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Inequivalent topologies of chaos in simple equations
Chaos, Solitons and Fractals 28 (2006) 337–360 www.elsevier.com/locate/chaos
Inequivalent topologies of chaos in simple equations Christophe Letellier a
a,*
, Elise Roulin a, Otto E. Ro¨ssler
b
CORIA UMR 6614, Universite´ de Rouen, Av. de l’Universite´, BP 12, F-76801 Saint-Etienne du Rouvray Cedex, France b Division of Theoretical Chemistry, University of Tu¨bingen, D-72076 Tu¨bingen, Germany Accepted 6 May 2005
Abstract In the 1970, one of us introduced a few simple sets of ordinary differential equations as examples showing different types of chaos. Most of them are now more or less forgotten with the exception of the so-called Ro¨ssler system published in [Ro¨ssler OE. An equation for continuous chaos. Phys Lett A 1976;57(5):397–8]. In the present paper, we review most of the original systems and classify them using the tools of modern topological analysis, that is, using the templates and the bounding tori recently introduced by Tsankov and Gilmore in [Tsankov TD, Gilmore R. Strange attractors are classified by bounding tori. Phys Rev Lett 2003;91(13):134104]. Thus, examples of inequivalent topologies of chaotic attractors are provided in modern spirit. Ó 2005 Elsevier Ltd. All rights reserved.
The menagerie of the simplest surprising flows in three-dimensional space is still a botanical task waiting to be completed. Even though these prototype flowers have neither smell nor color, one can, for example, listen to them while watching them do their thing. In the following, only their anatomical features will be presented in a new, more complete classification.
1. Introduction Although all the necessary conditions on the algebraic structure of ordinary differential equations which ensure that a chaotic solution is observed in a given system are still lacking, there are numerous examples of different sets of these equations which have a chaotic solution. Nevertheless, at least in three-dimensional phase spaces, there are not so various types of chaos. Such a limited number of different types of chaos results from the fact that, in three-dimensional phase spaces, there are quite strong topological constraints—the case of higher-dimensional phase spaces being still more or less an unattacked problem. It is therefore not surprising that a topological analysis using templates [1] and the recently introduced bounding tori [2,3] has been developed to characterize—and classify—chaotic attractors embedded in three-dimensional spaces. Nevertheless, a complete classification of all possible types of chaotic attractors with their corresponding topological properties is still lacking. From our knowledge, the single attempt—using the properties of the first-return maps—was *
Corresponding author. E-mail address: [email protected] (C. Letellier).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.036
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proposed by one of us in the mid of 70s [4]. The aim of the present paper is thus to address this problem using the recent advances in the topological analysis, one of our main attention being to provide an explicit set of ordinary differential equations with a chaotic attractor for each type of chaos identified. The subsequent part of this paper is organized as follows. Section 2 provides a brief review of the topological concepts—mainly templates, linking numbers and bounding tori—used to characterize the topology of chaotic attractors. Section 3 is devoted to chaotic attractors associated with unimodal maps—the simplest types of chaos—and bounded by genus-1 tori. In Section 4 are discussed different examples of combinations of the previous types to provide multimodal chaotic attractors still bounded by genus-1 tori. A few examples of chaotic attractors bounded by genus-3 tori are presented in Section 5. Section 6 gives a conclusion.
2. Templates and bounding tori 2.1. Templates Chaotic attractors can be classified by branched manifolds, also called templates or knot-holders. A template is a branched manifold which can encode the topological properties of chaotic attractor. The structure of chaotic attractors is mainly based on the stretching and folding—or cutting as we will discuss later—mechanisms which are the relevant ingredients for generating chaos. Roughly speaking, the stretching induces the sensitivity to initial conditions and the folding or cutting produces the mixing between trajectories. These two ingredients are required to have solutions with an underlying deterministic—therefore well-defined—structure but which are long-term unpredictable. A typical template with these two ingredients was probably drawn for the first time in 1976 [5] as a ‘‘chaotic blender’’. Slightly later, the same author proposed another view of such a template as a ‘‘paper model’’ or ‘‘origami’’ in [6]. The latter version, close to those now used, is shown in Fig. 1a added with a more schematic view (Fig. 1b) which can be found in works by Gilmore and co-workers [7,8] or Letellier et al. [9]. Both are topologically equivalent, that is, allow one to predict the same topological invariants between pairs of periodic orbits, such as linking numbers or relative rotation rates [1]. We should note that Birman and Williams [10,11] were the first to consider templates as ‘‘knot-holders’’, that is, as a branched manifold in which periodic orbits can be drawn with their relative organization. Such an approach was also followed by Holmes and Williams [12] and then developed by Gilmore and his co-workers [7,8]. An important point is that the first-return map to a Poincare´ section is not sufficient to determine all the properties of the associated template. In particular, the template shown in Fig. 1 can be associated both with the unimodal map with a differentiable maximum—the ‘‘walkin-stick’’ map in the terminology of [13]—(Fig. 2a) and with the ‘‘Lorenz map’’ with its characteristic cusp (Fig. 2b) [14]. Although Ro¨ssler classified these two types of map long ago [15] as ‘‘bent’’ (Fig. 1a) and ‘‘cut’’ maps (Fig. 1b), it is only recently that these maps were associated with the folding and cutting mechanisms, respectively [16]. On the other hand, a ‘‘cut’’ map requires a singularity in the flow [16]. While Ro¨ssler proposed a significant collection of different types of chaos with very suggestive names like ‘‘spiral’’, ‘‘screw’’ or ‘‘sandwich’’ type of chaos, we will review and clas-
Fig. 1. Two topologically equivalent templates. (a) Template adapted from first one drawn by Ro¨ssler in [5] and (b) a more schematic view. Both encode the two main ingredients for generating chaotic attractors, that is, stretching and folding.
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Fig. 2. Two different unimodal first-return maps which can be associated with the templates shown in Fig. 1.
sify these systems with the more recent terminology. We will also include other systems in order to have a more complete list of examples of chaotic behaviors with inequivalent topologies. Before that, we will need to briefly review the bounding tori, recently introduced by Tsankov and Gilmore [2,3], in order to be able to give a more general classification. 2.2. Bounding tori It is well known that a chaotic attractor can be bounded by a semi-permeable surface defining a domain of the phase space from which a trajectory cannot escape [18]. Such a bounding surface is oriented, that is, any orbit that passes through from the outside to the inside remains trapped inside forever. In general, such a surface is a genus-g surface where g is the number of holes in the boundary. The surface with g = 0 is called a sphere and that with g = 1 is commonly called a torus. Although the flow on the branched manifold has no fixed points in an open neighborhood of the branched manifold, restricted to the surface itself there are fixed point singularities. All are generalized saddles. As a result, the number of singularities is related to the genus g, and the number is 2(g 1) [3]. The flow, restricted to the bounding torusÕ surface, can be put into canonical form. For genus g = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . ., there are 0, 1, 0, 1, 1, 2, 2, 5, 6, . . .inequivalent canonical forms [3]. The canonical form for g = 3 is shown in Fig. 3. Typically, the central hole––with singularities—is associated with a saddle fixed point and the two other holes with focus fixed points (F±). This does not mean that a system with a chaotic attractor bounded by this genus-3 torus has necessarily only three fixed points but that these are only the three fixed points which are surrounded by the flow. The other fixed points are ‘‘outside’’ the outer disk boundary. For instance, the Lorenz attractor (Fig. 4) is bounded by a genus-3 torus as shown in Fig. 3. In three dimensions, a Poincare´ surface is a minimal two-dimensional surface with the property that all points in the attractor intersect this surface transversally an infinite number of times under the flow. A Poincare´ section is not necessarily constituted of a single component. In particular, when a dynamical system has an order-g symmetry, it is useful to introduce g disconnected components in order to properly compute the Poincare´ section [17,19,20]. More generally, the Poincare´ section is disjoint union of one or more nonoverlapping disks [2,3]. In fact, the number of disconnected
Fig. 3. The canonical form for the genus-3 bounding torus consists of an outer disk boundary and three interior holes. The four singularities are confined to the central interior holes. Location of the two components of the Poincare´ section are also shown (0 and 1).
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Fig. 4. The Lorenz attractor is bounded by a genus-3 torus. The central interior hole associated with the saddle fixed point—with its four singularities—has been enlarged for clarity.
components is equal to (g 1). Thus, the Poincare´ surface for a genus-1 bounding torus consists of a single disk that is transverse to the flow. The Poincare´ surface for a canonical genus-g bounding torus consists of the union of g 1 disjoint disks [3]. The locations of the two disks for the genus-3 canonical form has been shown in Fig. 3. Many different templates can be described by the same genus-g canonical form. Each branch line in any of these branched manifolds can be moved so that it is contained in one of the g 1 components of the global Poincare´ section. As a result, any branched manifold enclosed by a genus-g bounding torus has exactly g 1 branch lines (for g > 1). The first-return maps for branched manifolds enclosed by a genus-1 bounding torus are equivalent to maps from the single branch line that exists in the single component of the Poincare´ section that runs back onto itself. That is, in this case the return maps are exactly maps of the interval to itself. In the genus g P 3 case, the first-return maps have g 1 branch lines [16]. The initial conditions on any branch line flow to exactly two other branch lines. The return map is constructed as follows: Each branch line is represented as an interval. These intervals are laid out along a horizontal axis (initial conditions). Each branch line is oriented, from the interior to the exterior of the projection of the bounding torus onto a plane (cf., Fig. 1). As in the genus-1 case, the images are arranged along the vertical axis. To each point on the horizontal axis (consisting of g 1 disjoint oriented segments) there is a unique image. The first-return map of this type has been computed for the Lorenz attractor in Fig. 5. We will see that the most common type of chaos associated with a genus-1 torus has a first-return map with a differentiable extrema that reflect the foldings that take place between adjacent branches. When attractors bounded by a
Fig. 5. First-return map to the Poincare´ section constituted by two disconnected components of the Lorenz system. Parameter values: R = 28, r = 10 and b = 8/3.
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genus-3 tori are considered, the canonical form for such bounding tori (Fig. 3) has a central internal hole with four singularities which surrounds a saddle type singularity. Thus, there exists some point along each branch line—or component of the Poincare´ section—where the initial conditions on one side evolve to one branch line and the initial conditions on the other side evolve to the other branch line. This is a so-called ‘‘cutting point’’. At this point, the return map is discontinuous and, most often, with a slope discontinuity. The cutting occurs in the neighborhood of saddle points or other singularities that deflect the flow in a small neighborhood toward divergent directions [16].
3. Unimodal chaos bounded by a genus-1 torus Chaotic attractors involve, at least, one unimodal map, that is, a first-return map constituted of two monotonic branches separated by a single critical point—differentiable or not. In this case, there exists a universal order for the possible bifurcations under which periodic orbits can be created. We will designate the chaotic behavior characterized by a unimodal map as ‘‘unimodal chaos’’. We will therefore start out by describing the different type of unimodal chaos since multimodal chaos, later discussed, can be viewed as combination of unimodal chaos. 3.1. Unimodal folded chaos A unimodal folded chaos is characterized by a template as shown in Fig. 1 and an everywhere differentiable firstreturn map with a single critical point (Fig. 2a). This is the most common—and simplest—type of chaotic attractor that can be encountered. It always follows a period-doubling cascade as a route to chaos. The first system generating unimodal folded chaos was proposed in the beginning of 76 by Ro¨ssler [5]. This is a set of three ordinary differential equations: 8 ðk 3 y þ k 4 zÞx > > x_ ¼ k 1 þ k 2 x > > < xþK ð1Þ y_ ¼ k 5 x k 6 y > > > k z 10 > : l_z ¼ k 7 x þ k 8 z k 9 z2 z þ K0 This system was invented in order to obtain a simple chemical reaction able to generate chaotic behavior. This research was motivated by a discussion Ro¨ssler had with Art Winfree in 1975. A figure quite similar to Fig. 6a was obtained [5]. Ro¨ssler named this type of chaos ‘‘spiral chaos’’ but, as he later developed in his classification of different types of chaos by using the shape of the first-return map, we will prefer to use the term ‘‘folded chaos’’. The differentiable maximum of the first-return map to a Poincare´ section reveals the folding associated with this chaotic attractor (Fig. 6b).
Fig. 6. Unimodal folded chaos solution to system (1). Parameter values: k1 = 37.8, k2 = 1.4, k3 = 2.8, k4 = 2.8, k5 = 2, k6 = 1, k7 = 8, k8 = 1.84, k9 = 0.0616, k10 = 100, K = 0.05, K 0 = 0.02, l = 1/25; x0 = 7, y0 = 12, z0 = 0.2.
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The nature of the solution found in system (1) obviously depends on the chosen parameter values. We arbitrarily choose to vary one of the parameters, k6, to compute a bifurcation diagram (Fig. 7) showing the period-doubling cascade as a route to chaos. Varying parameter k6 does not allow one to find a different type of chaos before the boundary crisis (k6 0.987) which ejects the trajectory to infinity occurs. A slightly simpler system was proposed few months later by the same author [21]. It reads: 8 ðk 3 y þ k 4 zÞx > > x_ ¼ k 1 þ k 2 x > > < xþK ð2Þ y_ ¼ k 5 x k 6 y > > > k 8z > : z_ ¼ k 7 x z þ K0 Two terms were deleted from system (1). Nevertheless, this system still has rational right-hand-side members which can complexify its analysis. This system has a chaotic attractor shown in Fig. 8. In order to obtain a polynomial system, Ro¨ssler empirically simplified the equations of system (2) to obtain a still simpler system which continues generating a chaotic attractor topologically equivalent to the attractor shown in Fig. 6. This equation has, as a ‘‘model of a model’’, no longer an immediate physicochemical interpretation. The proposed system reads [22]:
Fig. 7. Bifurcation diagram of system (1) in dependence on parameter k6. The other parameters have the same values as in caption of Fig. 6.
Fig. 8. Unimodal folded chaos solution to system (2). Parameter values: k1 = 22, k2 = 2.2, k3 = k4 = 4.4, k5 = 1.2, k6 = 1, k7 = 14, k8 = 140, K = 0.01, K 0 = 0.05. Initial conditions: x0 = 7, y0 = 6, z0 = 0.1.
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Fig. 9. Unimodal folded chaos solution to the Ro¨ssler system (3). Parameter values: a = 0.420, b = 2 and c = 4.
8 > < x_ ¼ ðy þ zÞ y_ ¼ x þ ay > : z_ ¼ b þ zðx cÞ
ð3Þ
which is the so-called Ro¨ssler system. Only a single nonlinear term (xz) remains in this system. Ro¨ssler published a unimodal folded chaotic attractor with the parameter values (a, b, c) = (0.2, 0.2, 5.7). For convenience in the next subsection, other parameter values are preferred [9]. A chaotic attractor solution to system (3) is shown in Fig. 9. Many different systems with a unimodal folded chaos were later published in many different fields. Among others, a unimodal floded chaotic attractor was found by Buchler and Goupil [23] in astrophysics, Decroly and Goldbeter [24,25] in biochemistry, Upadhyay and co-workers [26,27] in population model, and Sprott in his systematic study [28]. 3.2. Inverted unimodal folded chaos During his attempts to simplify the equations for generating chaotic attractors, Ro¨ssler obtained another type of chaotic attractor with the following system [21]: 8 k 3 yx > > < x_ ¼ k 1 þ k 2 x x þ K ð4Þ y_ ¼ k 4 x k 5 y þ k 6 zy 2 > > : z_ ¼ k 7 k 6 zy 2 He obtained a chaotic attractor (Fig. 10a) that he termed ‘‘screw attractor’’. Although the term ‘‘screw’’ was motivated by the shape of the attractor, a first-return map to a Poincare´ section is a differentiable unimodal map (Fig. 10b). The mechanism generating the chaotic attractor is therefore not very different from that for the previous examples. Nevertheless, a careful topological analysis reveals that the attractor is actually characterized by a template with a negative half-turn on the other side as a global torsion (Fig. 11a). This corresponds to what is often called an ‘‘inverted horseshoe’’ since the odd number of half-turns inverts the map. According to our terminology, we will designate such a chaotic attractor as inverted unimodal folded chaos. In the case of system (4), this is an additive global torsion (Fig. 11a) since in the sense that the template can be redrawn with two branches having one and two negative half-turns, respectively, that both branches have one additional half-turn. This is not the case when the global torsion has a different sign than the local torsion (Fig. 11b). As for the ‘‘direct’’ unimodal folded chaos, the associated route to chaos is once more a period-doubling cascade. This can be confirmed with a bifurcation diagram versus the parameter k5 (Fig. 12). As observed with other systems having inverted unimodal folded chaos, the chaotic windows are considerably less developed than for ‘‘direct’’ unimodal folded chaos. The substractive case has been observed, for instance, in SprottÕs systems D, E and L according to the names used in [28]. The chaotic attractor solutions to system D which reads: 8 > < x_ ¼ y y_ ¼ x þ z ð5Þ > : 2 z_ ¼ xz þ ay is shown in Fig. 13.
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Fig. 10. Inverted unimodal folded chaos solution to system (4). Parameter values: k1 = 2, k2 = 0.9, k3 = k4 = 1, k5 = 0.5, k6 = 0.005, k7 = 1.95, K = 0.1. Initial conditions: x0 = 0.25, y0 = 3.1, z0 = 17.
Fig. 11. Two different templates for an inverted unimodal folded chaotic attractor. (a) The global torsion has the same sign as the local torsion and, consequently, this is an ‘‘additive’’ case. (b) The global torsion has an opposite sign to that of the local torsion, and this is therefore a substractive case.
3.3. Unimodal cutting chaos The term ‘‘sandwich chaos’’ was introduced by Ro¨ssler [32] to designate a chaotic attractor characterized by a map constituted of two monotonic branches with a discontinuity between them. Among this type of chaos, a unimodal cutting chaos corresponds to a chaotic attractor with a topology characterized by the template shown in Fig. 1 but with a Lorenz map as a first-return map to a Poincare´ section (Fig. 2b). From our knowledge, no simple set of polynomial equations was identified to generate a chaotic attractor bounded by a genus-1 torus and possessing a Lorenz map.
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Fig. 12. Bifurcation diagram versus k5 for system (4) with the other parameter values as in Fig. 10.
Fig. 13. Inverted folded chaotic attractor solution to SprottÕs system D. Parameter value: a = 4.0.
The first system with rational right members was published by Ro¨ssler [29] as an isothermal abstract reaction system. The systems reads: 8 ðdz þ eÞx > > x_ ¼ ax þ by cxy > > x þ K1 < jxy ð6Þ y_ ¼ f þ gz hy > > > y þ K 2 > : z_ ¼ k þ lxz mz This abstract chemical reaction generates a unimodal cut chaotic attractor as shown in Fig. 14a. A first-return map to a Poincare´ section (Fig. 14b) has the shape of the Lorenz map as expected. The l-value is slightly modified to obtain a Lorenz map without a gap between the two monotonic branches as originally published [29]. As seen in Fig. 14a, the saddle fixed point is not surrounded by the flow. Consequently, there is no internal hole, associated with this saddle point, in the corresponding bounding torus. The attractor is thus bounded by a genus-1 torus. We may say that only half of the saddle is active on the flow, as we will discuss later. This early abstract chemical reaction system is interesting also because it allows a continuous transition between unimodal folded chaos and unimodal cutting chaos. This is shown with a bifurcation diagram versus parameter l (Fig. 15) where the inverse period-doubling cascade is easily identified. The transition between these two types of chaos occurs around l = 5.3. Thus, with 4.795 < l < 5.3, the asymptotic behavior is a unimodal cut chaos and when 5.3 < l < 5.716, it is a unimodal folded chaos which is observed. Such a continuous transition was previously identified already in a 5dimensional laser model [30]. The other known example of unimodal cut chaos is the image of the Lorenz system [31], that is, the system which results from the 2 # 1 coordinate transformation introduced to ‘‘modd out’’ the symmetry properties of the Lorenz system [31]. This transformation reads:
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Fig. 14. Unimodal cut chaotic attractor solution to system (6). Parameter values: a = 33, b = 150, c = 1, d = 3.5, e = 4815, f = 410, g = 0.59, h = 4, j = 2.5, k = 2.5, l = 5.29, m = 750, K1 = 0.01 and K2 = 0.01.
Fig. 15. Bifurcation diagram versus l of the abstract chemical reaction system (6). Other parameter values as in Fig. 14.
u ¼ Reðx þ iyÞ2 ¼ x2 y 2 W ¼ v ¼ Imðx þ iyÞ2 ¼ 2xy w ¼ z
ð7Þ
The image dynamical equations are thus: 8 > < u_ ¼ ðr 1Þu þ ðr RÞv þ vw þ ð1 rÞq v_ ¼ ðR rÞu ðr þ 1Þv uw þ ðR þ rÞq qw ð8Þ > : 1 w_ ¼ bw þ 2 v pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ u2 þ v2 . This image of the Lorenz system generates a chaotic attractor (Fig. 16a) bounded by a genus-1 torus. A first-return map to a Poincare´ section (Fig. 16b) is equivalent to the Lorenz map (Fig. 2b). Its maximum is associated with a singularity in the flow—at the origin of phase space where the flow is not defined—which produces the cutting mechanism. At first sight, the image Lorenz attractor should be bounded by a genus-2 torus as shown in Fig. 17. A blow-up of the flow around the origin of phase space (not shown), reveals that it is a hole with two singularities which should be present. Note that this is in agreement with the fact that the number of singularities is equal to 2(g 1) since with g = 2, only two singularities can be obtained. Consequently, as pointed out by Tsankov and Gilmore [3], a genus-2 bounding torus does not exist, since its internal hole is associated with two singularities that must be removed, thus leading to a genus-1 torus [3].
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Fig. 16. Unimodal cut chaotic attractor solution to the image system (8) of the Lorenz system. Parameter values: R = 28, r = 10 and b = 8/3.
Fig. 17. A genus-2 torus that could be associated with the Lorenz image attractor, but the hole with two singularities surrounding the w-axis can be removed because there is no qualitative change of flow in its neighborhood, thus leading to a genus-1 torus. This is why there is no canonical form for genus-2 tori.
3.4. Unimodal half-inverted cut chaos The second type of ‘‘sandwich’’ chaos is the ‘‘unimodal half-cut chaos’’ thus termed since only one branch of the first-return map is inverted. He thus proposed a set of equations which can be viewed as a modified Lorenz system with a broken symmetry. It reads as follows [32]: 8 > < x_ ¼ x xy z y_ ¼ x2 ay ð9Þ > : z_ ¼ bx cz þ d where the symmetry is broken when d 5 0. An example of a chaotic attractor of the ‘‘sandwich’’ type is shown in shown in Fig. 18a. At first sight, the attractor should be bounded by a genus-3 torus but there is a path which is forbidden (Fig. 19a). Consequently, the two internal holes (at the right part of Fig. 19a) can be merged into a single hole, thus leading to a hole with two singularities (Fig. 19b). This resulting hole can therefore be removed. Half-inverted cut chaos can also be found in the system of [4]: 8 > < x_ ¼ x xy z y_ ¼ x2 ay ð10Þ > : z_ ¼ bx cz þ d with parameter values as a = 0.1, b = 0.08, c = 0.38 and d = 0.0015. Another example was also found by perturbing the Lorenz equations according to: 8 > < x_ ¼ rðy xÞ þ a y_ ¼ xðR zÞ y ð11Þ > : z_ ¼ xy bz
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Fig. 18. Unimodal half-inverted cut chaos solution to system (9). Parameter values a = 0.1, b = 0.09375, c = 0.38 and d = 0.0015.
Fig. 19. Perestroika in a genus-3 torus with a forbidden path leading to a genus-1 torus (see text).
with parameter values a = 2.2, r = 4, R = 80 and b = 8/3. Later, Ro¨ssler proposed an abstract chemical reaction scheme generating a chaotic attractor as close as possible to the Lorenz attractor [29]. The system reads: 8 dzx > > > x_ ¼ ax þ by cxy x þ K > 1 < hxy ð12Þ 2 y_ ¼ e þ fx gy > > > y þ K 2 > : z_ ¼ j þ kx lz In fact, it produces a unimodal half-inverted cut chaos if parameter values are a = c = d = f = 1, b = 2, e = h = 4, g = 0.1, j = 0.5415, k = 0.06, l = 0.33, K1 = K2 = 0.02.
4. Multimodal chaos bounded by a genus-1 torus 4.1. Multimodal ‘‘snail’’ chaos Multimodal chaotic attractors are associated with first-return maps consisting of more than two monotonic branches, that is, with more than one critical point. Consequently, when parameters are varied, sequences of bifurcations can be due to different modes, one mode per critical point. This is what is called ‘‘antimonotonicity’’ [33,34]. Most of the systems previously described can also generate multimodal chaos when parameter values are varied. In his early searches for different type of chaos, Ro¨ssler did not realize that a single system can have different type of chaos. It was only when he varied some parameters of system (3) on his analog computer that he realized that feature. This ‘‘late’’ understanding—only a few months after his first chaotic attractor—explains why he proposed so many different sets of equations. We will start with what we call the multimodal snail chaos which can be found in the Ro¨ssler system (3) when a 2 [0.43295; 0.556] with b = 2 and c = 4, as investigated in [9]. Ro¨ssler identified this multimodal chaos which he called ‘‘screw’’ chaos in [32]. When a > 0.43295, a third branch occurs in the first-return map. Then a fourth branch occurs
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when a > 0.492 and so on, up to more than 47 branches [35] for a = 0.5566334488—just before the boundary crisis which ejects the trajectory to infinity [9]. A typical attractor of the series is shown in Fig. 20 with its first-return map. The term ‘‘snail’’ is chosen in view of a Poincare´ section of the system (Fig. 21) which has the shape of snail shell. The snail shape of the Poincare´ section induces ordered locations for the critical points of the first-return map within the visited interval. For instance, along the y-axis as used for building the first-return map, a scaling law obeys according to [35]: y y n1 dl ¼ lim n ¼ 1.68 0.04 ð13Þ n!1 y nþ1 y n where yn is the y-coordinate of the nth critical point. The convergence toward the scaling number d1 is shown in Fig. 22. A deep difference between ascending critical points Cn (odd n) and descending critical points Cn (even n) is seen in this figure. Another scaling law is associated with the successive a-values that correspond to the occurrence of critical points in the first-return map. This second scaling law reads [35]: an an1 da ¼ lim ¼ 1.70 0.08 ð14Þ n!1 anþ1 an where an is the a-value at which the nth critical point Cn occurs in the first-return map. The scaling number da is equal to the above location scaling number dl within estimated uncertainties. Consequently, the appearance of the critical points
Fig. 20. Multimodal snail chaotic attractor solution to the Ro¨ssler system (3). Parameter values: a = 0.5566334488, b = 2 and c = 4.
Fig. 21. Poincare´ section of the Ro¨ssler system with the shape of a ‘‘snail’’ shell. Parameter values: a = 0.5566334488, b = 2 and c = 4.
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Fig. 22. Convergence toward the scaling number dl versus n, for a = 0.5566334488 in the case where the first-return map presents 47 monotonic branches.
is connected to their location in the first-return map, and vice versa. After the accumulation points of the scaling law, the second fixed point of the Ro¨ssler system, located in the boundary of the attraction basin, collides with the attractor. Due to this crisis, the trajectory diverges to infinity after a transient regime. 4.2. Cover of a folded chaotic attractor Another type of chaos can be observed in the twofold cover of the Ro¨ssler system, that is, in a Ro¨ssler attractor with a rotation symmetry by p around the z-axis, here designated as Rz ðpÞ. This rotation symmetry is obtained as follows. First, we start from a centered Ro¨ssler system through a rigid displacement, that is, the inner fixed point is moved to the origin of a new phase space R3 ðu; v; wÞ. In the translated coordinate system, the equations for this image system become 8 > < u_ ¼ v w v0 w0 v_ ¼ u þ av þ u0 þ av0 ð15Þ > : w_ ¼ b þ wðu þ u0 cÞ þ w0 u þ w0 ðu0 cÞ pffiffiffiffiffiffiffiffiffiffi 2 where u0 ¼ v0 ¼ aw0 ¼ c c2 4ab are the coordinates of the inner fixed point of the original Ro¨ssler system (3). Moreover, when the origin of the coordinates is displaced along the u-axis by a quantity equal to l, the equations for the image system in the translated coordinates are 8 > < u_ ¼ v w ð16Þ v_ ¼ u þ av þ l > : w_ ¼ ~bðu þ lÞ þ wðu ~c þ lÞ where ~b ¼ w0 and ~c ¼ c u0 . Then, applying the inverse of the coordinate transformation (7), we obtain the dynamical equations for the twofold cover of the Ro¨ssler system [31]: 8 1 2 2 > < x_ ¼ 2r2 ½r y þ xð2ay zÞ þ ly 1 2 2 y_ ¼ 2r2 ½r x þ yð2ax þ zÞ þ lx ð17Þ > : 2 2 2 2 ~ z_ ¼ bðx y þ lÞ þ zðx y ~c þ lÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ x2 þ y 2 . When l = 0, that is, when the rotation axis is located at the inner fixed point of the centered Ro¨ssler system, this twofold cover generates a chaotic attractor with two symmetry-related foldings (Fig. 23a). Its first-return map has 22 = 4 branches (Fig. 23b) and can be viewed as a ‘‘double unimodal folded chaos’’. The important point of having n-folded chaos by using the n-fold cover of a unimodal folded chaotic attractor is that there is no global torsion. This transpires from the template of the twofold cover of the Ro¨ssler system (Fig. 24). For instance, this process can be repeated with a fivefold cover of the Ro¨ssler system. In this case, the chaotic attractor has five symmetry-related foldings as shown in Fig. 25a. The first-return map has thus 25 = 32 monotonic branches (Fig. 25b). All maxima have exactly the same ordinate.
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Fig. 23. Twice folded chaotic attractor solutions to the twofold cover (17) of the Ro¨ssler system. Parameter values: a = 0.43294, b = 2.0 and c = 4.0.
Fig. 24. Template of the twofold cover of the Ro¨ssler system.
Fig. 25. Five folded chaotic attractor solutions to the fivefold cover of the Ro¨ssler system. Parameter values: a = 0.43294, b = 2.0 and c = 4.0.
Twice-folded chaos has been observed with an inversion symmetry in the Kremliovsky system [31] and in an integrodifferential equation proposed by Lu et al. [37]. This equation reads as:
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Fig. 26. Twice folded chaos solution to the integro-differential equation (18). Parameter values: a = 0.001, b = 3.8, c = 2.85 and d = 3.
x_ ¼ ax bðjxs þ 1j jxs 1jÞ þ d ðjxs þ d j jxs djÞ
ð18Þ
where xs = x(t s). This equations has the chaotic solution shown in Fig. 26. 4.3. Cover of the snail chaos Using the twofold cover (17) of the Ro¨ssler system with other parameter values, it is also possible to obtain a chaotic attractor corresponding to a twofold cover of the snail type of chaos (Fig. 27a). The first-return to a Poincare´ section is thus constituted by numerous monotonic branches (Fig. 27b). The structure of the map, although still characterized an alternating sequence of increasing and decreasing branches, has a quite complicated structure—from the point of view of critical point locations, for example. In particular, maxima have different ordinates. 4.4. Cover of inverted folded chaos The twofold cover of inverted folded chaos can be observed in the algebraically simplest equivariant jerk system proposed by Malasoma. It consists of three terms including one quadratic nonlinearity. It reads as:
x ¼ a€x þ x_x2 x
ð19Þ
Fig. 27. Twofold cover of the snail chaotic attractor solution to the twofold cover (17) of the Ro¨ssler system. Parameter values: a = 0.556, b = 2, c = 4 and l = 0.
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This system can be rewritten as a set of three ordinary differential equations [36]: 8 > < x_ ¼ y y_ ¼ z > : z_ ¼ az þ xy 2 x
ð20Þ
This simplest equivariant system generates the ‘‘double folded chaotic attractor’’ shown in Fig. 28a. The topology of this chaotic attractor is different than the topology of the two folded chaotic attractor of Fig. 24 since there is one global torsion by one half-turn between each folding (Fig. 29). The global torsion produces a different quantitative arrangement between the monotonic branches in the first-return map (Fig. 28b) which is characteristic of such twofold cover of an inverted folded chaos. Note that the symmetry of this system is an inversion but this has no relevant effect on the shape of the first-return map as we shall see later with the Burke and Shaw system. We can check on a higher order cover that the quantitative organisation of the first-return map is indeed different than for the snail chaos. This is done with the threefold cover of the simplest equivariant system—which is therefore the sixth-cover of the image of system (20). The attractor is shown in Fig. 30a where the six symmetry related foldings are easily identified. Between each of them, there is a global torsion by a half-turn. The first-return map to a Poincare´ section consists of 26 = 64 monotonic branches (Fig. 30b). The regular organization of the branches signifies a departure from the snail chaos (Fig. 20b), the n-fold cover of unimodal folded chaos (Fig. 25b) and from the twofold cover of the snail chaos (Fig. 27b). Another example of a twofold cover of a unimodal inverted folded chaotic attractor can be observed in the Burke and Shaw system [38,20]:
Fig. 28. Twice-inverted folded chaotic attractor solution to the simplest equivariant jerk system (20) just before the attractor suffers a merging crisis. Parameter value: a = 2.027717.
Fig. 29. Templates of two different twofold covers of unimodal inverted chaos.
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Fig. 30. Chaotic attractor with six inverted foldings of the threefold cover of the simplest equivariant jerk system (20). Parameter value: a = 2.027717.
Fig. 31. Chaotic attractor with two inverted foldings in the Burke and Shaw system (21). Parameter values: V = 4.271 and S = 10.0.
8 > < x_ ¼ Sðx þ yÞ y_ ¼ y Sxz > : z_ ¼ Sxy þ V
ð21Þ
where S and V are the parameters. The first-return map to a Poincare´ section (not shown) of the chaotic attractor of Fig. 31 has the same shape as the map to a Poincare´ section of the attractor of the simplest equivariant system (Fig. 28b). The topology of the Burke and Shaw attractor is described by the template shown in Fig. 29b since the attractor is globally invariant under a rotation symmetry Rz ðpÞ. Consequently, the nature of the symmetry is not so important for this type of chaos. 4.5. Cover of the half-inverted cut chaos A twofold cover of the half-inverted cut chaos is now investigated for system (9). The coordinate transformation x ¼ X 2 Z2 ð22Þ W2 ¼ y ¼ Y z ¼ 2XZ is inverted and applied to system (9). A chaotic attractor with two half-inverted cut mechanisms is obtained: Fig. 32a (‘‘laughing girl attractor’’). A first-return map to a Poincare´ section has 22 = 4 branches with slopes of the same sign (Fig. 32b).
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Fig. 32. Twofold cover of the half-inverted cut chaotic attractor. Parameter values: a = 0.1, b = 0.082, c = 0.38, d = 0.0015.
Fig. 33. Twofold cover of the cut chaotic attractor shown in Fig. 14a. Parameter values: a = 0.1, b = 0.082, c = 0.38, d = 0.0015.
4.6. Cover of unimodal cut chaos Using a twofold cover of system (6) proposed by Ro¨ssler and Ortoleva, one can obtain a chaotic attractor with two cutting mechanisms. The chaotic attractor is shown in Fig. 33 together with its first-return map.
5. Chaos bounded by a genus-3 torus When a chaotic attractor is bounded by a genus-3 torus, it has always the configuration shown in Fig. 3 since there is a unique canonical form for a genus-3 bounding torus [3]. Thus, there is always a central internal hole with singularities surrounding a saddle fixed point. There is therefore always a cutting mechanism involved in such an attractor. 5.1. Pure cutting mechanism The simplest attractor bounded by a genus-3 torus is the Lorenz attractor shown in Fig. 4. Since the Poincare´ section consists of (g 1) disjoint components when g > 1, this means that for all systems discussed in this section, two components are required for computing the Poincare´ section. From the canonical form of the genus-3 torus shown in Fig. 3, the flow has at least two possibilities when it comes from one component: (i) to return to the same component or (ii) to
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go to the other component. One monotonic branch is associated with each possibility. There are, therefore, at least four monotonic branches in the first-return map shown in Fig. 5. There are in fact two conjugated mechanisms—two cuttings—produced by a single fixed point of the saddle type. An important point to see is that the saddle, surrounded by the flow, needs to act through two different modes—two singularities per mode. Otherwise, as in the case of the abstract chemical reaction (6), only one half-side of the saddle is active in the flow. In the case of the image of the Lorenz system (8), the flow is strictly speaking not defined where the cutting occurs, but a blow-up of the flow in this area clearly reveals that there is only a ‘‘half-saddle’’ which is active, leading to a hole with two singularities which can therefore be removed. 5.2. Cutting combined with unimodal folding The system which generates the simplest example of a cutting mechanism associated with two foldings is the twofold cover of the Ro¨ssler system when the rotation axis crosses the image attractor [31], for example, when l = 2.083. In this case, the chaotic attractor is bounded by a genus-3 torus as shown in Fig. 34a. The first-return map to a Poincare´ section reveals that the cutting acts in the attractor at the same place as the folding mechanism does since the discontinuity of the first-return map occurs where the slope of the branch is nearly zero (Fig. 34b). 5.3. Cutting combined with several multimodal foldings In his attempt to list the different types of chaos, Ro¨ssler proposed a system which has the symmetry of KhaikinÕs ‘‘universal’’ circuit, that is, there are two symmetric foci related by fast jumps. The equations for this type of chaos are [4]: 8 x_ ¼ ax yð1 x2 Þ > > < ð23Þ y_ ¼ lðy þ 0.3x 2zÞ > > : z_ ¼ lðx þ 2y 0.5zÞ Two plane projections of the chaotic attractor are shown in Fig. 35a and b. The structure of the attractor is quite complex. A first-return map is computed using two components for the Poincare´ section as required for a chaotic attractor bounded by a genus-3 torus. The return map is made up of 16 branches (Fig. 35c). The corresponding template therefore has 16 branches (not shown). This system is topologically equivalent to the Matsumoto–Chua system discovered a few years later in an electronic circuit [39,40]. This electronic circuit is an RLC circuit with four linear elements (two capacitors, one resistor and one inductor) and a nonlinear diode, and can be modeled by a system of three differential equations. The equations for the Matsumoto–Chua circuit are:
Fig. 34. Chaotic attractor with a cutting combined with two foldings. This attractor is a solution to the twofold cover of the Ro¨ssler system (17). Parameter values: a = 0.432, b = 2.0, c = 4.0 and l = 2.083.
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Fig. 35. Phase portrait and first-return map associated with a chaotic solution to system (23). Parameter values: a = 0.03 and l = 0.1 (and not 10 at reported in the Ro¨sslerÕs paper). Initial conditions: x0 = 1, y0 = 0.55 and z0 = 0.12.
8 > < x_ ¼ a½y x hðxÞ y_ ¼ x y þ z > : z_ ¼ by
ð24Þ
where hðxÞ ¼ m1 x þ
ðm0 m1 Þ ½jx þ 1j jx 1j 2
ð25Þ
A typical attractor solution to system (24) is shown in Fig. 36. It is slightly less developed than the chaotic solution to system (23) since its first-return map to a Poincare´ section (not shown) has a smaller number of monotonic branches. Varying the a-parameter in system (23) or the a-parameter in system (24) would allow one to obtain two mutually closer dynamics. The topology of this circuit and other similar circuits has been investigated in [41]. 5.4. Layered cut chaos To end this classification of chaotic attractors, a surprising type of chaos can be finally considered. In 1977, Ro¨ssler proposed another example of ‘‘Lorenzian chaos’’ [15], that is in our terminology, pure cut chaos. The equations then proposed can be viewed as a combination of a focus-plus-saddle subsystem with a switching variable when both the focus-plus-saddle subsystem and the switching variable are simplified up to linearity. The system thus reads as:
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Fig. 36. Chaotic attractor solution to the Matsumoto–Chua circuit. Parameter values: b = 100/7, m0 = 8/7, m1 = 5/7 and a = 9.
Fig. 37. Layered cut chaos solution to system (26). The map reveals that this is a layered Lorenzian chaos (compare the map with Fig. 5). Parameter values: a = 0.1, b = 0.08 and c = 0.125. Initial conditions: x0 = 0, y0 = 0.2 and c = 106.
8 x_ ¼ x xy z > < y_ ¼ x2 ay > : z_ ¼ bðcx zÞ
ð26Þ
Here, the first two lines determine a double-focus-plus-saddle system. The chaotic attractor solution to this system is shown in Fig. 37a. The surprising property of this attractor is revealed by the first-return map (Fig. 37b). Rather than a cloud of points as could be expected from a first inspection of the attractor, a well-ordered layered map is obtained. Many different branches—which obey the symmetry properties of the flow—are obtained. Their shapes seem to converge toward certain points of the map. It is quite surprising to obtain such a layered flow in a three-dimensional system. This system therefore constitutes a challenging example for accurate topological analysis.
6. Conclusions The introduction of templates and symmetry transformations into the topological and numerical investigations of chaotic attractors has brought on a new, more consistent picture. The smiling little girl of Fig. 32a illustrates the new easiness.
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This does not mean that all problems have already been solved in the context of generic three-dimensional chaotic flows. The topological analysis presented above implicitly presupposes that the flows are very strongly dissipative, so that the Poincare´ sections are essentially one-dimensional. A more Hamiltonian-systems like ‘‘second zoo’’ may therefore still waiting to be discovered and analyzed in detail.
Acknowledgments A part of this work was done during a stay by C. L. at University of Tu¨bingen. We would like to thank Bob Gilmore for his being a fountain of stimulation and insight over more than a decade already. For J.O.R.
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Inequivalent topologies of chaos in simple equations Christophe Letellier a
a,*
, Elise Roulin a, Otto E. Ro¨ssler
b
CORIA UMR 6614, Universite´ de Rouen, Av. de l’Universite´, BP 12, F-76801 Saint-Etienne du Rouvray Cedex, France b Division of Theoretical Chemistry, University of Tu¨bingen, D-72076 Tu¨bingen, Germany Accepted 6 May 2005
Abstract In the 1970, one of us introduced a few simple sets of ordinary differential equations as examples showing different types of chaos. Most of them are now more or less forgotten with the exception of the so-called Ro¨ssler system published in [Ro¨ssler OE. An equation for continuous chaos. Phys Lett A 1976;57(5):397–8]. In the present paper, we review most of the original systems and classify them using the tools of modern topological analysis, that is, using the templates and the bounding tori recently introduced by Tsankov and Gilmore in [Tsankov TD, Gilmore R. Strange attractors are classified by bounding tori. Phys Rev Lett 2003;91(13):134104]. Thus, examples of inequivalent topologies of chaotic attractors are provided in modern spirit. Ó 2005 Elsevier Ltd. All rights reserved.
The menagerie of the simplest surprising flows in three-dimensional space is still a botanical task waiting to be completed. Even though these prototype flowers have neither smell nor color, one can, for example, listen to them while watching them do their thing. In the following, only their anatomical features will be presented in a new, more complete classification.
1. Introduction Although all the necessary conditions on the algebraic structure of ordinary differential equations which ensure that a chaotic solution is observed in a given system are still lacking, there are numerous examples of different sets of these equations which have a chaotic solution. Nevertheless, at least in three-dimensional phase spaces, there are not so various types of chaos. Such a limited number of different types of chaos results from the fact that, in three-dimensional phase spaces, there are quite strong topological constraints—the case of higher-dimensional phase spaces being still more or less an unattacked problem. It is therefore not surprising that a topological analysis using templates [1] and the recently introduced bounding tori [2,3] has been developed to characterize—and classify—chaotic attractors embedded in three-dimensional spaces. Nevertheless, a complete classification of all possible types of chaotic attractors with their corresponding topological properties is still lacking. From our knowledge, the single attempt—using the properties of the first-return maps—was *
Corresponding author. E-mail address: [email protected] (C. Letellier).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.05.036
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proposed by one of us in the mid of 70s [4]. The aim of the present paper is thus to address this problem using the recent advances in the topological analysis, one of our main attention being to provide an explicit set of ordinary differential equations with a chaotic attractor for each type of chaos identified. The subsequent part of this paper is organized as follows. Section 2 provides a brief review of the topological concepts—mainly templates, linking numbers and bounding tori—used to characterize the topology of chaotic attractors. Section 3 is devoted to chaotic attractors associated with unimodal maps—the simplest types of chaos—and bounded by genus-1 tori. In Section 4 are discussed different examples of combinations of the previous types to provide multimodal chaotic attractors still bounded by genus-1 tori. A few examples of chaotic attractors bounded by genus-3 tori are presented in Section 5. Section 6 gives a conclusion.
2. Templates and bounding tori 2.1. Templates Chaotic attractors can be classified by branched manifolds, also called templates or knot-holders. A template is a branched manifold which can encode the topological properties of chaotic attractor. The structure of chaotic attractors is mainly based on the stretching and folding—or cutting as we will discuss later—mechanisms which are the relevant ingredients for generating chaos. Roughly speaking, the stretching induces the sensitivity to initial conditions and the folding or cutting produces the mixing between trajectories. These two ingredients are required to have solutions with an underlying deterministic—therefore well-defined—structure but which are long-term unpredictable. A typical template with these two ingredients was probably drawn for the first time in 1976 [5] as a ‘‘chaotic blender’’. Slightly later, the same author proposed another view of such a template as a ‘‘paper model’’ or ‘‘origami’’ in [6]. The latter version, close to those now used, is shown in Fig. 1a added with a more schematic view (Fig. 1b) which can be found in works by Gilmore and co-workers [7,8] or Letellier et al. [9]. Both are topologically equivalent, that is, allow one to predict the same topological invariants between pairs of periodic orbits, such as linking numbers or relative rotation rates [1]. We should note that Birman and Williams [10,11] were the first to consider templates as ‘‘knot-holders’’, that is, as a branched manifold in which periodic orbits can be drawn with their relative organization. Such an approach was also followed by Holmes and Williams [12] and then developed by Gilmore and his co-workers [7,8]. An important point is that the first-return map to a Poincare´ section is not sufficient to determine all the properties of the associated template. In particular, the template shown in Fig. 1 can be associated both with the unimodal map with a differentiable maximum—the ‘‘walkin-stick’’ map in the terminology of [13]—(Fig. 2a) and with the ‘‘Lorenz map’’ with its characteristic cusp (Fig. 2b) [14]. Although Ro¨ssler classified these two types of map long ago [15] as ‘‘bent’’ (Fig. 1a) and ‘‘cut’’ maps (Fig. 1b), it is only recently that these maps were associated with the folding and cutting mechanisms, respectively [16]. On the other hand, a ‘‘cut’’ map requires a singularity in the flow [16]. While Ro¨ssler proposed a significant collection of different types of chaos with very suggestive names like ‘‘spiral’’, ‘‘screw’’ or ‘‘sandwich’’ type of chaos, we will review and clas-
Fig. 1. Two topologically equivalent templates. (a) Template adapted from first one drawn by Ro¨ssler in [5] and (b) a more schematic view. Both encode the two main ingredients for generating chaotic attractors, that is, stretching and folding.
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Fig. 2. Two different unimodal first-return maps which can be associated with the templates shown in Fig. 1.
sify these systems with the more recent terminology. We will also include other systems in order to have a more complete list of examples of chaotic behaviors with inequivalent topologies. Before that, we will need to briefly review the bounding tori, recently introduced by Tsankov and Gilmore [2,3], in order to be able to give a more general classification. 2.2. Bounding tori It is well known that a chaotic attractor can be bounded by a semi-permeable surface defining a domain of the phase space from which a trajectory cannot escape [18]. Such a bounding surface is oriented, that is, any orbit that passes through from the outside to the inside remains trapped inside forever. In general, such a surface is a genus-g surface where g is the number of holes in the boundary. The surface with g = 0 is called a sphere and that with g = 1 is commonly called a torus. Although the flow on the branched manifold has no fixed points in an open neighborhood of the branched manifold, restricted to the surface itself there are fixed point singularities. All are generalized saddles. As a result, the number of singularities is related to the genus g, and the number is 2(g 1) [3]. The flow, restricted to the bounding torusÕ surface, can be put into canonical form. For genus g = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . ., there are 0, 1, 0, 1, 1, 2, 2, 5, 6, . . .inequivalent canonical forms [3]. The canonical form for g = 3 is shown in Fig. 3. Typically, the central hole––with singularities—is associated with a saddle fixed point and the two other holes with focus fixed points (F±). This does not mean that a system with a chaotic attractor bounded by this genus-3 torus has necessarily only three fixed points but that these are only the three fixed points which are surrounded by the flow. The other fixed points are ‘‘outside’’ the outer disk boundary. For instance, the Lorenz attractor (Fig. 4) is bounded by a genus-3 torus as shown in Fig. 3. In three dimensions, a Poincare´ surface is a minimal two-dimensional surface with the property that all points in the attractor intersect this surface transversally an infinite number of times under the flow. A Poincare´ section is not necessarily constituted of a single component. In particular, when a dynamical system has an order-g symmetry, it is useful to introduce g disconnected components in order to properly compute the Poincare´ section [17,19,20]. More generally, the Poincare´ section is disjoint union of one or more nonoverlapping disks [2,3]. In fact, the number of disconnected
Fig. 3. The canonical form for the genus-3 bounding torus consists of an outer disk boundary and three interior holes. The four singularities are confined to the central interior holes. Location of the two components of the Poincare´ section are also shown (0 and 1).
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Fig. 4. The Lorenz attractor is bounded by a genus-3 torus. The central interior hole associated with the saddle fixed point—with its four singularities—has been enlarged for clarity.
components is equal to (g 1). Thus, the Poincare´ surface for a genus-1 bounding torus consists of a single disk that is transverse to the flow. The Poincare´ surface for a canonical genus-g bounding torus consists of the union of g 1 disjoint disks [3]. The locations of the two disks for the genus-3 canonical form has been shown in Fig. 3. Many different templates can be described by the same genus-g canonical form. Each branch line in any of these branched manifolds can be moved so that it is contained in one of the g 1 components of the global Poincare´ section. As a result, any branched manifold enclosed by a genus-g bounding torus has exactly g 1 branch lines (for g > 1). The first-return maps for branched manifolds enclosed by a genus-1 bounding torus are equivalent to maps from the single branch line that exists in the single component of the Poincare´ section that runs back onto itself. That is, in this case the return maps are exactly maps of the interval to itself. In the genus g P 3 case, the first-return maps have g 1 branch lines [16]. The initial conditions on any branch line flow to exactly two other branch lines. The return map is constructed as follows: Each branch line is represented as an interval. These intervals are laid out along a horizontal axis (initial conditions). Each branch line is oriented, from the interior to the exterior of the projection of the bounding torus onto a plane (cf., Fig. 1). As in the genus-1 case, the images are arranged along the vertical axis. To each point on the horizontal axis (consisting of g 1 disjoint oriented segments) there is a unique image. The first-return map of this type has been computed for the Lorenz attractor in Fig. 5. We will see that the most common type of chaos associated with a genus-1 torus has a first-return map with a differentiable extrema that reflect the foldings that take place between adjacent branches. When attractors bounded by a
Fig. 5. First-return map to the Poincare´ section constituted by two disconnected components of the Lorenz system. Parameter values: R = 28, r = 10 and b = 8/3.
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genus-3 tori are considered, the canonical form for such bounding tori (Fig. 3) has a central internal hole with four singularities which surrounds a saddle type singularity. Thus, there exists some point along each branch line—or component of the Poincare´ section—where the initial conditions on one side evolve to one branch line and the initial conditions on the other side evolve to the other branch line. This is a so-called ‘‘cutting point’’. At this point, the return map is discontinuous and, most often, with a slope discontinuity. The cutting occurs in the neighborhood of saddle points or other singularities that deflect the flow in a small neighborhood toward divergent directions [16].
3. Unimodal chaos bounded by a genus-1 torus Chaotic attractors involve, at least, one unimodal map, that is, a first-return map constituted of two monotonic branches separated by a single critical point—differentiable or not. In this case, there exists a universal order for the possible bifurcations under which periodic orbits can be created. We will designate the chaotic behavior characterized by a unimodal map as ‘‘unimodal chaos’’. We will therefore start out by describing the different type of unimodal chaos since multimodal chaos, later discussed, can be viewed as combination of unimodal chaos. 3.1. Unimodal folded chaos A unimodal folded chaos is characterized by a template as shown in Fig. 1 and an everywhere differentiable firstreturn map with a single critical point (Fig. 2a). This is the most common—and simplest—type of chaotic attractor that can be encountered. It always follows a period-doubling cascade as a route to chaos. The first system generating unimodal folded chaos was proposed in the beginning of 76 by Ro¨ssler [5]. This is a set of three ordinary differential equations: 8 ðk 3 y þ k 4 zÞx > > x_ ¼ k 1 þ k 2 x > > < xþK ð1Þ y_ ¼ k 5 x k 6 y > > > k z 10 > : l_z ¼ k 7 x þ k 8 z k 9 z2 z þ K0 This system was invented in order to obtain a simple chemical reaction able to generate chaotic behavior. This research was motivated by a discussion Ro¨ssler had with Art Winfree in 1975. A figure quite similar to Fig. 6a was obtained [5]. Ro¨ssler named this type of chaos ‘‘spiral chaos’’ but, as he later developed in his classification of different types of chaos by using the shape of the first-return map, we will prefer to use the term ‘‘folded chaos’’. The differentiable maximum of the first-return map to a Poincare´ section reveals the folding associated with this chaotic attractor (Fig. 6b).
Fig. 6. Unimodal folded chaos solution to system (1). Parameter values: k1 = 37.8, k2 = 1.4, k3 = 2.8, k4 = 2.8, k5 = 2, k6 = 1, k7 = 8, k8 = 1.84, k9 = 0.0616, k10 = 100, K = 0.05, K 0 = 0.02, l = 1/25; x0 = 7, y0 = 12, z0 = 0.2.
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The nature of the solution found in system (1) obviously depends on the chosen parameter values. We arbitrarily choose to vary one of the parameters, k6, to compute a bifurcation diagram (Fig. 7) showing the period-doubling cascade as a route to chaos. Varying parameter k6 does not allow one to find a different type of chaos before the boundary crisis (k6 0.987) which ejects the trajectory to infinity occurs. A slightly simpler system was proposed few months later by the same author [21]. It reads: 8 ðk 3 y þ k 4 zÞx > > x_ ¼ k 1 þ k 2 x > > < xþK ð2Þ y_ ¼ k 5 x k 6 y > > > k 8z > : z_ ¼ k 7 x z þ K0 Two terms were deleted from system (1). Nevertheless, this system still has rational right-hand-side members which can complexify its analysis. This system has a chaotic attractor shown in Fig. 8. In order to obtain a polynomial system, Ro¨ssler empirically simplified the equations of system (2) to obtain a still simpler system which continues generating a chaotic attractor topologically equivalent to the attractor shown in Fig. 6. This equation has, as a ‘‘model of a model’’, no longer an immediate physicochemical interpretation. The proposed system reads [22]:
Fig. 7. Bifurcation diagram of system (1) in dependence on parameter k6. The other parameters have the same values as in caption of Fig. 6.
Fig. 8. Unimodal folded chaos solution to system (2). Parameter values: k1 = 22, k2 = 2.2, k3 = k4 = 4.4, k5 = 1.2, k6 = 1, k7 = 14, k8 = 140, K = 0.01, K 0 = 0.05. Initial conditions: x0 = 7, y0 = 6, z0 = 0.1.
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Fig. 9. Unimodal folded chaos solution to the Ro¨ssler system (3). Parameter values: a = 0.420, b = 2 and c = 4.
8 > < x_ ¼ ðy þ zÞ y_ ¼ x þ ay > : z_ ¼ b þ zðx cÞ
ð3Þ
which is the so-called Ro¨ssler system. Only a single nonlinear term (xz) remains in this system. Ro¨ssler published a unimodal folded chaotic attractor with the parameter values (a, b, c) = (0.2, 0.2, 5.7). For convenience in the next subsection, other parameter values are preferred [9]. A chaotic attractor solution to system (3) is shown in Fig. 9. Many different systems with a unimodal folded chaos were later published in many different fields. Among others, a unimodal floded chaotic attractor was found by Buchler and Goupil [23] in astrophysics, Decroly and Goldbeter [24,25] in biochemistry, Upadhyay and co-workers [26,27] in population model, and Sprott in his systematic study [28]. 3.2. Inverted unimodal folded chaos During his attempts to simplify the equations for generating chaotic attractors, Ro¨ssler obtained another type of chaotic attractor with the following system [21]: 8 k 3 yx > > < x_ ¼ k 1 þ k 2 x x þ K ð4Þ y_ ¼ k 4 x k 5 y þ k 6 zy 2 > > : z_ ¼ k 7 k 6 zy 2 He obtained a chaotic attractor (Fig. 10a) that he termed ‘‘screw attractor’’. Although the term ‘‘screw’’ was motivated by the shape of the attractor, a first-return map to a Poincare´ section is a differentiable unimodal map (Fig. 10b). The mechanism generating the chaotic attractor is therefore not very different from that for the previous examples. Nevertheless, a careful topological analysis reveals that the attractor is actually characterized by a template with a negative half-turn on the other side as a global torsion (Fig. 11a). This corresponds to what is often called an ‘‘inverted horseshoe’’ since the odd number of half-turns inverts the map. According to our terminology, we will designate such a chaotic attractor as inverted unimodal folded chaos. In the case of system (4), this is an additive global torsion (Fig. 11a) since in the sense that the template can be redrawn with two branches having one and two negative half-turns, respectively, that both branches have one additional half-turn. This is not the case when the global torsion has a different sign than the local torsion (Fig. 11b). As for the ‘‘direct’’ unimodal folded chaos, the associated route to chaos is once more a period-doubling cascade. This can be confirmed with a bifurcation diagram versus the parameter k5 (Fig. 12). As observed with other systems having inverted unimodal folded chaos, the chaotic windows are considerably less developed than for ‘‘direct’’ unimodal folded chaos. The substractive case has been observed, for instance, in SprottÕs systems D, E and L according to the names used in [28]. The chaotic attractor solutions to system D which reads: 8 > < x_ ¼ y y_ ¼ x þ z ð5Þ > : 2 z_ ¼ xz þ ay is shown in Fig. 13.
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Fig. 10. Inverted unimodal folded chaos solution to system (4). Parameter values: k1 = 2, k2 = 0.9, k3 = k4 = 1, k5 = 0.5, k6 = 0.005, k7 = 1.95, K = 0.1. Initial conditions: x0 = 0.25, y0 = 3.1, z0 = 17.
Fig. 11. Two different templates for an inverted unimodal folded chaotic attractor. (a) The global torsion has the same sign as the local torsion and, consequently, this is an ‘‘additive’’ case. (b) The global torsion has an opposite sign to that of the local torsion, and this is therefore a substractive case.
3.3. Unimodal cutting chaos The term ‘‘sandwich chaos’’ was introduced by Ro¨ssler [32] to designate a chaotic attractor characterized by a map constituted of two monotonic branches with a discontinuity between them. Among this type of chaos, a unimodal cutting chaos corresponds to a chaotic attractor with a topology characterized by the template shown in Fig. 1 but with a Lorenz map as a first-return map to a Poincare´ section (Fig. 2b). From our knowledge, no simple set of polynomial equations was identified to generate a chaotic attractor bounded by a genus-1 torus and possessing a Lorenz map.
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Fig. 12. Bifurcation diagram versus k5 for system (4) with the other parameter values as in Fig. 10.
Fig. 13. Inverted folded chaotic attractor solution to SprottÕs system D. Parameter value: a = 4.0.
The first system with rational right members was published by Ro¨ssler [29] as an isothermal abstract reaction system. The systems reads: 8 ðdz þ eÞx > > x_ ¼ ax þ by cxy > > x þ K1 < jxy ð6Þ y_ ¼ f þ gz hy > > > y þ K 2 > : z_ ¼ k þ lxz mz This abstract chemical reaction generates a unimodal cut chaotic attractor as shown in Fig. 14a. A first-return map to a Poincare´ section (Fig. 14b) has the shape of the Lorenz map as expected. The l-value is slightly modified to obtain a Lorenz map without a gap between the two monotonic branches as originally published [29]. As seen in Fig. 14a, the saddle fixed point is not surrounded by the flow. Consequently, there is no internal hole, associated with this saddle point, in the corresponding bounding torus. The attractor is thus bounded by a genus-1 torus. We may say that only half of the saddle is active on the flow, as we will discuss later. This early abstract chemical reaction system is interesting also because it allows a continuous transition between unimodal folded chaos and unimodal cutting chaos. This is shown with a bifurcation diagram versus parameter l (Fig. 15) where the inverse period-doubling cascade is easily identified. The transition between these two types of chaos occurs around l = 5.3. Thus, with 4.795 < l < 5.3, the asymptotic behavior is a unimodal cut chaos and when 5.3 < l < 5.716, it is a unimodal folded chaos which is observed. Such a continuous transition was previously identified already in a 5dimensional laser model [30]. The other known example of unimodal cut chaos is the image of the Lorenz system [31], that is, the system which results from the 2 # 1 coordinate transformation introduced to ‘‘modd out’’ the symmetry properties of the Lorenz system [31]. This transformation reads:
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Fig. 14. Unimodal cut chaotic attractor solution to system (6). Parameter values: a = 33, b = 150, c = 1, d = 3.5, e = 4815, f = 410, g = 0.59, h = 4, j = 2.5, k = 2.5, l = 5.29, m = 750, K1 = 0.01 and K2 = 0.01.
Fig. 15. Bifurcation diagram versus l of the abstract chemical reaction system (6). Other parameter values as in Fig. 14.
u ¼ Reðx þ iyÞ2 ¼ x2 y 2 W ¼ v ¼ Imðx þ iyÞ2 ¼ 2xy w ¼ z
ð7Þ
The image dynamical equations are thus: 8 > < u_ ¼ ðr 1Þu þ ðr RÞv þ vw þ ð1 rÞq v_ ¼ ðR rÞu ðr þ 1Þv uw þ ðR þ rÞq qw ð8Þ > : 1 w_ ¼ bw þ 2 v pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ u2 þ v2 . This image of the Lorenz system generates a chaotic attractor (Fig. 16a) bounded by a genus-1 torus. A first-return map to a Poincare´ section (Fig. 16b) is equivalent to the Lorenz map (Fig. 2b). Its maximum is associated with a singularity in the flow—at the origin of phase space where the flow is not defined—which produces the cutting mechanism. At first sight, the image Lorenz attractor should be bounded by a genus-2 torus as shown in Fig. 17. A blow-up of the flow around the origin of phase space (not shown), reveals that it is a hole with two singularities which should be present. Note that this is in agreement with the fact that the number of singularities is equal to 2(g 1) since with g = 2, only two singularities can be obtained. Consequently, as pointed out by Tsankov and Gilmore [3], a genus-2 bounding torus does not exist, since its internal hole is associated with two singularities that must be removed, thus leading to a genus-1 torus [3].
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Fig. 16. Unimodal cut chaotic attractor solution to the image system (8) of the Lorenz system. Parameter values: R = 28, r = 10 and b = 8/3.
Fig. 17. A genus-2 torus that could be associated with the Lorenz image attractor, but the hole with two singularities surrounding the w-axis can be removed because there is no qualitative change of flow in its neighborhood, thus leading to a genus-1 torus. This is why there is no canonical form for genus-2 tori.
3.4. Unimodal half-inverted cut chaos The second type of ‘‘sandwich’’ chaos is the ‘‘unimodal half-cut chaos’’ thus termed since only one branch of the first-return map is inverted. He thus proposed a set of equations which can be viewed as a modified Lorenz system with a broken symmetry. It reads as follows [32]: 8 > < x_ ¼ x xy z y_ ¼ x2 ay ð9Þ > : z_ ¼ bx cz þ d where the symmetry is broken when d 5 0. An example of a chaotic attractor of the ‘‘sandwich’’ type is shown in shown in Fig. 18a. At first sight, the attractor should be bounded by a genus-3 torus but there is a path which is forbidden (Fig. 19a). Consequently, the two internal holes (at the right part of Fig. 19a) can be merged into a single hole, thus leading to a hole with two singularities (Fig. 19b). This resulting hole can therefore be removed. Half-inverted cut chaos can also be found in the system of [4]: 8 > < x_ ¼ x xy z y_ ¼ x2 ay ð10Þ > : z_ ¼ bx cz þ d with parameter values as a = 0.1, b = 0.08, c = 0.38 and d = 0.0015. Another example was also found by perturbing the Lorenz equations according to: 8 > < x_ ¼ rðy xÞ þ a y_ ¼ xðR zÞ y ð11Þ > : z_ ¼ xy bz
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Fig. 18. Unimodal half-inverted cut chaos solution to system (9). Parameter values a = 0.1, b = 0.09375, c = 0.38 and d = 0.0015.
Fig. 19. Perestroika in a genus-3 torus with a forbidden path leading to a genus-1 torus (see text).
with parameter values a = 2.2, r = 4, R = 80 and b = 8/3. Later, Ro¨ssler proposed an abstract chemical reaction scheme generating a chaotic attractor as close as possible to the Lorenz attractor [29]. The system reads: 8 dzx > > > x_ ¼ ax þ by cxy x þ K > 1 < hxy ð12Þ 2 y_ ¼ e þ fx gy > > > y þ K 2 > : z_ ¼ j þ kx lz In fact, it produces a unimodal half-inverted cut chaos if parameter values are a = c = d = f = 1, b = 2, e = h = 4, g = 0.1, j = 0.5415, k = 0.06, l = 0.33, K1 = K2 = 0.02.
4. Multimodal chaos bounded by a genus-1 torus 4.1. Multimodal ‘‘snail’’ chaos Multimodal chaotic attractors are associated with first-return maps consisting of more than two monotonic branches, that is, with more than one critical point. Consequently, when parameters are varied, sequences of bifurcations can be due to different modes, one mode per critical point. This is what is called ‘‘antimonotonicity’’ [33,34]. Most of the systems previously described can also generate multimodal chaos when parameter values are varied. In his early searches for different type of chaos, Ro¨ssler did not realize that a single system can have different type of chaos. It was only when he varied some parameters of system (3) on his analog computer that he realized that feature. This ‘‘late’’ understanding—only a few months after his first chaotic attractor—explains why he proposed so many different sets of equations. We will start with what we call the multimodal snail chaos which can be found in the Ro¨ssler system (3) when a 2 [0.43295; 0.556] with b = 2 and c = 4, as investigated in [9]. Ro¨ssler identified this multimodal chaos which he called ‘‘screw’’ chaos in [32]. When a > 0.43295, a third branch occurs in the first-return map. Then a fourth branch occurs
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when a > 0.492 and so on, up to more than 47 branches [35] for a = 0.5566334488—just before the boundary crisis which ejects the trajectory to infinity [9]. A typical attractor of the series is shown in Fig. 20 with its first-return map. The term ‘‘snail’’ is chosen in view of a Poincare´ section of the system (Fig. 21) which has the shape of snail shell. The snail shape of the Poincare´ section induces ordered locations for the critical points of the first-return map within the visited interval. For instance, along the y-axis as used for building the first-return map, a scaling law obeys according to [35]: y y n1 dl ¼ lim n ¼ 1.68 0.04 ð13Þ n!1 y nþ1 y n where yn is the y-coordinate of the nth critical point. The convergence toward the scaling number d1 is shown in Fig. 22. A deep difference between ascending critical points Cn (odd n) and descending critical points Cn (even n) is seen in this figure. Another scaling law is associated with the successive a-values that correspond to the occurrence of critical points in the first-return map. This second scaling law reads [35]: an an1 da ¼ lim ¼ 1.70 0.08 ð14Þ n!1 anþ1 an where an is the a-value at which the nth critical point Cn occurs in the first-return map. The scaling number da is equal to the above location scaling number dl within estimated uncertainties. Consequently, the appearance of the critical points
Fig. 20. Multimodal snail chaotic attractor solution to the Ro¨ssler system (3). Parameter values: a = 0.5566334488, b = 2 and c = 4.
Fig. 21. Poincare´ section of the Ro¨ssler system with the shape of a ‘‘snail’’ shell. Parameter values: a = 0.5566334488, b = 2 and c = 4.
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Fig. 22. Convergence toward the scaling number dl versus n, for a = 0.5566334488 in the case where the first-return map presents 47 monotonic branches.
is connected to their location in the first-return map, and vice versa. After the accumulation points of the scaling law, the second fixed point of the Ro¨ssler system, located in the boundary of the attraction basin, collides with the attractor. Due to this crisis, the trajectory diverges to infinity after a transient regime. 4.2. Cover of a folded chaotic attractor Another type of chaos can be observed in the twofold cover of the Ro¨ssler system, that is, in a Ro¨ssler attractor with a rotation symmetry by p around the z-axis, here designated as Rz ðpÞ. This rotation symmetry is obtained as follows. First, we start from a centered Ro¨ssler system through a rigid displacement, that is, the inner fixed point is moved to the origin of a new phase space R3 ðu; v; wÞ. In the translated coordinate system, the equations for this image system become 8 > < u_ ¼ v w v0 w0 v_ ¼ u þ av þ u0 þ av0 ð15Þ > : w_ ¼ b þ wðu þ u0 cÞ þ w0 u þ w0 ðu0 cÞ pffiffiffiffiffiffiffiffiffiffi 2 where u0 ¼ v0 ¼ aw0 ¼ c c2 4ab are the coordinates of the inner fixed point of the original Ro¨ssler system (3). Moreover, when the origin of the coordinates is displaced along the u-axis by a quantity equal to l, the equations for the image system in the translated coordinates are 8 > < u_ ¼ v w ð16Þ v_ ¼ u þ av þ l > : w_ ¼ ~bðu þ lÞ þ wðu ~c þ lÞ where ~b ¼ w0 and ~c ¼ c u0 . Then, applying the inverse of the coordinate transformation (7), we obtain the dynamical equations for the twofold cover of the Ro¨ssler system [31]: 8 1 2 2 > < x_ ¼ 2r2 ½r y þ xð2ay zÞ þ ly 1 2 2 y_ ¼ 2r2 ½r x þ yð2ax þ zÞ þ lx ð17Þ > : 2 2 2 2 ~ z_ ¼ bðx y þ lÞ þ zðx y ~c þ lÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ x2 þ y 2 . When l = 0, that is, when the rotation axis is located at the inner fixed point of the centered Ro¨ssler system, this twofold cover generates a chaotic attractor with two symmetry-related foldings (Fig. 23a). Its first-return map has 22 = 4 branches (Fig. 23b) and can be viewed as a ‘‘double unimodal folded chaos’’. The important point of having n-folded chaos by using the n-fold cover of a unimodal folded chaotic attractor is that there is no global torsion. This transpires from the template of the twofold cover of the Ro¨ssler system (Fig. 24). For instance, this process can be repeated with a fivefold cover of the Ro¨ssler system. In this case, the chaotic attractor has five symmetry-related foldings as shown in Fig. 25a. The first-return map has thus 25 = 32 monotonic branches (Fig. 25b). All maxima have exactly the same ordinate.
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Fig. 23. Twice folded chaotic attractor solutions to the twofold cover (17) of the Ro¨ssler system. Parameter values: a = 0.43294, b = 2.0 and c = 4.0.
Fig. 24. Template of the twofold cover of the Ro¨ssler system.
Fig. 25. Five folded chaotic attractor solutions to the fivefold cover of the Ro¨ssler system. Parameter values: a = 0.43294, b = 2.0 and c = 4.0.
Twice-folded chaos has been observed with an inversion symmetry in the Kremliovsky system [31] and in an integrodifferential equation proposed by Lu et al. [37]. This equation reads as:
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Fig. 26. Twice folded chaos solution to the integro-differential equation (18). Parameter values: a = 0.001, b = 3.8, c = 2.85 and d = 3.
x_ ¼ ax bðjxs þ 1j jxs 1jÞ þ d ðjxs þ d j jxs djÞ
ð18Þ
where xs = x(t s). This equations has the chaotic solution shown in Fig. 26. 4.3. Cover of the snail chaos Using the twofold cover (17) of the Ro¨ssler system with other parameter values, it is also possible to obtain a chaotic attractor corresponding to a twofold cover of the snail type of chaos (Fig. 27a). The first-return to a Poincare´ section is thus constituted by numerous monotonic branches (Fig. 27b). The structure of the map, although still characterized an alternating sequence of increasing and decreasing branches, has a quite complicated structure—from the point of view of critical point locations, for example. In particular, maxima have different ordinates. 4.4. Cover of inverted folded chaos The twofold cover of inverted folded chaos can be observed in the algebraically simplest equivariant jerk system proposed by Malasoma. It consists of three terms including one quadratic nonlinearity. It reads as:
x ¼ a€x þ x_x2 x
ð19Þ
Fig. 27. Twofold cover of the snail chaotic attractor solution to the twofold cover (17) of the Ro¨ssler system. Parameter values: a = 0.556, b = 2, c = 4 and l = 0.
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This system can be rewritten as a set of three ordinary differential equations [36]: 8 > < x_ ¼ y y_ ¼ z > : z_ ¼ az þ xy 2 x
ð20Þ
This simplest equivariant system generates the ‘‘double folded chaotic attractor’’ shown in Fig. 28a. The topology of this chaotic attractor is different than the topology of the two folded chaotic attractor of Fig. 24 since there is one global torsion by one half-turn between each folding (Fig. 29). The global torsion produces a different quantitative arrangement between the monotonic branches in the first-return map (Fig. 28b) which is characteristic of such twofold cover of an inverted folded chaos. Note that the symmetry of this system is an inversion but this has no relevant effect on the shape of the first-return map as we shall see later with the Burke and Shaw system. We can check on a higher order cover that the quantitative organisation of the first-return map is indeed different than for the snail chaos. This is done with the threefold cover of the simplest equivariant system—which is therefore the sixth-cover of the image of system (20). The attractor is shown in Fig. 30a where the six symmetry related foldings are easily identified. Between each of them, there is a global torsion by a half-turn. The first-return map to a Poincare´ section consists of 26 = 64 monotonic branches (Fig. 30b). The regular organization of the branches signifies a departure from the snail chaos (Fig. 20b), the n-fold cover of unimodal folded chaos (Fig. 25b) and from the twofold cover of the snail chaos (Fig. 27b). Another example of a twofold cover of a unimodal inverted folded chaotic attractor can be observed in the Burke and Shaw system [38,20]:
Fig. 28. Twice-inverted folded chaotic attractor solution to the simplest equivariant jerk system (20) just before the attractor suffers a merging crisis. Parameter value: a = 2.027717.
Fig. 29. Templates of two different twofold covers of unimodal inverted chaos.
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Fig. 30. Chaotic attractor with six inverted foldings of the threefold cover of the simplest equivariant jerk system (20). Parameter value: a = 2.027717.
Fig. 31. Chaotic attractor with two inverted foldings in the Burke and Shaw system (21). Parameter values: V = 4.271 and S = 10.0.
8 > < x_ ¼ Sðx þ yÞ y_ ¼ y Sxz > : z_ ¼ Sxy þ V
ð21Þ
where S and V are the parameters. The first-return map to a Poincare´ section (not shown) of the chaotic attractor of Fig. 31 has the same shape as the map to a Poincare´ section of the attractor of the simplest equivariant system (Fig. 28b). The topology of the Burke and Shaw attractor is described by the template shown in Fig. 29b since the attractor is globally invariant under a rotation symmetry Rz ðpÞ. Consequently, the nature of the symmetry is not so important for this type of chaos. 4.5. Cover of the half-inverted cut chaos A twofold cover of the half-inverted cut chaos is now investigated for system (9). The coordinate transformation x ¼ X 2 Z2 ð22Þ W2 ¼ y ¼ Y z ¼ 2XZ is inverted and applied to system (9). A chaotic attractor with two half-inverted cut mechanisms is obtained: Fig. 32a (‘‘laughing girl attractor’’). A first-return map to a Poincare´ section has 22 = 4 branches with slopes of the same sign (Fig. 32b).
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Fig. 32. Twofold cover of the half-inverted cut chaotic attractor. Parameter values: a = 0.1, b = 0.082, c = 0.38, d = 0.0015.
Fig. 33. Twofold cover of the cut chaotic attractor shown in Fig. 14a. Parameter values: a = 0.1, b = 0.082, c = 0.38, d = 0.0015.
4.6. Cover of unimodal cut chaos Using a twofold cover of system (6) proposed by Ro¨ssler and Ortoleva, one can obtain a chaotic attractor with two cutting mechanisms. The chaotic attractor is shown in Fig. 33 together with its first-return map.
5. Chaos bounded by a genus-3 torus When a chaotic attractor is bounded by a genus-3 torus, it has always the configuration shown in Fig. 3 since there is a unique canonical form for a genus-3 bounding torus [3]. Thus, there is always a central internal hole with singularities surrounding a saddle fixed point. There is therefore always a cutting mechanism involved in such an attractor. 5.1. Pure cutting mechanism The simplest attractor bounded by a genus-3 torus is the Lorenz attractor shown in Fig. 4. Since the Poincare´ section consists of (g 1) disjoint components when g > 1, this means that for all systems discussed in this section, two components are required for computing the Poincare´ section. From the canonical form of the genus-3 torus shown in Fig. 3, the flow has at least two possibilities when it comes from one component: (i) to return to the same component or (ii) to
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go to the other component. One monotonic branch is associated with each possibility. There are, therefore, at least four monotonic branches in the first-return map shown in Fig. 5. There are in fact two conjugated mechanisms—two cuttings—produced by a single fixed point of the saddle type. An important point to see is that the saddle, surrounded by the flow, needs to act through two different modes—two singularities per mode. Otherwise, as in the case of the abstract chemical reaction (6), only one half-side of the saddle is active in the flow. In the case of the image of the Lorenz system (8), the flow is strictly speaking not defined where the cutting occurs, but a blow-up of the flow in this area clearly reveals that there is only a ‘‘half-saddle’’ which is active, leading to a hole with two singularities which can therefore be removed. 5.2. Cutting combined with unimodal folding The system which generates the simplest example of a cutting mechanism associated with two foldings is the twofold cover of the Ro¨ssler system when the rotation axis crosses the image attractor [31], for example, when l = 2.083. In this case, the chaotic attractor is bounded by a genus-3 torus as shown in Fig. 34a. The first-return map to a Poincare´ section reveals that the cutting acts in the attractor at the same place as the folding mechanism does since the discontinuity of the first-return map occurs where the slope of the branch is nearly zero (Fig. 34b). 5.3. Cutting combined with several multimodal foldings In his attempt to list the different types of chaos, Ro¨ssler proposed a system which has the symmetry of KhaikinÕs ‘‘universal’’ circuit, that is, there are two symmetric foci related by fast jumps. The equations for this type of chaos are [4]: 8 x_ ¼ ax yð1 x2 Þ > > < ð23Þ y_ ¼ lðy þ 0.3x 2zÞ > > : z_ ¼ lðx þ 2y 0.5zÞ Two plane projections of the chaotic attractor are shown in Fig. 35a and b. The structure of the attractor is quite complex. A first-return map is computed using two components for the Poincare´ section as required for a chaotic attractor bounded by a genus-3 torus. The return map is made up of 16 branches (Fig. 35c). The corresponding template therefore has 16 branches (not shown). This system is topologically equivalent to the Matsumoto–Chua system discovered a few years later in an electronic circuit [39,40]. This electronic circuit is an RLC circuit with four linear elements (two capacitors, one resistor and one inductor) and a nonlinear diode, and can be modeled by a system of three differential equations. The equations for the Matsumoto–Chua circuit are:
Fig. 34. Chaotic attractor with a cutting combined with two foldings. This attractor is a solution to the twofold cover of the Ro¨ssler system (17). Parameter values: a = 0.432, b = 2.0, c = 4.0 and l = 2.083.
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Fig. 35. Phase portrait and first-return map associated with a chaotic solution to system (23). Parameter values: a = 0.03 and l = 0.1 (and not 10 at reported in the Ro¨sslerÕs paper). Initial conditions: x0 = 1, y0 = 0.55 and z0 = 0.12.
8 > < x_ ¼ a½y x hðxÞ y_ ¼ x y þ z > : z_ ¼ by
ð24Þ
where hðxÞ ¼ m1 x þ
ðm0 m1 Þ ½jx þ 1j jx 1j 2
ð25Þ
A typical attractor solution to system (24) is shown in Fig. 36. It is slightly less developed than the chaotic solution to system (23) since its first-return map to a Poincare´ section (not shown) has a smaller number of monotonic branches. Varying the a-parameter in system (23) or the a-parameter in system (24) would allow one to obtain two mutually closer dynamics. The topology of this circuit and other similar circuits has been investigated in [41]. 5.4. Layered cut chaos To end this classification of chaotic attractors, a surprising type of chaos can be finally considered. In 1977, Ro¨ssler proposed another example of ‘‘Lorenzian chaos’’ [15], that is in our terminology, pure cut chaos. The equations then proposed can be viewed as a combination of a focus-plus-saddle subsystem with a switching variable when both the focus-plus-saddle subsystem and the switching variable are simplified up to linearity. The system thus reads as:
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Fig. 36. Chaotic attractor solution to the Matsumoto–Chua circuit. Parameter values: b = 100/7, m0 = 8/7, m1 = 5/7 and a = 9.
Fig. 37. Layered cut chaos solution to system (26). The map reveals that this is a layered Lorenzian chaos (compare the map with Fig. 5). Parameter values: a = 0.1, b = 0.08 and c = 0.125. Initial conditions: x0 = 0, y0 = 0.2 and c = 106.
8 x_ ¼ x xy z > < y_ ¼ x2 ay > : z_ ¼ bðcx zÞ
ð26Þ
Here, the first two lines determine a double-focus-plus-saddle system. The chaotic attractor solution to this system is shown in Fig. 37a. The surprising property of this attractor is revealed by the first-return map (Fig. 37b). Rather than a cloud of points as could be expected from a first inspection of the attractor, a well-ordered layered map is obtained. Many different branches—which obey the symmetry properties of the flow—are obtained. Their shapes seem to converge toward certain points of the map. It is quite surprising to obtain such a layered flow in a three-dimensional system. This system therefore constitutes a challenging example for accurate topological analysis.
6. Conclusions The introduction of templates and symmetry transformations into the topological and numerical investigations of chaotic attractors has brought on a new, more consistent picture. The smiling little girl of Fig. 32a illustrates the new easiness.
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This does not mean that all problems have already been solved in the context of generic three-dimensional chaotic flows. The topological analysis presented above implicitly presupposes that the flows are very strongly dissipative, so that the Poincare´ sections are essentially one-dimensional. A more Hamiltonian-systems like ‘‘second zoo’’ may therefore still waiting to be discovered and analyzed in detail.
Acknowledgments A part of this work was done during a stay by C. L. at University of Tu¨bingen. We would like to thank Bob Gilmore for his being a fountain of stimulation and insight over more than a decade already. For J.O.R.
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