 339. E-mail address: [email protected] (M. Giona).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.213
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negative (positive) times under the differential of the transformation, i.e. kDfnjx Æ vk < Aknkvk, and kDfnjx Æ wk < Aknkwk, where A > 0 and k < 1 for any x 2 M (the wording uniformly hyperbolic indicates precisely that the constants A and k entering the bounds of vectors norm dynamics hold for the entire manifold M). Uniformly hyperbolic systems possess ‘‘nice’’ and interesting features that can be proved rigorously such as positivity of topological and metric entropies, mixing property, existence of global stable and unstable invariant manifolds that are dense in M. and everywhere transverse to each other, shadowing property, e.g. ensuring that there exist trajectories of the system e-close to (computer-generated) pseudotrajectories [4]. Classical archetypal examples of Anosov systems are hyperbolic toral automorphisms, which are represented by the action of hyperbolic matrices with integer entries and unit modulus determinant on the unit two-dimensional torus [1]. However, the Anosov paradigm can account solely for some of the dynamical features of generic nonlinear systems that are considered physically interesting. In generic nonlinear systems, the most evident indication of departure from uniform hyperbolic behavior is that chaos is generally ‘‘massive but not ubiquitous’’, meaning that there exists points of the phase space with zero Lyapunov exponent (e.g. elliptic or parabolic periodic points or quasiperiodic trajectories) that coexist with a set of positive measure of hyperbolic trajectories. This picture agrees with the categorization introduced by Pesin in the mid 70s [5], who defined the class of nonuniformly hyperbolic systems. In a two-dimensional phase space, nonuniformly hyperbolic systems are defined as those systems for which the set of all hyperbolic trajectories (referred to as the Pesin set, say P) has positive measure. The presence of nonhyperbolic orbits can influence geometric and statistical properties associated with points of the Pesin set P. For instance, the existence of a strictly positive constant A entering the bounds of vectors norm dynamics is granted only pointwise (i.e. one cannot find a constant that works for the entire set P), and so is the lowerbound for the angle between stable and unstable invariant directions. The latter property implies that in the Pesin region of a nonuniformly hyperbolic system, it is possible to find points at which the angle between stable and unstable directions becomes arbitrarily small, i.e. the stable and unstable manifolds become almost tangent to each other. In the physics literature, systems that exhibit invariant manifold tangencies have also been referred to as nonhyperbolic [6]. In this article, we stick to Pesins categorization as we find that it could be misleading to use the adjective nonhyperbolic for systems that show hyperbolic behavior in a set of points of positive measure, possibly even of full measure (In Section 4, we provide an archetypal example of nonuniform hyperbolic system for which nonhyperbolic orbits form a set of zero Lebesgue measure). The focus of this article is to analyze how the presence of nonhyperbolic points influences the local and global geometry of invariant manifolds associated with points of the Pesin set P Specifically, we show that the presence of an isolated parabolic point that lies at the boundary of P implies that there are points of the Pesin region where stable and unstable manifolds possess arbitrarily high curvature. Furthermore, the analysis of the local structure of the stable and unstable manifolds at, and in the vicinity of, the parabolic point allows us to derive a geometric property of invariant manifolds, referred to as recursive folding, which we believe to be typical of physically interesting nonlinear models, but which cannot occur in strictly uniformly hyperbolic area-preserving systems.
2. Case study As an illustrative example of nonuniformly hyperbolic system, we consider the Standard Map (SM) family [7] x þ y þ s sinð2pxÞ f s ðxÞ ¼ ðmod 1Þ; ð1Þ y þ s sinð2pxÞ where (x, y) = x are coordinates on the unit square interval [0, 1) · [0, 1) representing a global projection chart of the two-torus (the coordinates of the image point are considered ‘‘mod 1’’), and s is a real parameter. It is well established that the maps of this family display the basic phenomenology that characterizes conservative chaotic dynamics of twodimensional physically interesting nonlinear systems [8], including stroboscopic maps of physically realizable time-periodic incompressible flows in closed domains [9]. Specifically, we consider the map f1(x) corresponding to the value s = 1, whose phase space is characterized by a main chaotic region, henceforth referred to as the Pesin set P, that invades the entire torus with the exception of two small elliptic islands centered around the period-two elliptic orbit constituted by the two points (1/4,1/2) and (3/4,1/2) (see Fig. 1(A)). Fig. 1(B) shows the structure of the unstable manifold, Wu0 , of the origin (an hyperbolic fixed point for the map f1(x)), computed by tracking for n = 7 periods a small segment aligned with the local unstable direction at the point. Likewise those associated with hyperbolic toral homeomorphism, the manifold Wu0 is dense in the chaotic region, which, however, in the present case does not coincide with the entire manifold. On the other hand, the geometry of Wu0 is different and remarkably more complicated than that of invariant
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Fig. 1. Panel (A) Poincare´ section of a portion (N = 105 iterations) of a typical hyperbolic trajectory associated with the map f1(x). The two small regions devoid of points are quasiperiodic islands surrounding the period-two elliptic points (1/4,1/2) and (3/4,1/2). Panel (B) structure of the unstable manifold, Wu0 , of the origin.
manifolds of hyperbolic toral automorphism, (or of any Anosov toral diffeomorphisms smoothly conjugated to an Anosov toral automorphism, for that matter [10]) this difference being precisely the recursive folding property. Let us first define this property in the general context of a continuous transformation h of a smooth two-dimensional (orientable) manifold M. Let W be an h-invariant manifold (be it a global stable or unstable manifold) that is dense in a chaotic region P 2 M. Let an orientation along W be established (e.g. by introducing the natural parametrization in terms of the arc length s from an arbitrary point x0 2 W and by considering increasing values of s). By the fact that W is dense in P, it will intersect with the boundary oBe of any e-ball embedded in the chaotic region at infinitely many points xl 2 oBe. At each point xl, it is well defined whether, following the assigned orientation along the curve, the crossing occurs while entering or leaving Be. For the map to be recursively folding, we ask that for each crossing point xl where W enters Be there is a crossing point xm arbitrarily close to it where W leaves Be (see Fig. 2). As a confirmation that the map f1 is recursively folding, let us consider the intersections of Wu0 with a reference line, say the horizontal segment {0 6 x < 1, y = 1/4}, which represents a circumference in the torus topology. Fig. 3 shows the signed intersections of Wu0 with this circumference. The sign associated with the intersection is taken as positive or negative according to whether the crossing of the reference line along Wu0 occurs upwards or downwards (i.e. while increasing or decreasing the value of the y-coordinate), respectively. As can be noted, there are opposedly signed intersections arbitrarily close to each other. The recursive folding property should not be confused with the occurrence of simple ‘‘macroscopic’’ folding of the invariant manifolds, which, clearly, can occur even in uniformly hyperbolic systems. As an illustration, Fig. 4 shows the structure and the signed intersections of the unstable manifold at the origin for the uniformly hyperbolic diffeomorphism g(x), constructed by using the SM f1(x) as a change of coordinates of an Anosov toral homeomorphism 1 1 a(x) = a(x, y) = (x + y, x + 2y) [mod 1], i.e. gðxÞ ¼ f 1 1 a f 1 ðxÞ ¼ f 1 ða½f 1 ðxÞÞ, f 1 being the inverse of the map f1. This example, that is much less artificial than as it may appear at first glance [20], clearly indicates that recursive folding does not occur in a uniformly hyperbolic system on the two-torus. Thus, the presence of recursive folding must be
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Fig. 2. Sketch of the recursively folding property of a global invariant manifold W. An oriented arc of W intersects any e-ball embedded in the chaotic region at nearby points with opposite orientations.
Fig. 3. Signed intersections of a continuous arc of the unstable manifold Wu0 at the origin with the reference line {(x, y) j 0 6 x < 1, y = 1/4}. Intersection points are represented as (xint, ±1) where xint is the intersection abscissa and the sign is chosen as positive or negative according to whether the crossing occurs while increasing or decreasing the y coordinate, respectively. Consecutive points are joined by a dotted line to enhance visualization of the high frequency oscillations between ±1, which indicate that the transformation f1 is recursively folding.
strictly related to that of nonhyperbolic trajectories. Specifically, parabolic points that lay at the boundary of the Pesin set appear as the most natural candidates for analyzing the influence of regular orbits on the geometric invariant structure of the chaotic region. Let us consider the parabolic fixed point xp = (1/4,0) of the SM f1(x). The Poincare´ section depicted in Fig. 1(A) suggests that xp possess vanishing distance from the Pesin set. At xp, the Jacobian matrix Dxjxp 1 1 given by Dfjxp ¼ possesses a double unit eigenvalue to which corresponds the eigenspace spanned by 0 1 ex = (1, 0). Let us analyze the dynamics in a neighborhood of xp by considering the evolute of a small segment Ie = {(x, y)jjx 1/4j < e, y = 0} centered at xp and aligned along the (unique) invariant direction. Fig. 5(A) and (B) shows the local structure of the images of such segment up to ten iterates under the forward and inverse mappings, f1 and f 1 1 , respectively. In both cases, the qualitative behavior of the segment images is the same, namely the segment folds onto itself at the parabolic point and collapses onto an invariant curve (a curve Cxp for forward iterations of the map, and a curve cxp for backward iterations). The global outlook of these two invariant curves is indistinguishable from that of any global unstable manifold associate with hyperbolic points (not shown in the interest of brevity). Also, n distances along Cxp and cxp are shrunk to zero by f n 1 and f 1 , respectively. Thus, the curves Cxp and cxp fulfill all the requirements that qualify global invariant manifolds. There is, however, one important difference that distinguishes these manifolds from those associated with hyperbolic points, namely the fact that they originate at the point rather than running through it. Because of this peculiar property, we refer to these invariant curves as one-sided stable and
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Fig. 4. Panel (A): Unstable manifold of the origin associated with the transformation gðxÞ ¼ f 1 1 a f 1 ðxÞ. By construction, this curve is topologically conjugate to a straight line that wraps densely around the torus. Panel (B): Signed intersection of the unstable manifold with the reference line {0 6 x < 1, y = 1/4}. The transformation g(x) is not recursively folding.
unstable manifolds associated with the parabolic point xp (see Fig. 5(C)). It is also worth noting that, unlike the case of hyperbolic periodic points, the curvature of Cxp and cxp is, in a dynamical sense, undefined. Specifically, while the local curvature of a generic filament through a hyperbolic periodic point xh converges to the local curvature of the unstable (stable) manifold of xh for positive (negative) times (see, e.g. [12]), the local curvature of a filament through xp diverges to infinity. Since the local curvature of the invariant stable and unstable manifold is well defined along any point of hyperbolic trajectories, this peculiar phenomenon suggests that the local curvature of invariant stable and unstable manifolds at hyperbolic points e-close to xp can assume arbitrarily large values. This issue is analyzed in the next section. To sum up, as it concerns the existence of stable and unstable manifolds, the parabolic point xp behaves ‘‘almost’’ as a generic hyperbolic point, the difference being the feature that invariant manifolds are one-sided in the meaning discussed above.
3. Lobes, recursive folding and singular curvature of invariant manifolds The next question to be answered is then how the peculiar geometric structure of invariant manifolds at the parabolic points influences the structure of invariant manifolds in points of the Pesin set that are close to xp, and how this local geometric picture is related to the recursive folding property. Fig. 6 shows a zoom-in of the local structure of the stable and unstable manifolds of the origin, Ws0 and Wu0 , in a small rectangular domain about xp (for visualization purposes, a different chart is used to represent this interval of the torus, obtained by translating of 1 units the y-coordinate of points falling in the lower part of the domain). A first straightforward observation is that unstable manifolds undergo a bend that encloses the parabolic point and the local one-sided manifolds. The most important feature, however, arises from the analysis of the mutual structure of the manifolds. Specifically, it is possible to find two continuous arcs, ‘s 2 Ws0 , and ‘u 2 Wu0 , with the following properties (see Fig. 7):
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Fig. 5. Dynamics of a horizontal segment, Ie, through the parabolic point xp of the map f1(x) under forward (Panel (A)) and backward (Panel (B)) iterations of f1(x). The images of Ie fold around the point developing an ever increasing curvature at the point xp until both branches of the folded curve collapse onto the invariant one-sided manifold. Panel (C): Tangency between the one-sided stable manifold cxp (upper curve), and unstable manifold Cxp (lower curve) at the parabolic point xp.
1. ‘s and ‘u intersect only at two points, say x and y. 2. ‘s [ ‘u are the boundary of a closed domain say L. The region L is referred to as a lobe [13]. With respect to the definition reported in [13], we make one further assumption on the lobe L, namely we suppose that L contains a dense set of hyperbolic trajectories. This assumption, suggested by the structure of the Poincare´ section depicted in Fig. 1(A), is used below to show that the mere existence of a lobe such as L has important implications as it regards the existence of geometric singularities associated with invariant manifolds of hyperbolic points.
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Fig. 6. Structure of Ws0 (Panel (A)), and Wu0 (Panel (B)) associated with the transformation f1 near the parabolic point xp. Both manifold undergo a bend that encloses xp.
s
Fig. 7. Schematic diagram explaining the construction of a lobe L from the arcs ‘s (dashed line connecting x and y), ‘~ (dashed line s connecting x* and y*) and ‘u (continuous line). The fact that the arc ‘~ contracts exponentially as the image of x* visits densely the Pesin region provides the connection between the local structure of invariant manifolds and the recursive folding property.
First, let us show that recursive folding follows as direct consequence of properties 1 and 2 defined above. In fact, suppose these properties hold true. On Wu0 consider a point, say x*, that is arbitrarily close to x and whose forward s orbit is dense in the Pesin set [21] (see Fig. 7). Consider the arc ‘~ originating at x* and intersecting Wu0 at a point s u * ~ y close to y. Fix an orientation along the arc ‘ and ‘ , e.g. by establishing a clockwise orientation on the boundary s oL. Consider the unit tangent vectors tu(x*), tu(y*) (ts(x*), ts(y*)) to ‘u ð‘~ Þ at the points x*, y* oriented according to the positive direction. From the geometric picture described above, it readily follows that vector products tu(x*) ^ ts(x*) and tu(y*) ^ ts(y*) are of opposite sign. By the fact that f1 is smooth together with its inverse f 1 1 , this property is pres served under the action of the map. But as n ! +1, the arc ‘~ shrinks to vanishing length, and its endpoints collapse onto a dense orbit. Thus, the arcs of Wu0 that run through the intersection points fn(x*) and fn(y*), will eventually visit any e-ball embedded in the Pesin set as they become arbitrarily close to each other, while still being opposedly oriented. Thus, the manifold Wu0 must be recursively folding.
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It is worth stressing out that lobes cannot exist in uniformly hyperbolic diffeomorphisms of the two-torus, which, on the other hand, do not possess nonhyperbolic points by definition. A natural question arises as to whether or not the presence of isolated parabolic points in a chaotic region of a two-dimensional toral diffeomorphism (or, possibly, of a surface diffeomorphism) always implies the occurrence of lobes and viceversa. While we cannot provide a general argument to prove (or disprove) this statement, we next connect the presence of lobes with singularities of local curvature associated with stable and unstable manifold of hyperbolic points. In order to see this, consider again Fig. 7 and let the lobe L be formed by ‘u and ‘s, i.e. L is the closed domain delimited by ‘u and ‘s containing xp (in the argument that s follows it is immaterial whether one chooses ‘s or ‘~ as delimiting the lobe). In L, let us define an unit vector field e(x) = (ex(x), ey(x)) (ke(x)k = 1) satisfying the following conditions: (i) e(x) be smooth in L, (ii) e(x)jx2‘u = tu(x) for x 2 ‘u , i.e. on ‘u the vector e(x) coincides with the unit tangent vector associated with the assigned orientation established on oL, (iii) e(x) be transverse to the local stable direction Esx at all the points x 2 P \ L. By Stokes theorem applied to the field e(x) in L, it follows that Z Z Z Z eðxÞ t ds ¼ eðxÞ t ds þ eðxÞ t ds ¼ curl2 ðeðxÞÞ dx dy; ð2Þ oL
‘u
‘s
L
where oL = ‘s [ ‘u, curl2(e(x)) = oey/ox oex/oy, and where t(x) is the unit tangent vector on oL (which is defined almost everywhere, i.e. everywhere but at the points x and y). By the fact that t(x) = e(x) for x 2 ‘u, one immediately R derives that ‘u eðxÞ t ds ¼ Lengthð‘u Þ. Let us now address what are the consequences of Eq. (2) if one considers for~ ¼ f n1 ðxÞ with ward iterates Ln ¼ f n1 ðLÞ of the lobe L. The vector field e(x) at x maps to the vector field 1ð~ xÞ at x n ~ 2 Ln a new unit vector field 1ð~ xÞ ¼ Df 1 jx eðxÞ. By renormalizing vector lengths, one can define at the image points x ~, exen ð~ xÞ ¼ 1ð~ xÞ=k1ð~ xÞk. This unit vector field satisfies, with respect to the mapped lobe Ln and the new coordinates x actly the same properties (i),-(ii), and (iii) that were verified by e(x) in L. Also, for large n; en ð~ xÞ is aligned with the local ~ 2 Ln that belong to the Pesin set. Thus, if x ~ 2 P; en ð~ unstable direction at all the points x xÞ is almost tangent to the ~. Applying the Stokes theorem again in this new situation, we obtain unstable manifold through x Z Z curl2 ðeð~ xÞÞ d~x d~y ¼ Lengthð‘un Þ þ en ð~ xÞ t d~s; ð3Þ Ln
‘sn
where ~s denotes the arc length along oLn and t represents a tangent vector along ‘sn . At large n, the contribution of the line integral along ‘sn at the r.h.s. of Eq. (3) becomes immaterial, since the arc ‘sn shrinks exponentially fast to vanishing length. Therefore, the value of the l.h.s. of Eq. (3) approaches the length ‘un as n increases. Since the measure (i.e. the area) of the lobe Ln is constant, this implies that fixed a value j arbitrarily large, there exists an integer m such that for n > m one can find points of Ln where the absolute value of the integrand curl2 ðeð~ xÞÞ results greater than j. If the Pesin set P is dense in L and therefore in Ln (as the Poincare` section of Fig. 1(A) suggests), the above result implies that one can find hyperbolic points where the value jcurl2(e(x))j is arbitrarily large. But at these points, for n large enough, the field of unit vectors en ð~ xÞ collapses onto the unstable directions Eux~ , which is tangent to the local unstable manifolds, i.e. u en ð~ xÞ ! e ð~ xÞ, where eu ð~ xÞ 2 Eux~ . Besides, it is straightforward to show (see Appendix A) that the absolute value of u ~ , i.e. jcurl2 ðe ð~ xÞÞj equals the modulus of the local curvature ju ðxÞ ¼ kku ð~ xÞk of the unstable manifolds at the point x jcurl2 ðeu ð~ xÞÞj ¼ keu ð~ xÞ reun ð~ xÞk ¼ kku ð~ xÞk. Thus, the above argument allows us to conclude that the presence of a lobe such as that depicted in Fig. 7, formed by two continuous arcs, ‘s and ‘u, of a stable and an unstable manifold that intersect only at two points and embrace a dense set of hyperbolic trajectories implies that there are points of the Pesin set at which the modulus of the local curvature associated with the local stable and unstable invariant manifolds assumes arbitrarily large values.
4. An archetypal example of nonuniformly hyperbolic behavior Next, we describe an archetypal model, the toral homeomorphism H, which, as it regards the geometric picture discussed above, can be envisioned as a topological analogous of the transformation f1. A detailed account of all the statistical and geometric properties of H (most of which can be derived analytically) has been presented elsewhere [14]. Here, we focus on the geometric aspects of the transformation that are strictly pertinent to the focus of the present discussion. The homeomorphism HðxÞ ¼ Hðx; yÞ ¼ ðx0 ; y 0 Þ is a piecewise linear, globally continuous transformation of the two-torus defined by 0 x ¼ x þ f ðyÞ ðmod 1Þ; ð4Þ y 0 ¼ x þ y þ f ðyÞ
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where (x, y),(x 0 , y 0 ) 2 [0, 1) · [0, 1) are coordinates on the two-torus, and f(n) : [0, 1] ! [0, 1] is the tent map defined by f(n) = 2n for 0 6 n < 1/2 and f(n) = 2 2n whenever 1/2 6 n < 1. In Ref. [14], it is shown that one can break down the torus into two disjoint subsets say R1 and R2 (see Fig. 8(A)), where the invariant stable and unstable directions, Esx and Esx respectively, are constant. For x 2 R1, these directions, say Es1 and Eu1 , are given by the contracting and 1 2 , i.e. Es1 ¼ spanðs1 Þ and Eu1 ¼ spanðu1 Þ where expanding eigenspace associated with the hyperbolic matrix H 1 ¼ 1 3 pffiffiffi pffiffiffi u1 ¼ ð2; 1 þ 3Þ, and s1 ¼ ð2; 1 3Þ. The invariant directions Es2 and Eu2 at the points x 2 R2 are given by the images 1 2 of the eigenspaces associated with points of R1 through the matrix H 0 ¼ , i.e. Es2 ¼ spanðH 0 s1 Þ, and 1 1 Eu2 ¼ spanðH 0 u1 Þ. Fig. 8(B) and (C) shows the structure of the stable and unstable manifold of the origin 0, Ws0 and Wu0 , respectively. These curves are globally continuous, piecewise-smooth, H -invariant, and everywhere dense on the torus. They are nonsmooth only at the boundary oR2 of the set R2. To the focus of the present discussion, an interesting geometric phenomenon, similar to that observed in the analysis of the map f1, occurs at the (fixed) points xp,1 = (1, 1/2) and xp,2 = (1, 1). Fig. 8(D) shows a zoom-in of the region near the point xp,1 = (1, 1/2) (a local chart is used in the figure for rendering the continuity of the structure). The invariant manifolds, Wu0 (continuous line) and Ws0 (dashed line), undergo a ‘‘U-turn’’ around the fixed point in question, thus making it possible to identify a lobe (in fact, a family of lobes) similar to that associated with the map f1. Clearly, in this case, the analogous of the arc ‘u is no longer smooth (it intersects the boundary of R). Also, the curvature of invariant manifolds is not defined at the point xp,1. Yet, the essence of the geometric peculiarity that was observed in the SM can still be appreciated here, e.g. by identifying as a ‘‘generalized’’ local curvature in the neighborhood of the periodic point the ‘‘width of the U-turn’’, i.e. the gap separating the two parallel arms of the arc of Wu0 as it goes around xp,1. With this identification, it is clear that the generalized curvature assumes arbitrarily high values in that one can find lobes arbitrarily small that still contain xp,1. The identification of the lobe also indicates the recursive folding property
Fig. 8. Panel (A): Decomposition of the torus into two sets R1 (grey shaded area) and R2 (white area) where the expanding and dilating directions are constant (see main text for details). Panel (B) and (C): Arcs of the unstable and stable manifold of the origin for the map H, respectively. Panel (D): Zoom-in of the neighborhood of the point xP,1 = (1, 1/2) (a local chart is used for rendering continuity of the manifolds). The stable and unstable manifolds are represented as continuous and dashed line, respectively. Note that one can define quadrangular lobes.
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of H. Clearly, the lack of global smoothness of the model (in particular the fact that H is not differentiable at xp,1 and xp,2 makes it impossible to push the analogy further. For instance, while the ‘‘one-sided’’ character of the stable and unstable manifolds at xp,1 and xp,2 still holds true for the transformation H, their local tangency does not (the curves are not even differentiable at these points). However, the analysis [14] of a class of smooth maps that are e-close to H (in the C0 topology) shows that the geometric structure of invariant manifolds in the vicinity of the two points xp,1, xp,2 (which are unchanged by the smooth perturbation) is actually completely analogous to that of the transformation f1 in the neighborhood of the parabolic periodic point xp (or of any other isolated parabolic point at the boundary of the Pesin set, for that matter). Thus, as it regards the focus of the present article, the availability of an archetype such as H (and its smooth perturbations) is important in that it provides a concrete example of nonuniformly hyperbolic system where chaotic behavior is ‘‘almost sure’’, i.e. the Lebesgue measure of the Pesin set coincides with that of the entire manifold. This suggests that the escape from uniformly hyperbolic behavior must be characterized in a topological rather than a measure-theoretical framework in that the invariant geometric features associated with the hyperbolic (Pesin) region can be significantly affected by the mere presence of a nonempty set of nonhyperbolic points, even when this set possesses zero Lebesgue measure. Our standpoint is that geometric singularities can occur whenever the Pesin region is not compact (roughly speaking, it has accumulation points that do not belong to the set). For instance, the lack of compactness makes it possible to reconciliate the fact that curvature of invariant manifolds is well defined and continuous at any point of the Pesin region with the result that one can find a sequence of lobes such that the average value of curvature modulus kkuk over the lobe region diverges. In fact, were the hyperbolic set compact and k(x) continuous with x 2 P, then, by Weiertrass theorem, the modulus of curvature would admit absolute maximum, and geometric singular behavior could not occur.
5. Concluding remarks: Physical significance of recursive folding and curvature singularities The main focus of this article was to show that the presence of a lobe (as defined in Section 3) containing a dense set of hyperbolic orbits implies certain peculiar geometric features of invariant manifolds associated with hyperbolic points, namely the existence of points where the local curvature assume arbitrarily high values, and the recursive folding property. We also pointed out that these features cannot occur in a uniformly hyperbolic system on the two-torus T2 in that, by the results of Franks and Manning [11], every Anosov toral diffeomorphism is topologically conjugate to a linear hyperbolic system. This implies that unstable manifolds of a uniformly hyperbolic system on T2 are topologically conjugate to straight lines (the diffeomorphism g(x) discussed in Section 3 provides an illustration of this property). We remark that the identification of the lobe associated with the map f1(x) was originally motivated by the analysis of the invariant manifold structure near the parabolic fixed point xp = (1/4,0). This raises the question whether any lobe must necessarily contain nonhyperbolic points (and, in particular, parabolic points). In the case where the mixing space is the two-torus, the answer to this question appears to be affirmative, since the arguments developed in this article prove that the existence of a lobe implies the existence of nonhyperbolic points, even though we did not strictly prove that some of these nonhyperbolic points must necessary fall inside the lobe itself. Besides, generalizing this statement to the case of a generic diffeomorphism of a two-dimensional domain (think, e.g., of the stroboscopic map of a time-periodic incompressible flow in a closed cavity) may be nontrivial. Yet, results of numerical simulations conducted over a wealth of two-dimensional systems defined in closed two-dimensional domains with boundaries show that invariant manifolds possess the typical sharply bent structure that results from recursive folding (see, e.g. [15]). This observation supports the conjecture that a bounded simply connected domain of R2 cannot support an Anosov diffeomorphism. In turn, this statements suggest that the shortcoming of the Anosov paradigm as a model of physical systems (and of physically interesting models such as the SM family) lies in a misrepresentation of the folding action of this type of transformations. In fact, if one considers the simplest model of Anosov systems, namely the class of hyperbolic toral automorphisms, it is easy to realize that the curvature necessary to embed a curve whose length increases exponentially (e.g. a piece of unstable manifold) into a bounded two-dimensional manifold is provided by the folded structure of the phase space (the two-torus) rather than by the actual folding action of the map. Besides, other popular archetypes of mixing such as the Smales horseshoe [16] and the Bakers maps [17] are characterized by analogous shortcomings as it regards modeling of folding. Specifically, in the horseshoe map folding takes place outside of the ‘‘region of interest’’ (e.g. a rectangle), whereas in the Bakers map (be it the original archetype or any generalization characterized by multiple stretching exponents) folding is modeled by an unphysical ‘‘cutting and staking’’ procedure. From the physical standpoint, curvature dynamics and recursive folding can have important effects on processes that are influenced by chaotic advection. For instance, it has been observed that bubble and droplet break up and coalescence in physically realizable chaotic flows can be influenced by the local curvature of invariant manifolds [18]. As it
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regards recursive folding, it is sensible to expect that this property be strictly related to the phenomenon of cancellation associated with advecting–diffusing magnetic fields in the fast dynamo approximation [19]. In summary, we think that a complete understanding of mixing in two-dimensional systems cannot do away with an attentive investigation of the physical implications of nonuniform hyperbolicity. A first step in this direction could be that of establishing, in the general context of area-preserving diffeomorphisms of a bounded two-dimensional domain onto itself, whether the occurrence of one of the specific features analyzed in this work—existence of lobes, recursive folding, existence of parabolic points, singular behavior of local curvature of invariant manifolds—implies all the others.
Appendix A. Local curvature of a unit modulus vector field In this Appendix, we define the pointwise curvature vector field, k(x) = (k1(x), k2(x)) associated with a unit modulus vector field e(x) = (e1(x), e2(x)) where x = (x1, x2) belongs to a region D of R2 . We assume that e(x) is C2 ðDÞ. We want to show that the modulus of k(x) is numerically equal to the absolute value of curl2(e(x)) = oe2/ox1 oe1/ox2. Let us define the autonomous dynamical system dx/ds = e(x), whose trajectories are curves everywhere tangent to e(x) param~ as the local ‘‘curvature vector’’ associated with the trajectory etrized in terms of arc length s. We define kð~ xÞ at x ~. Let x(s) = (x1(s), x2(s)) be the parametric representation of this trajectory, with xð~sÞ ¼ x ~. The local curvature through x ~ is defined as kð~ kð~ xÞ at the point x xÞ ¼ d2 xðsÞ=ds2 js¼~s . In fact, by this definition, one can readily see that curvature is a skew-symmetric second-order tensor (thus specified by three independent components) which can be conveniently represented at any given point by the vector k, whose modulus represents the inverse radius of the osculating circle to the curve at the given point, and whose direction (normal to the tangent vector) is oriented towards the center of this circle. We use quotation marks for addressing k as a vector to underline that k is only a handy representation of a tensor quantity. As, by definition, dx/ds = e(x), it results k = de(x(s))/ds = [e(x) Æ $]e(x). Thus, ki = ejoei/oxj, where i, j = 1, 2 and a repeated index mean summation over that index. In order to show that kkk = jcurl2(e(x))j = joe2/ox1 oe1/ox2j, let us represent the unit vector field e(x) as e(x) = (cosh(x), sinh(x)) where h(x) = h(x1, x2) 2 [0, 2p) is the angle through which e(x) must rotate clockwise to become aligned with the positive direction of the x1-axis. Let us denote with hi = oh/ox1. With this notation, k is given by ! k1 sin h cos h sin hh1 sin2 hh2 ¼ ½cos hh ¼ þ sin hh . ðA:1Þ 1 2 cos h k2 cos2 hh1 þ sin h cos hh2 On the other hand, jcurl2 (e(x))j = joe2/ox1 oe1/ox2j = jcos h h1 + sin h h2j which is equal the factor in between square brackets in Eq. (A.1). Since this factor multiplies a unit modulus vector, it results kk(x)k = jcurl2(e(x))j.
References [1] Katok A, Hasselblatt B. Introduction to the modern theory of dynamical systems. Cambridge: Cambridge University Press; 1995. [2] Anosov DV. Geodesic flows on Riemann manifolds with negative curvature. Proc Stekl Inst Math, vol. 90. Providence, RI: American Mathematical Society; 1967. [3] Smale S. Bull Am Math Soc 1967;73:747. [4] Grebogi C, Hammel SM, Yorke JA, Sauer T. Phys Rev Lett 1990;65:1527. [5] Pesin YaB. Izv Akad Nauk SSSR Ser Mat 1976;40:1332; Pesin YaB. Russ Math Surv 1976;32:55. The notion of nonuniformly hyperbolic system introduced in these papers, which has by now become a cornerstone of the mathematical theory of dynamical systems, has been substantially ignored by the physics and applied science communities throughout the last 25 years. [6] Kantz H, Grebogi C, Prasad A, Lai YC, Sinde E. Phys Rev E 2002;65:026209. [7] Chirikov BV. Phys Rep 1979;52:263. [8] Ott E. Chaos in dynamical systems. Cambridge: Cambridge University Press; 2001; Casati G, Chirikov B. Quantum chaos. Cambridge: Cambridge University Press; 1995; Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: SpringerVerlag; 1990. [9] Chien WL, Rising M, Ottino JM. J Fluid Mech 1986;170:355. [10] Cerbelli S, Giona M, Adrover A, Alvarez M, Muzzio FJ. Chaos, Solitons & Fractals 2000;11:607. [11] Franks J. Trans Am Math Soc 1969;145:117; Manning A. Am J Math 1974;96:422. [12] Cerbelli S, Zalc J, Muzzio FJ. Chem Eng Sci 2000;55:363.
S. Cerbelli, M. Giona / Chaos, Solitons and Fractals 29 (2006) 36–47 [13] [14] [15] [16]
[17] [18] [19]
[20]
[21]
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Wiggins S. Chaotic transport in dynamical systems. New York: Springer-Verlag; 1992. Cerbelli S, Giona M. J Nonlinear Sci, in press. Giona M, Adrover A, Muzzio FJ, Cerbelli S, Alvarez MM. Physica D 1999;132:298. Smales horseshoe (see [3]) provides a classical example of a dynamical system possessing an hyperbolic attractor that has the structure of a Cantor set. In its original formulation the action of the map restricted to the region of interest is both discontinuous and dissipative. Finn JM, Ott E. Phys Rev Lett 1988;60:760; Finn JM, Ott E. Phys Fluids 1988;31:2992. Tjahjadi M, Ottino JM. J Fluid Mech 1991;232:191. Childress S, Gilbert AD. Stretch, twist, fold: the fast dynamo. New York: Springer-Verlag; 1995. The fast dynamo problem studies what type of kinematic fluid motion can induce amplification of an advected magnetic field seed. The gradient-erasing action of diffusion is greatly favoured whenever the advected magnetic field possesses a structure with rapid changes of sign in space (i.e. whenever the exist nearby zones where the magnetic vectors are parallel but opposedly oriented). In fact, the class of systems that are topologically conjugate with a linear hyperbolic system (i.e. a hyperbolic toral automorphism) on the two-torus coincides with the class of uniformly hyperbolic systems defined on this manifold [11]. In other words, any Anosov diffeomorphism on the two-torus is topologically conjugate to a hyperbolic toral automorphism. The existence of a point such as x* follows from the topological transitivity of the dynamics in the Pesin set. In fact, consider a trajectory O ¼ fxj gi2N that is dense in P and a point xl 2 O GO that is arbitrarily close to x. Let Wsxl;loc be the local stable manifold through xl. This local manifold will intersect ‘u at a point x* e-close to x. Since distances on Wsxp;loc shrink to zero exponentially fast, the forward orbit of x* will collapse onto that of xl. As a consequence, the forward trajectory of x* is also dense.
One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior S. Cerbelli, M. Giona
*
Centro Interuniversitario sui Sistemi Disordinati, e sui Frattali nell’Ingegneria Chimica, Dipartimento di Ingegneria Chimica, Universita` di Roma ‘‘La Sapienza’’, Via Eudossiana, 18-00154 Roma, Italy Accepted 8 August 2005
Abstract Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a ‘‘massive’’ hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The theoretical understanding and interpretation of chaos in low-dimensional conservative dynamics is still largely based on the paradigm of uniformly hyperbolic systems (also referred to as Anosov systems [1]) that was laid out in the 60s by Anosov [2] and Smale [3]. Shortly, a uniformly hyperbolic transformation f of a smooth manifold M. is a measure-preserving map for which there exists, at any point x of M, a splitting of the tangent space T x ¼ Eux Esx , invariant under the differential Df of transformation, that is, Dfjx ðEax Þ ¼ EafðxÞ , where a = s, u. The splitting is uniquely identified by the property that the norm of vectors v (w) belonging to the subspace Eux ðEsx Þ shrinks exponentially fast to zero for *
Corresponding author. Tel.: +39 6 445 85 892; fax: +39 6 445 85 339. E-mail address: [email protected] (M. Giona).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.213
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negative (positive) times under the differential of the transformation, i.e. kDfnjx Æ vk < Aknkvk, and kDfnjx Æ wk < Aknkwk, where A > 0 and k < 1 for any x 2 M (the wording uniformly hyperbolic indicates precisely that the constants A and k entering the bounds of vectors norm dynamics hold for the entire manifold M). Uniformly hyperbolic systems possess ‘‘nice’’ and interesting features that can be proved rigorously such as positivity of topological and metric entropies, mixing property, existence of global stable and unstable invariant manifolds that are dense in M. and everywhere transverse to each other, shadowing property, e.g. ensuring that there exist trajectories of the system e-close to (computer-generated) pseudotrajectories [4]. Classical archetypal examples of Anosov systems are hyperbolic toral automorphisms, which are represented by the action of hyperbolic matrices with integer entries and unit modulus determinant on the unit two-dimensional torus [1]. However, the Anosov paradigm can account solely for some of the dynamical features of generic nonlinear systems that are considered physically interesting. In generic nonlinear systems, the most evident indication of departure from uniform hyperbolic behavior is that chaos is generally ‘‘massive but not ubiquitous’’, meaning that there exists points of the phase space with zero Lyapunov exponent (e.g. elliptic or parabolic periodic points or quasiperiodic trajectories) that coexist with a set of positive measure of hyperbolic trajectories. This picture agrees with the categorization introduced by Pesin in the mid 70s [5], who defined the class of nonuniformly hyperbolic systems. In a two-dimensional phase space, nonuniformly hyperbolic systems are defined as those systems for which the set of all hyperbolic trajectories (referred to as the Pesin set, say P) has positive measure. The presence of nonhyperbolic orbits can influence geometric and statistical properties associated with points of the Pesin set P. For instance, the existence of a strictly positive constant A entering the bounds of vectors norm dynamics is granted only pointwise (i.e. one cannot find a constant that works for the entire set P), and so is the lowerbound for the angle between stable and unstable invariant directions. The latter property implies that in the Pesin region of a nonuniformly hyperbolic system, it is possible to find points at which the angle between stable and unstable directions becomes arbitrarily small, i.e. the stable and unstable manifolds become almost tangent to each other. In the physics literature, systems that exhibit invariant manifold tangencies have also been referred to as nonhyperbolic [6]. In this article, we stick to Pesins categorization as we find that it could be misleading to use the adjective nonhyperbolic for systems that show hyperbolic behavior in a set of points of positive measure, possibly even of full measure (In Section 4, we provide an archetypal example of nonuniform hyperbolic system for which nonhyperbolic orbits form a set of zero Lebesgue measure). The focus of this article is to analyze how the presence of nonhyperbolic points influences the local and global geometry of invariant manifolds associated with points of the Pesin set P Specifically, we show that the presence of an isolated parabolic point that lies at the boundary of P implies that there are points of the Pesin region where stable and unstable manifolds possess arbitrarily high curvature. Furthermore, the analysis of the local structure of the stable and unstable manifolds at, and in the vicinity of, the parabolic point allows us to derive a geometric property of invariant manifolds, referred to as recursive folding, which we believe to be typical of physically interesting nonlinear models, but which cannot occur in strictly uniformly hyperbolic area-preserving systems.
2. Case study As an illustrative example of nonuniformly hyperbolic system, we consider the Standard Map (SM) family [7] x þ y þ s sinð2pxÞ f s ðxÞ ¼ ðmod 1Þ; ð1Þ y þ s sinð2pxÞ where (x, y) = x are coordinates on the unit square interval [0, 1) · [0, 1) representing a global projection chart of the two-torus (the coordinates of the image point are considered ‘‘mod 1’’), and s is a real parameter. It is well established that the maps of this family display the basic phenomenology that characterizes conservative chaotic dynamics of twodimensional physically interesting nonlinear systems [8], including stroboscopic maps of physically realizable time-periodic incompressible flows in closed domains [9]. Specifically, we consider the map f1(x) corresponding to the value s = 1, whose phase space is characterized by a main chaotic region, henceforth referred to as the Pesin set P, that invades the entire torus with the exception of two small elliptic islands centered around the period-two elliptic orbit constituted by the two points (1/4,1/2) and (3/4,1/2) (see Fig. 1(A)). Fig. 1(B) shows the structure of the unstable manifold, Wu0 , of the origin (an hyperbolic fixed point for the map f1(x)), computed by tracking for n = 7 periods a small segment aligned with the local unstable direction at the point. Likewise those associated with hyperbolic toral homeomorphism, the manifold Wu0 is dense in the chaotic region, which, however, in the present case does not coincide with the entire manifold. On the other hand, the geometry of Wu0 is different and remarkably more complicated than that of invariant
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Fig. 1. Panel (A) Poincare´ section of a portion (N = 105 iterations) of a typical hyperbolic trajectory associated with the map f1(x). The two small regions devoid of points are quasiperiodic islands surrounding the period-two elliptic points (1/4,1/2) and (3/4,1/2). Panel (B) structure of the unstable manifold, Wu0 , of the origin.
manifolds of hyperbolic toral automorphism, (or of any Anosov toral diffeomorphisms smoothly conjugated to an Anosov toral automorphism, for that matter [10]) this difference being precisely the recursive folding property. Let us first define this property in the general context of a continuous transformation h of a smooth two-dimensional (orientable) manifold M. Let W be an h-invariant manifold (be it a global stable or unstable manifold) that is dense in a chaotic region P 2 M. Let an orientation along W be established (e.g. by introducing the natural parametrization in terms of the arc length s from an arbitrary point x0 2 W and by considering increasing values of s). By the fact that W is dense in P, it will intersect with the boundary oBe of any e-ball embedded in the chaotic region at infinitely many points xl 2 oBe. At each point xl, it is well defined whether, following the assigned orientation along the curve, the crossing occurs while entering or leaving Be. For the map to be recursively folding, we ask that for each crossing point xl where W enters Be there is a crossing point xm arbitrarily close to it where W leaves Be (see Fig. 2). As a confirmation that the map f1 is recursively folding, let us consider the intersections of Wu0 with a reference line, say the horizontal segment {0 6 x < 1, y = 1/4}, which represents a circumference in the torus topology. Fig. 3 shows the signed intersections of Wu0 with this circumference. The sign associated with the intersection is taken as positive or negative according to whether the crossing of the reference line along Wu0 occurs upwards or downwards (i.e. while increasing or decreasing the value of the y-coordinate), respectively. As can be noted, there are opposedly signed intersections arbitrarily close to each other. The recursive folding property should not be confused with the occurrence of simple ‘‘macroscopic’’ folding of the invariant manifolds, which, clearly, can occur even in uniformly hyperbolic systems. As an illustration, Fig. 4 shows the structure and the signed intersections of the unstable manifold at the origin for the uniformly hyperbolic diffeomorphism g(x), constructed by using the SM f1(x) as a change of coordinates of an Anosov toral homeomorphism 1 1 a(x) = a(x, y) = (x + y, x + 2y) [mod 1], i.e. gðxÞ ¼ f 1 1 a f 1 ðxÞ ¼ f 1 ða½f 1 ðxÞÞ, f 1 being the inverse of the map f1. This example, that is much less artificial than as it may appear at first glance [20], clearly indicates that recursive folding does not occur in a uniformly hyperbolic system on the two-torus. Thus, the presence of recursive folding must be
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Fig. 2. Sketch of the recursively folding property of a global invariant manifold W. An oriented arc of W intersects any e-ball embedded in the chaotic region at nearby points with opposite orientations.
Fig. 3. Signed intersections of a continuous arc of the unstable manifold Wu0 at the origin with the reference line {(x, y) j 0 6 x < 1, y = 1/4}. Intersection points are represented as (xint, ±1) where xint is the intersection abscissa and the sign is chosen as positive or negative according to whether the crossing occurs while increasing or decreasing the y coordinate, respectively. Consecutive points are joined by a dotted line to enhance visualization of the high frequency oscillations between ±1, which indicate that the transformation f1 is recursively folding.
strictly related to that of nonhyperbolic trajectories. Specifically, parabolic points that lay at the boundary of the Pesin set appear as the most natural candidates for analyzing the influence of regular orbits on the geometric invariant structure of the chaotic region. Let us consider the parabolic fixed point xp = (1/4,0) of the SM f1(x). The Poincare´ section depicted in Fig. 1(A) suggests that xp possess vanishing distance from the Pesin set. At xp, the Jacobian matrix Dxjxp 1 1 given by Dfjxp ¼ possesses a double unit eigenvalue to which corresponds the eigenspace spanned by 0 1 ex = (1, 0). Let us analyze the dynamics in a neighborhood of xp by considering the evolute of a small segment Ie = {(x, y)jjx 1/4j < e, y = 0} centered at xp and aligned along the (unique) invariant direction. Fig. 5(A) and (B) shows the local structure of the images of such segment up to ten iterates under the forward and inverse mappings, f1 and f 1 1 , respectively. In both cases, the qualitative behavior of the segment images is the same, namely the segment folds onto itself at the parabolic point and collapses onto an invariant curve (a curve Cxp for forward iterations of the map, and a curve cxp for backward iterations). The global outlook of these two invariant curves is indistinguishable from that of any global unstable manifold associate with hyperbolic points (not shown in the interest of brevity). Also, n distances along Cxp and cxp are shrunk to zero by f n 1 and f 1 , respectively. Thus, the curves Cxp and cxp fulfill all the requirements that qualify global invariant manifolds. There is, however, one important difference that distinguishes these manifolds from those associated with hyperbolic points, namely the fact that they originate at the point rather than running through it. Because of this peculiar property, we refer to these invariant curves as one-sided stable and
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Fig. 4. Panel (A): Unstable manifold of the origin associated with the transformation gðxÞ ¼ f 1 1 a f 1 ðxÞ. By construction, this curve is topologically conjugate to a straight line that wraps densely around the torus. Panel (B): Signed intersection of the unstable manifold with the reference line {0 6 x < 1, y = 1/4}. The transformation g(x) is not recursively folding.
unstable manifolds associated with the parabolic point xp (see Fig. 5(C)). It is also worth noting that, unlike the case of hyperbolic periodic points, the curvature of Cxp and cxp is, in a dynamical sense, undefined. Specifically, while the local curvature of a generic filament through a hyperbolic periodic point xh converges to the local curvature of the unstable (stable) manifold of xh for positive (negative) times (see, e.g. [12]), the local curvature of a filament through xp diverges to infinity. Since the local curvature of the invariant stable and unstable manifold is well defined along any point of hyperbolic trajectories, this peculiar phenomenon suggests that the local curvature of invariant stable and unstable manifolds at hyperbolic points e-close to xp can assume arbitrarily large values. This issue is analyzed in the next section. To sum up, as it concerns the existence of stable and unstable manifolds, the parabolic point xp behaves ‘‘almost’’ as a generic hyperbolic point, the difference being the feature that invariant manifolds are one-sided in the meaning discussed above.
3. Lobes, recursive folding and singular curvature of invariant manifolds The next question to be answered is then how the peculiar geometric structure of invariant manifolds at the parabolic points influences the structure of invariant manifolds in points of the Pesin set that are close to xp, and how this local geometric picture is related to the recursive folding property. Fig. 6 shows a zoom-in of the local structure of the stable and unstable manifolds of the origin, Ws0 and Wu0 , in a small rectangular domain about xp (for visualization purposes, a different chart is used to represent this interval of the torus, obtained by translating of 1 units the y-coordinate of points falling in the lower part of the domain). A first straightforward observation is that unstable manifolds undergo a bend that encloses the parabolic point and the local one-sided manifolds. The most important feature, however, arises from the analysis of the mutual structure of the manifolds. Specifically, it is possible to find two continuous arcs, ‘s 2 Ws0 , and ‘u 2 Wu0 , with the following properties (see Fig. 7):
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Fig. 5. Dynamics of a horizontal segment, Ie, through the parabolic point xp of the map f1(x) under forward (Panel (A)) and backward (Panel (B)) iterations of f1(x). The images of Ie fold around the point developing an ever increasing curvature at the point xp until both branches of the folded curve collapse onto the invariant one-sided manifold. Panel (C): Tangency between the one-sided stable manifold cxp (upper curve), and unstable manifold Cxp (lower curve) at the parabolic point xp.
1. ‘s and ‘u intersect only at two points, say x and y. 2. ‘s [ ‘u are the boundary of a closed domain say L. The region L is referred to as a lobe [13]. With respect to the definition reported in [13], we make one further assumption on the lobe L, namely we suppose that L contains a dense set of hyperbolic trajectories. This assumption, suggested by the structure of the Poincare´ section depicted in Fig. 1(A), is used below to show that the mere existence of a lobe such as L has important implications as it regards the existence of geometric singularities associated with invariant manifolds of hyperbolic points.
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Fig. 6. Structure of Ws0 (Panel (A)), and Wu0 (Panel (B)) associated with the transformation f1 near the parabolic point xp. Both manifold undergo a bend that encloses xp.
s
Fig. 7. Schematic diagram explaining the construction of a lobe L from the arcs ‘s (dashed line connecting x and y), ‘~ (dashed line s connecting x* and y*) and ‘u (continuous line). The fact that the arc ‘~ contracts exponentially as the image of x* visits densely the Pesin region provides the connection between the local structure of invariant manifolds and the recursive folding property.
First, let us show that recursive folding follows as direct consequence of properties 1 and 2 defined above. In fact, suppose these properties hold true. On Wu0 consider a point, say x*, that is arbitrarily close to x and whose forward s orbit is dense in the Pesin set [21] (see Fig. 7). Consider the arc ‘~ originating at x* and intersecting Wu0 at a point s u * ~ y close to y. Fix an orientation along the arc ‘ and ‘ , e.g. by establishing a clockwise orientation on the boundary s oL. Consider the unit tangent vectors tu(x*), tu(y*) (ts(x*), ts(y*)) to ‘u ð‘~ Þ at the points x*, y* oriented according to the positive direction. From the geometric picture described above, it readily follows that vector products tu(x*) ^ ts(x*) and tu(y*) ^ ts(y*) are of opposite sign. By the fact that f1 is smooth together with its inverse f 1 1 , this property is pres served under the action of the map. But as n ! +1, the arc ‘~ shrinks to vanishing length, and its endpoints collapse onto a dense orbit. Thus, the arcs of Wu0 that run through the intersection points fn(x*) and fn(y*), will eventually visit any e-ball embedded in the Pesin set as they become arbitrarily close to each other, while still being opposedly oriented. Thus, the manifold Wu0 must be recursively folding.
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It is worth stressing out that lobes cannot exist in uniformly hyperbolic diffeomorphisms of the two-torus, which, on the other hand, do not possess nonhyperbolic points by definition. A natural question arises as to whether or not the presence of isolated parabolic points in a chaotic region of a two-dimensional toral diffeomorphism (or, possibly, of a surface diffeomorphism) always implies the occurrence of lobes and viceversa. While we cannot provide a general argument to prove (or disprove) this statement, we next connect the presence of lobes with singularities of local curvature associated with stable and unstable manifold of hyperbolic points. In order to see this, consider again Fig. 7 and let the lobe L be formed by ‘u and ‘s, i.e. L is the closed domain delimited by ‘u and ‘s containing xp (in the argument that s follows it is immaterial whether one chooses ‘s or ‘~ as delimiting the lobe). In L, let us define an unit vector field e(x) = (ex(x), ey(x)) (ke(x)k = 1) satisfying the following conditions: (i) e(x) be smooth in L, (ii) e(x)jx2‘u = tu(x) for x 2 ‘u , i.e. on ‘u the vector e(x) coincides with the unit tangent vector associated with the assigned orientation established on oL, (iii) e(x) be transverse to the local stable direction Esx at all the points x 2 P \ L. By Stokes theorem applied to the field e(x) in L, it follows that Z Z Z Z eðxÞ t ds ¼ eðxÞ t ds þ eðxÞ t ds ¼ curl2 ðeðxÞÞ dx dy; ð2Þ oL
‘u
‘s
L
where oL = ‘s [ ‘u, curl2(e(x)) = oey/ox oex/oy, and where t(x) is the unit tangent vector on oL (which is defined almost everywhere, i.e. everywhere but at the points x and y). By the fact that t(x) = e(x) for x 2 ‘u, one immediately R derives that ‘u eðxÞ t ds ¼ Lengthð‘u Þ. Let us now address what are the consequences of Eq. (2) if one considers for~ ¼ f n1 ðxÞ with ward iterates Ln ¼ f n1 ðLÞ of the lobe L. The vector field e(x) at x maps to the vector field 1ð~ xÞ at x n ~ 2 Ln a new unit vector field 1ð~ xÞ ¼ Df 1 jx eðxÞ. By renormalizing vector lengths, one can define at the image points x ~, exen ð~ xÞ ¼ 1ð~ xÞ=k1ð~ xÞk. This unit vector field satisfies, with respect to the mapped lobe Ln and the new coordinates x actly the same properties (i),-(ii), and (iii) that were verified by e(x) in L. Also, for large n; en ð~ xÞ is aligned with the local ~ 2 Ln that belong to the Pesin set. Thus, if x ~ 2 P; en ð~ unstable direction at all the points x xÞ is almost tangent to the ~. Applying the Stokes theorem again in this new situation, we obtain unstable manifold through x Z Z curl2 ðeð~ xÞÞ d~x d~y ¼ Lengthð‘un Þ þ en ð~ xÞ t d~s; ð3Þ Ln
‘sn
where ~s denotes the arc length along oLn and t represents a tangent vector along ‘sn . At large n, the contribution of the line integral along ‘sn at the r.h.s. of Eq. (3) becomes immaterial, since the arc ‘sn shrinks exponentially fast to vanishing length. Therefore, the value of the l.h.s. of Eq. (3) approaches the length ‘un as n increases. Since the measure (i.e. the area) of the lobe Ln is constant, this implies that fixed a value j arbitrarily large, there exists an integer m such that for n > m one can find points of Ln where the absolute value of the integrand curl2 ðeð~ xÞÞ results greater than j. If the Pesin set P is dense in L and therefore in Ln (as the Poincare` section of Fig. 1(A) suggests), the above result implies that one can find hyperbolic points where the value jcurl2(e(x))j is arbitrarily large. But at these points, for n large enough, the field of unit vectors en ð~ xÞ collapses onto the unstable directions Eux~ , which is tangent to the local unstable manifolds, i.e. u en ð~ xÞ ! e ð~ xÞ, where eu ð~ xÞ 2 Eux~ . Besides, it is straightforward to show (see Appendix A) that the absolute value of u ~ , i.e. jcurl2 ðe ð~ xÞÞj equals the modulus of the local curvature ju ðxÞ ¼ kku ð~ xÞk of the unstable manifolds at the point x jcurl2 ðeu ð~ xÞÞj ¼ keu ð~ xÞ reun ð~ xÞk ¼ kku ð~ xÞk. Thus, the above argument allows us to conclude that the presence of a lobe such as that depicted in Fig. 7, formed by two continuous arcs, ‘s and ‘u, of a stable and an unstable manifold that intersect only at two points and embrace a dense set of hyperbolic trajectories implies that there are points of the Pesin set at which the modulus of the local curvature associated with the local stable and unstable invariant manifolds assumes arbitrarily large values.
4. An archetypal example of nonuniformly hyperbolic behavior Next, we describe an archetypal model, the toral homeomorphism H, which, as it regards the geometric picture discussed above, can be envisioned as a topological analogous of the transformation f1. A detailed account of all the statistical and geometric properties of H (most of which can be derived analytically) has been presented elsewhere [14]. Here, we focus on the geometric aspects of the transformation that are strictly pertinent to the focus of the present discussion. The homeomorphism HðxÞ ¼ Hðx; yÞ ¼ ðx0 ; y 0 Þ is a piecewise linear, globally continuous transformation of the two-torus defined by 0 x ¼ x þ f ðyÞ ðmod 1Þ; ð4Þ y 0 ¼ x þ y þ f ðyÞ
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where (x, y),(x 0 , y 0 ) 2 [0, 1) · [0, 1) are coordinates on the two-torus, and f(n) : [0, 1] ! [0, 1] is the tent map defined by f(n) = 2n for 0 6 n < 1/2 and f(n) = 2 2n whenever 1/2 6 n < 1. In Ref. [14], it is shown that one can break down the torus into two disjoint subsets say R1 and R2 (see Fig. 8(A)), where the invariant stable and unstable directions, Esx and Esx respectively, are constant. For x 2 R1, these directions, say Es1 and Eu1 , are given by the contracting and 1 2 , i.e. Es1 ¼ spanðs1 Þ and Eu1 ¼ spanðu1 Þ where expanding eigenspace associated with the hyperbolic matrix H 1 ¼ 1 3 pffiffiffi pffiffiffi u1 ¼ ð2; 1 þ 3Þ, and s1 ¼ ð2; 1 3Þ. The invariant directions Es2 and Eu2 at the points x 2 R2 are given by the images 1 2 of the eigenspaces associated with points of R1 through the matrix H 0 ¼ , i.e. Es2 ¼ spanðH 0 s1 Þ, and 1 1 Eu2 ¼ spanðH 0 u1 Þ. Fig. 8(B) and (C) shows the structure of the stable and unstable manifold of the origin 0, Ws0 and Wu0 , respectively. These curves are globally continuous, piecewise-smooth, H -invariant, and everywhere dense on the torus. They are nonsmooth only at the boundary oR2 of the set R2. To the focus of the present discussion, an interesting geometric phenomenon, similar to that observed in the analysis of the map f1, occurs at the (fixed) points xp,1 = (1, 1/2) and xp,2 = (1, 1). Fig. 8(D) shows a zoom-in of the region near the point xp,1 = (1, 1/2) (a local chart is used in the figure for rendering the continuity of the structure). The invariant manifolds, Wu0 (continuous line) and Ws0 (dashed line), undergo a ‘‘U-turn’’ around the fixed point in question, thus making it possible to identify a lobe (in fact, a family of lobes) similar to that associated with the map f1. Clearly, in this case, the analogous of the arc ‘u is no longer smooth (it intersects the boundary of R). Also, the curvature of invariant manifolds is not defined at the point xp,1. Yet, the essence of the geometric peculiarity that was observed in the SM can still be appreciated here, e.g. by identifying as a ‘‘generalized’’ local curvature in the neighborhood of the periodic point the ‘‘width of the U-turn’’, i.e. the gap separating the two parallel arms of the arc of Wu0 as it goes around xp,1. With this identification, it is clear that the generalized curvature assumes arbitrarily high values in that one can find lobes arbitrarily small that still contain xp,1. The identification of the lobe also indicates the recursive folding property
Fig. 8. Panel (A): Decomposition of the torus into two sets R1 (grey shaded area) and R2 (white area) where the expanding and dilating directions are constant (see main text for details). Panel (B) and (C): Arcs of the unstable and stable manifold of the origin for the map H, respectively. Panel (D): Zoom-in of the neighborhood of the point xP,1 = (1, 1/2) (a local chart is used for rendering continuity of the manifolds). The stable and unstable manifolds are represented as continuous and dashed line, respectively. Note that one can define quadrangular lobes.
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of H. Clearly, the lack of global smoothness of the model (in particular the fact that H is not differentiable at xp,1 and xp,2 makes it impossible to push the analogy further. For instance, while the ‘‘one-sided’’ character of the stable and unstable manifolds at xp,1 and xp,2 still holds true for the transformation H, their local tangency does not (the curves are not even differentiable at these points). However, the analysis [14] of a class of smooth maps that are e-close to H (in the C0 topology) shows that the geometric structure of invariant manifolds in the vicinity of the two points xp,1, xp,2 (which are unchanged by the smooth perturbation) is actually completely analogous to that of the transformation f1 in the neighborhood of the parabolic periodic point xp (or of any other isolated parabolic point at the boundary of the Pesin set, for that matter). Thus, as it regards the focus of the present article, the availability of an archetype such as H (and its smooth perturbations) is important in that it provides a concrete example of nonuniformly hyperbolic system where chaotic behavior is ‘‘almost sure’’, i.e. the Lebesgue measure of the Pesin set coincides with that of the entire manifold. This suggests that the escape from uniformly hyperbolic behavior must be characterized in a topological rather than a measure-theoretical framework in that the invariant geometric features associated with the hyperbolic (Pesin) region can be significantly affected by the mere presence of a nonempty set of nonhyperbolic points, even when this set possesses zero Lebesgue measure. Our standpoint is that geometric singularities can occur whenever the Pesin region is not compact (roughly speaking, it has accumulation points that do not belong to the set). For instance, the lack of compactness makes it possible to reconciliate the fact that curvature of invariant manifolds is well defined and continuous at any point of the Pesin region with the result that one can find a sequence of lobes such that the average value of curvature modulus kkuk over the lobe region diverges. In fact, were the hyperbolic set compact and k(x) continuous with x 2 P, then, by Weiertrass theorem, the modulus of curvature would admit absolute maximum, and geometric singular behavior could not occur.
5. Concluding remarks: Physical significance of recursive folding and curvature singularities The main focus of this article was to show that the presence of a lobe (as defined in Section 3) containing a dense set of hyperbolic orbits implies certain peculiar geometric features of invariant manifolds associated with hyperbolic points, namely the existence of points where the local curvature assume arbitrarily high values, and the recursive folding property. We also pointed out that these features cannot occur in a uniformly hyperbolic system on the two-torus T2 in that, by the results of Franks and Manning [11], every Anosov toral diffeomorphism is topologically conjugate to a linear hyperbolic system. This implies that unstable manifolds of a uniformly hyperbolic system on T2 are topologically conjugate to straight lines (the diffeomorphism g(x) discussed in Section 3 provides an illustration of this property). We remark that the identification of the lobe associated with the map f1(x) was originally motivated by the analysis of the invariant manifold structure near the parabolic fixed point xp = (1/4,0). This raises the question whether any lobe must necessarily contain nonhyperbolic points (and, in particular, parabolic points). In the case where the mixing space is the two-torus, the answer to this question appears to be affirmative, since the arguments developed in this article prove that the existence of a lobe implies the existence of nonhyperbolic points, even though we did not strictly prove that some of these nonhyperbolic points must necessary fall inside the lobe itself. Besides, generalizing this statement to the case of a generic diffeomorphism of a two-dimensional domain (think, e.g., of the stroboscopic map of a time-periodic incompressible flow in a closed cavity) may be nontrivial. Yet, results of numerical simulations conducted over a wealth of two-dimensional systems defined in closed two-dimensional domains with boundaries show that invariant manifolds possess the typical sharply bent structure that results from recursive folding (see, e.g. [15]). This observation supports the conjecture that a bounded simply connected domain of R2 cannot support an Anosov diffeomorphism. In turn, this statements suggest that the shortcoming of the Anosov paradigm as a model of physical systems (and of physically interesting models such as the SM family) lies in a misrepresentation of the folding action of this type of transformations. In fact, if one considers the simplest model of Anosov systems, namely the class of hyperbolic toral automorphisms, it is easy to realize that the curvature necessary to embed a curve whose length increases exponentially (e.g. a piece of unstable manifold) into a bounded two-dimensional manifold is provided by the folded structure of the phase space (the two-torus) rather than by the actual folding action of the map. Besides, other popular archetypes of mixing such as the Smales horseshoe [16] and the Bakers maps [17] are characterized by analogous shortcomings as it regards modeling of folding. Specifically, in the horseshoe map folding takes place outside of the ‘‘region of interest’’ (e.g. a rectangle), whereas in the Bakers map (be it the original archetype or any generalization characterized by multiple stretching exponents) folding is modeled by an unphysical ‘‘cutting and staking’’ procedure. From the physical standpoint, curvature dynamics and recursive folding can have important effects on processes that are influenced by chaotic advection. For instance, it has been observed that bubble and droplet break up and coalescence in physically realizable chaotic flows can be influenced by the local curvature of invariant manifolds [18]. As it
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regards recursive folding, it is sensible to expect that this property be strictly related to the phenomenon of cancellation associated with advecting–diffusing magnetic fields in the fast dynamo approximation [19]. In summary, we think that a complete understanding of mixing in two-dimensional systems cannot do away with an attentive investigation of the physical implications of nonuniform hyperbolicity. A first step in this direction could be that of establishing, in the general context of area-preserving diffeomorphisms of a bounded two-dimensional domain onto itself, whether the occurrence of one of the specific features analyzed in this work—existence of lobes, recursive folding, existence of parabolic points, singular behavior of local curvature of invariant manifolds—implies all the others.
Appendix A. Local curvature of a unit modulus vector field In this Appendix, we define the pointwise curvature vector field, k(x) = (k1(x), k2(x)) associated with a unit modulus vector field e(x) = (e1(x), e2(x)) where x = (x1, x2) belongs to a region D of R2 . We assume that e(x) is C2 ðDÞ. We want to show that the modulus of k(x) is numerically equal to the absolute value of curl2(e(x)) = oe2/ox1 oe1/ox2. Let us define the autonomous dynamical system dx/ds = e(x), whose trajectories are curves everywhere tangent to e(x) param~ as the local ‘‘curvature vector’’ associated with the trajectory etrized in terms of arc length s. We define kð~ xÞ at x ~. Let x(s) = (x1(s), x2(s)) be the parametric representation of this trajectory, with xð~sÞ ¼ x ~. The local curvature through x ~ is defined as kð~ kð~ xÞ at the point x xÞ ¼ d2 xðsÞ=ds2 js¼~s . In fact, by this definition, one can readily see that curvature is a skew-symmetric second-order tensor (thus specified by three independent components) which can be conveniently represented at any given point by the vector k, whose modulus represents the inverse radius of the osculating circle to the curve at the given point, and whose direction (normal to the tangent vector) is oriented towards the center of this circle. We use quotation marks for addressing k as a vector to underline that k is only a handy representation of a tensor quantity. As, by definition, dx/ds = e(x), it results k = de(x(s))/ds = [e(x) Æ $]e(x). Thus, ki = ejoei/oxj, where i, j = 1, 2 and a repeated index mean summation over that index. In order to show that kkk = jcurl2(e(x))j = joe2/ox1 oe1/ox2j, let us represent the unit vector field e(x) as e(x) = (cosh(x), sinh(x)) where h(x) = h(x1, x2) 2 [0, 2p) is the angle through which e(x) must rotate clockwise to become aligned with the positive direction of the x1-axis. Let us denote with hi = oh/ox1. With this notation, k is given by ! k1 sin h cos h sin hh1 sin2 hh2 ¼ ½cos hh ¼ þ sin hh . ðA:1Þ 1 2 cos h k2 cos2 hh1 þ sin h cos hh2 On the other hand, jcurl2 (e(x))j = joe2/ox1 oe1/ox2j = jcos h h1 + sin h h2j which is equal the factor in between square brackets in Eq. (A.1). Since this factor multiplies a unit modulus vector, it results kk(x)k = jcurl2(e(x))j.
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Wiggins S. Chaotic transport in dynamical systems. New York: Springer-Verlag; 1992. Cerbelli S, Giona M. J Nonlinear Sci, in press. Giona M, Adrover A, Muzzio FJ, Cerbelli S, Alvarez MM. Physica D 1999;132:298. Smales horseshoe (see [3]) provides a classical example of a dynamical system possessing an hyperbolic attractor that has the structure of a Cantor set. In its original formulation the action of the map restricted to the region of interest is both discontinuous and dissipative. Finn JM, Ott E. Phys Rev Lett 1988;60:760; Finn JM, Ott E. Phys Fluids 1988;31:2992. Tjahjadi M, Ottino JM. J Fluid Mech 1991;232:191. Childress S, Gilbert AD. Stretch, twist, fold: the fast dynamo. New York: Springer-Verlag; 1995. The fast dynamo problem studies what type of kinematic fluid motion can induce amplification of an advected magnetic field seed. The gradient-erasing action of diffusion is greatly favoured whenever the advected magnetic field possesses a structure with rapid changes of sign in space (i.e. whenever the exist nearby zones where the magnetic vectors are parallel but opposedly oriented). In fact, the class of systems that are topologically conjugate with a linear hyperbolic system (i.e. a hyperbolic toral automorphism) on the two-torus coincides with the class of uniformly hyperbolic systems defined on this manifold [11]. In other words, any Anosov diffeomorphism on the two-torus is topologically conjugate to a hyperbolic toral automorphism. The existence of a point such as x* follows from the topological transitivity of the dynamics in the Pesin set. In fact, consider a trajectory O ¼ fxj gi2N that is dense in P and a point xl 2 O GO that is arbitrarily close to x. Let Wsxl;loc be the local stable manifold through xl. This local manifold will intersect ‘u at a point x* e-close to x. Since distances on Wsxp;loc shrink to zero exponentially fast, the forward orbit of x* will collapse onto that of xl. As a consequence, the forward trajectory of x* is also dense.