- Email: [email protected]
Closed periodic orbits of convective solutions in rapidly rotating system: Double torus knots and links, DTK
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Chaos, Solitons and Fractals 36 (2008) 25–31 www.elsevier.com/locate/chaos
Closed periodic orbits of convective solutions in rapidly rotating system: Double torus knots and links, DTK A.A. Abdulrahman, E.A. Elrifai
*
Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia Accepted 7 June 2006
Communicated by Prof. L. Marek Crnjac
Abstract The classification of closed periodic orbits of convection in a rapidly rotating system is given. It is shown that double torus knots and links, DTK, do occur, which is a very wide and important class of knots and links. We also proved that there is no double torus Lorenz knots, this answers question 6 raised by Hill and Murasugi in [Peter Hill, On doubletorus knots 1. J Knot Theor Ramif 1999;8(8):1009–48]. It is also shown that the system produces torus knots and links, for some specific parameters. In fact this approach suggests the study of double torus knots and links through dynamical tools, such as symbolic dynamics and templates. Ó 2006 Published by Elsevier Ltd.
1. Introduction It is known that chaotic dynamics may be generated by many kinds of experimental systems from various fields of science, such as biology, chemistry, physics, and hydrodynamics. [1,2]. The characterization of the dynamics of such systems is separated into two main branches. The first is the study of the geometric properties of the attractors, knot holders or templates, on which the asymptotic motion settles down [3,4]. While the other is based on the knowledge of the population of periodic orbits on attractors [5,6]. Topologists studied knots and links as a periodic orbits in dynamical systems [4,7,8], where periodic orbits of a flow are embedded circles. When the flow is three-dimensional, closed periodic orbits are knots and the collection of them forms a link. In this paper, we shall study chaotic solutions of convection in rapidly rotating system. This problem has occupied a considerable attention of workers in many different fields, specially in planetary physics and astrophysics. Space missions have showed that many planets with gaseous envelopes in our solar system such as Jupiter and Saturn have turbulent atmospheres [9]. Planetary scientists believe that the atmosphere of Jupiter and Saturn are in state of convection [10]. Exploring nonlinear convection using simple models of rapidly rotating fluid is one of the best tools to understand the transition to turbulence [11,12]. In order to reach a high value of the controlling parameter in convection, the Rayleigh number Ra (a ratio of buoyancy to diffusive *
Corresponding author. E-mail address: [email protected] (E.A. Elrifai).
0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.049
26
A.A. Abdulrahman, E.A. Elrifai / Chaos, Solitons and Fractals 36 (2008) 25–31
forces), the model is often simplified from a sphere into an annulus [13]. The equations governing the fluid motion inside the annulus are ox oðw; xÞ ow oh oh oðw; hÞ ow þ b ¼ Ra þ r2 x; Pr þ ¼ þ r2 h; ot oðx; yÞ ox ox ot oðx; yÞ ox where h is the horizontal temperature deviation from the basic state of pure conduction, w is the z-averaged toroidal part of the flow, x denotes the vertical vorticity, and r2 w ¼ x with vx ¼
ow ow and vy ¼ : oy ox
The dimensionless parameters are defined as follows: Prandtl number, Rayleigh number and the parameter b, respectively Pr ¼
m ; j
Ra ¼
gaDTD3 ; jm
b¼
4gXD3 ; Lm
where D denotes the typical length scale, X the angular velocity, m the kinematic viscosity, DT the temperature difference between the walls of the annulus, j kinematic diffusivity, a the coefficient of volume expansion, and g the acceleration due to gravity. Due to strong nonlinearity of the system, it is obvious that direct treatment is very difficult. Therefore, we resort to a multiple scale analysis to reduce the system to be dependent on time and one-dimensional space. The result are an amplitude equation, ^ t þ ikuwÞ ^ ¼ a2 w ^ ^ yy þ a3 ðr2 þ r0 Prhy Þw; a 1 ðw where a1 ¼ k 2 þ
r0 Pr ðk ic0 PrÞ2
;
a2 ¼ 2k 2 ikc0 þ
r0 ðk ic0 PrÞ2
; and a3 ¼
k k ic0 Pr
together with the following mean equations ^ yy Þ; and Prht ¼ hyy þ ^w ^ w ^ w ut ¼ uyy þ ikðw yy
2k 2 Pr ^ y Þ: ^w ^ þ w ^ w ðw y k 2 þ c20 Pr2
k, r0, and c0 are evaluated from the following relations, 1=6 1 2 2 Pr ð1 þ PrÞ ; r0 ¼ 9k 2 ; c0 ¼ 41=3 k 2 ; k¼ 2 and Ra = 4r, where r = r0 + 2r2 + with 1. This system is then solved numerically under the assumption that the boundaries are stress free and perfect conductors of heat, so the temperature perturbations vanishes there; namely, ^ ¼ ou ¼ h ¼ 0 at y = 0,1. The nonlinear solutions of this system when rapidly rotated exhibit a sequence of bifurcations w oy from steady solutions to periodic solutions, to tori and 2-tori, typically ending in chaotic behavior. These bifurcations are manifested on two branches of asymmetric solutions and one branch of symmetric solutions. As r2 increased along the symmetric branch, the symmetric chaos bifurcates into a symmetric 2-torus, Fig. 5. Also there are two asymmetric
Fig. 1. The asymmetric solution for u projected on the mean velocity plane showing one of the asymmetric 2-tori at r2 = 46.
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27
aperiodic solutions which remain aperiodic for all the values r2 2 [40, 45]. As r2 is further increased the chaotic solutions restabilise into asymmetric 2-tori, one of these is shown in Fig. 1. Also frequency-locking was found when r2 2 [45.144, 45.153]. An example of a frequency locked solution at r2 = 45.15 is in Fig. 9, such these solutions continue to exist up to r2 = 46.7. The scheme of this paper is as follows, after the introduction we give some required terminologies and the main results. Then some examples and calculations. Finally we end the work with speculations.
2. Terminology and main results A knot is an embedded circle in S3 or R3. A link is a disjoint collection of knots. The unknot is a trivial circle S S2 S3. Two links L1 and L2 are ambient isotopy if there are a homotopy Ht:S3 ! S3, t 2 [0, 1] = I, such that H0 is the identity map, H1 sends L1 to L2 and Ht is a homeomorphism for every t 2 I. Knots and links are in the same knot and link type if and only if they are equivalent under ambient isotopy. All knots and links in this paper are oriented by the arrow of time. Knots and links are described geometrically via projections onto a plane in which over and under crossings are indicated by broken lines. A knot K is a composite knot if it is obtained by connecting diagrams of two non-trivial knots K1 and K2, the prime knot is a non-composite knot. For more details about knot and link theory, we refer to [14]. 1
Lemma 1. The asymmetric 2-tori in Fig. 1, is topologically equivalent to a standard double torus. Proof. The proof will be done combinatorially, by a continuous deformation. The two-dimensional manifold T, in Fig. 1, can be transformed to a standard double torus, as in Fig. 2. So the standard double torus is, topologically, an attractor for the closed periodic orbits of the rapidly rotating system. In fact this is a well-known result, where nonlinear systems can make a transition from quasiperiodic motion directly to chaos. The two asymmetric aperiodic solutions remain aperiodic for all the values 40 < r2 < 45. As r2 is further increased. The chaotic solutions destabilize into asymmetric 2-tori, one of these is shown at r2 = 46 in Fig. 1. A knot or link in S3 or in R3 is called a double torus knot or link (DTK), if L is contained in a Heegaard surface on genus two (an embedded surface in a closed 3-manifold M 3, which divides M 3 into two handlebodies [15]). The Heegaard surface of genus two is the standard double torus. Now let us give a formal definition of DTK as it is given in [15,16,17]. h Definition 1. Let K be a knot embedded in a double torus H = TL ] TR, which is the gluing of two once-punctured tori TL,TR along a circle C. On each tori KnC consists of at most three parallel classes, ðn1 ; n2 ; n3 ; n01 ; n02 ; n03 Þ for the numbers of the constituent arcs, then n1 þ n2 þ n3 ¼ n01 þ n02 þ n03 ¼ n. Let (p, q), (r, s) the slopes of the first and the second parallel classes of arcs in TL, where the slope of the third is (p + r, q + s). Also denote by (p 0 , q 0 ), (r 0 , s 0 ) the slopes of the
Fig. 2. A topological transformation of the asymmetric 2-torus in Fig. 1 to the standard double torus.
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two of the parallel classes in TR. Finally, in gluing the arcs along C, we have a choice, which is denoted by n < q 6 n. Then by arranging the above numbers as in K ¼ fðn1 ; n2 ; n3 ; n01 ; n02 ; n03 jqÞðp; q; r; sÞðp0 ; q0 ; r0 ; s0 Þg, we can express a double torus knot K, as in Fig. 3. Let K be a DTK with only one parallel classes of arcs on TL and TR, then K can be denoted by K = {(n, 0, 0; n, 0, 0jq)(p, q, ,)(p 0 , q 0 , ,)} and we say that K is of type (1, 1), [22]. Then from lemma 1 and definition 1, we have directly the following result. Corollary 1. Closed periodic orbits of the convective solutions in the rapidly rotating system, contain double torus knots and links. In fact double torus knots and links contains torus knots, 2-bridge knots, knots with genus one and bridge one, (1, 1) – decomposition tunnel number one knot, Berge’s doubly positive knots which are fibred, Dean’s twisted torus knots and Pretzel knots. Double torus knots is either of period 2 or strongly invertible and their tunnel number at most 2, [15,16]. As it is clear in Fig. 3, DTK can be obtained from two torus knots (each of which is embedded in a genus one surface) by gluing together along discs. But torus knots are fibred knots, so it is quite likely to fined fibred DTK. It is also known that a nonlinear system can make a transition from quasiperiodic motion directly to chaos, which is known as the quasiperiodic route to chaos. That was one of the first proposed mechanisms leading to the formation of a strange attractor. There are many routes to chaos, one of them is the formulation of a double torus, Fig. 4. In the torus doubling route to chaos, the original torus (which is a closed curve in a cross section) appears to split into two circles at the torus doubling bifurcation point [18]. Proposition 1. Torus knots and links do occur as closed orbits of the convective solutions in the rapidly rotating system. Proof. Numerically, it is got a symmetric 2-torus, T2, for u at r2 = 42, As in Fig. 5.
h
Fig. 3. A double torus knot, K{(3, 3, 3; 3, 3, 3j4) (1, 0, 1, 1)(0, 1, 1, 1)}.
Fig. 4. Schematic of a torus doubling bifurcation.
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Fig. 5. The symmetric 2-torus for u at r2 = 42.
Fig. 6. (a) A recurrent flow around a hyperbolic fixed point. (b) A close recurrence of a chaotic trajectory and gently adjusting a segment of the chaotic trajectory we can locate a nearby periodic orbit.
Remark 1. There are knots which are not DTK for example knots 816 and 817 are not DTK, where this enumeration as in Rolfson’s book [19]. But here we show that Lorenz knots are non-DTK. In fact the following result is an answer of question 6 in the open problems which are given in [16] by Hill and Murasugi. Theorem 1. There is no double torus Lorenz knots. Proof. It is known that the only possible satellites of a Lorenz knot are parallels with possible twists [7]. It is also known that Lorenz knots are prime [20]. But in [21] Osawa answered question 12 (a) in [16], where he classified satellite DTK and showed that if a satellite double torus knot is not a cable knot, then it has torus knot companion. On the other hand Lorenz knots are fibred, while fibred DTKs [17] are either band sum of two torus knots, which are composite knots, or a closure of some positive (or negative) braids, which are the Murasugi sum of several generalized Pretzel knots. Therefore there is no double torus Lorenz knots. h
3. Calculations Remark 2. Consider a three-dimensional flow in the vicinity of a hyperbolic orbit, Fig. 6a. Since the flow is recurrent, we can choose a surface of section in the vicinity of the fixed point. This section gives a compact map of the disk onto itself with at least one fixed point. In the vicinity of this fixed point a chaotic limit set, a horseshoe, containing an infinite number of unstable periodic points can exist. A single chaotic trajectory meanders around this chaotic limit set in an ergodic manner, passing arbitrarily close to every point in the set including its starting point and each periodic point. If we consider a small segment of the chaotic trajectory that returns close to some periodic points. Intuitively, we expect to be able to gently adjust the starting point of the segment so that the segment precisely returns to its initial starting point, thereby creating a periodic orbit, Fig. 6b, [23]. Now applying the algorithm in the above remark on the trajectories in Fig. 6, we have the knots in Figs. 7–9.
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Fig. 7. The symmetric 2-torus, Ts, for u at one selection of time frames showing its progression to final stage for the value r2 = 42. The horizontal is u (0.25, t) and the vertical is u (0.75, t).
Fig. 8. Knots as periodic orbits in the solution of rapidly rotating system.
Fig. 9. The frequency in the 2-torus is now locked at r2 = 45.15.
4. Speculations In this work we have find a torus doubling as a chaotic attractor. A central problem of topological analysis is to determine a template describing the structure of this attractor. Where the template is specified by the branched manifold and the semi flow that result from the Birman–Williams work [3,6]. In fact their work provides a bridge between phase space where chaotic attractors live and abstract space of equivalent classes where templates live. This can be down as
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resulting from smooth deformation which preserves periodic orbits and does induce crossings between them. Therefore knot and braid invariants of any set of periodic orbits in the real phase space and of its counterpart in the abstract space are identical. In a revolutionary and very surprising work of Ghrist [24], he proved that any knot or link can be realized as a periodic orbit on any one of three kinds of templates. As a future work, we hope to study double torus knots and links via theory of template.
References [1] Guckenheimer John, Holmes Philip. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied mathematical sciences, vol. 42. Springer-Verlag; 1983. [2] Devaney Robert L. An introduction to chaotic dynamical systems. Addison-Wesley Publishing Company, Inc.; 1989. [3] Birman J, Williams RF. Knotted periodic orbits in dynamical systems-11: Knot holders for fibered knots. Cont Math 1983;20:1–60. [4] Ghrist RW, Holmes PJ, Sullivan MC. Knots and links in three dimensional flows. Lecture noted in mathematics, vol. 1654. NY: Springer-Verlag; 1997. [5] Daniel S, Silver Susan G. Williams, knots invariants from symbolic dynamical systems. Trans Amer Math Soc 1999;351:3243–65. [6] Birman J, Williams RF. Knotted periodic orbits in dynamical systems-1: Lorenz’s equations. Topology 1983;22(1):47–82. [7] Elrifai EA. Necessary and sufficient condition for Lorenz knots to be closed under satellite construction. Chaos Solitons & Fractals 1999;10(1):137–46. [8] Elrifai EA, Ahmed E. Knotted periodic orbits in Rossler’s equations. J Math Phys 1995;36(2). [9] Ingersoll AP, Beebe RF, Mitchell JL, Garneau GW, Yagi GM, Muller JP. Interaction of eddies and mean zonal flow on jupiter as inferred from Voyager 1 and 2 images. Geophys Res 1981;86A(10):8733–43. [10] Busse FH. Convection driven zonal flows and vortices in the major planets. Chaos 1994;4:123–34. [11] Busse FH. A model zonal flow in the major planets. Geophys Astrophys Fluid Dynam 1983;23:153–74. [12] Knobloch E. Bifurcations in rotating systems. In: Proctor MRE, Gilbert AD, editors. Lectures on solar and planetary dynamos. Cambridge University Press; 1994. p. 331–72. [13] Busse FH. Thermal instabilities in rapidly rotating systems. J Fluid Mech 1970;44:441–60. [14] Burde G, Zieschang H. Knots. De-Gruyter studies in Math, vol. 5. New York: Watter De-Gruyter; 1985. [15] Peter Hill. On double-torus knots 1. J Knot Theor Ramif 1999;8(8):1009–48. [16] Peter Hill, Kunio Murasugi. On double-torus knots 11. J Knot Theor Ramif 2000;9(5):617–67. [17] Mikami Hirasawa, Kunio Murasugi. Fibered double torus knots which are band-sums of torus knots. Osaka J Math 2007. [18] Argoul F, Arnedo A. From quasiperiodicity to chaos: An unstable scenario via period doubling bifurcation or tori. J de Mecanique Theorique et Appliquee, Numerero Special 1984:241–88. [19] Rolfson Dale. Knots and links. Publish or Perish; 1977. [20] Williams RF. Lorenz knots are prime. Ergod Th and Dynam Sys 1983;4:147–63. [21] Ozawa M. Satellite double torus knots. J Knot Theor Ramif 2001;10(1):133–42. [22] Ozawa M. Tangle decomposition of double torus knots and links. J Knot Theor Ramif 1999;8(7):931–9. [23] Cvitanovic P. Chaos for cyclists in noise and chaos in nonlinear dynamical systems. Cambridge: Cambridge University Press; 1988. [24] Ghrist RW. Branched two manifolds supporting all knots. Topology 1997;36(2):423–48.
Chaos, Solitons and Fractals 36 (2008) 25–31 www.elsevier.com/locate/chaos
Closed periodic orbits of convective solutions in rapidly rotating system: Double torus knots and links, DTK A.A. Abdulrahman, E.A. Elrifai
*
Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia Accepted 7 June 2006
Communicated by Prof. L. Marek Crnjac
Abstract The classification of closed periodic orbits of convection in a rapidly rotating system is given. It is shown that double torus knots and links, DTK, do occur, which is a very wide and important class of knots and links. We also proved that there is no double torus Lorenz knots, this answers question 6 raised by Hill and Murasugi in [Peter Hill, On doubletorus knots 1. J Knot Theor Ramif 1999;8(8):1009–48]. It is also shown that the system produces torus knots and links, for some specific parameters. In fact this approach suggests the study of double torus knots and links through dynamical tools, such as symbolic dynamics and templates. Ó 2006 Published by Elsevier Ltd.
1. Introduction It is known that chaotic dynamics may be generated by many kinds of experimental systems from various fields of science, such as biology, chemistry, physics, and hydrodynamics. [1,2]. The characterization of the dynamics of such systems is separated into two main branches. The first is the study of the geometric properties of the attractors, knot holders or templates, on which the asymptotic motion settles down [3,4]. While the other is based on the knowledge of the population of periodic orbits on attractors [5,6]. Topologists studied knots and links as a periodic orbits in dynamical systems [4,7,8], where periodic orbits of a flow are embedded circles. When the flow is three-dimensional, closed periodic orbits are knots and the collection of them forms a link. In this paper, we shall study chaotic solutions of convection in rapidly rotating system. This problem has occupied a considerable attention of workers in many different fields, specially in planetary physics and astrophysics. Space missions have showed that many planets with gaseous envelopes in our solar system such as Jupiter and Saturn have turbulent atmospheres [9]. Planetary scientists believe that the atmosphere of Jupiter and Saturn are in state of convection [10]. Exploring nonlinear convection using simple models of rapidly rotating fluid is one of the best tools to understand the transition to turbulence [11,12]. In order to reach a high value of the controlling parameter in convection, the Rayleigh number Ra (a ratio of buoyancy to diffusive *
Corresponding author. E-mail address: [email protected] (E.A. Elrifai).
0960-0779/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.06.049
26
A.A. Abdulrahman, E.A. Elrifai / Chaos, Solitons and Fractals 36 (2008) 25–31
forces), the model is often simplified from a sphere into an annulus [13]. The equations governing the fluid motion inside the annulus are ox oðw; xÞ ow oh oh oðw; hÞ ow þ b ¼ Ra þ r2 x; Pr þ ¼ þ r2 h; ot oðx; yÞ ox ox ot oðx; yÞ ox where h is the horizontal temperature deviation from the basic state of pure conduction, w is the z-averaged toroidal part of the flow, x denotes the vertical vorticity, and r2 w ¼ x with vx ¼
ow ow and vy ¼ : oy ox
The dimensionless parameters are defined as follows: Prandtl number, Rayleigh number and the parameter b, respectively Pr ¼
m ; j
Ra ¼
gaDTD3 ; jm
b¼
4gXD3 ; Lm
where D denotes the typical length scale, X the angular velocity, m the kinematic viscosity, DT the temperature difference between the walls of the annulus, j kinematic diffusivity, a the coefficient of volume expansion, and g the acceleration due to gravity. Due to strong nonlinearity of the system, it is obvious that direct treatment is very difficult. Therefore, we resort to a multiple scale analysis to reduce the system to be dependent on time and one-dimensional space. The result are an amplitude equation, ^ t þ ikuwÞ ^ ¼ a2 w ^ ^ yy þ a3 ðr2 þ r0 Prhy Þw; a 1 ðw where a1 ¼ k 2 þ
r0 Pr ðk ic0 PrÞ2
;
a2 ¼ 2k 2 ikc0 þ
r0 ðk ic0 PrÞ2
; and a3 ¼
k k ic0 Pr
together with the following mean equations ^ yy Þ; and Prht ¼ hyy þ ^w ^ w ^ w ut ¼ uyy þ ikðw yy
2k 2 Pr ^ y Þ: ^w ^ þ w ^ w ðw y k 2 þ c20 Pr2
k, r0, and c0 are evaluated from the following relations, 1=6 1 2 2 Pr ð1 þ PrÞ ; r0 ¼ 9k 2 ; c0 ¼ 41=3 k 2 ; k¼ 2 and Ra = 4r, where r = r0 + 2r2 + with 1. This system is then solved numerically under the assumption that the boundaries are stress free and perfect conductors of heat, so the temperature perturbations vanishes there; namely, ^ ¼ ou ¼ h ¼ 0 at y = 0,1. The nonlinear solutions of this system when rapidly rotated exhibit a sequence of bifurcations w oy from steady solutions to periodic solutions, to tori and 2-tori, typically ending in chaotic behavior. These bifurcations are manifested on two branches of asymmetric solutions and one branch of symmetric solutions. As r2 increased along the symmetric branch, the symmetric chaos bifurcates into a symmetric 2-torus, Fig. 5. Also there are two asymmetric
Fig. 1. The asymmetric solution for u projected on the mean velocity plane showing one of the asymmetric 2-tori at r2 = 46.
A.A. Abdulrahman, E.A. Elrifai / Chaos, Solitons and Fractals 36 (2008) 25–31
27
aperiodic solutions which remain aperiodic for all the values r2 2 [40, 45]. As r2 is further increased the chaotic solutions restabilise into asymmetric 2-tori, one of these is shown in Fig. 1. Also frequency-locking was found when r2 2 [45.144, 45.153]. An example of a frequency locked solution at r2 = 45.15 is in Fig. 9, such these solutions continue to exist up to r2 = 46.7. The scheme of this paper is as follows, after the introduction we give some required terminologies and the main results. Then some examples and calculations. Finally we end the work with speculations.
2. Terminology and main results A knot is an embedded circle in S3 or R3. A link is a disjoint collection of knots. The unknot is a trivial circle S S2 S3. Two links L1 and L2 are ambient isotopy if there are a homotopy Ht:S3 ! S3, t 2 [0, 1] = I, such that H0 is the identity map, H1 sends L1 to L2 and Ht is a homeomorphism for every t 2 I. Knots and links are in the same knot and link type if and only if they are equivalent under ambient isotopy. All knots and links in this paper are oriented by the arrow of time. Knots and links are described geometrically via projections onto a plane in which over and under crossings are indicated by broken lines. A knot K is a composite knot if it is obtained by connecting diagrams of two non-trivial knots K1 and K2, the prime knot is a non-composite knot. For more details about knot and link theory, we refer to [14]. 1
Lemma 1. The asymmetric 2-tori in Fig. 1, is topologically equivalent to a standard double torus. Proof. The proof will be done combinatorially, by a continuous deformation. The two-dimensional manifold T, in Fig. 1, can be transformed to a standard double torus, as in Fig. 2. So the standard double torus is, topologically, an attractor for the closed periodic orbits of the rapidly rotating system. In fact this is a well-known result, where nonlinear systems can make a transition from quasiperiodic motion directly to chaos. The two asymmetric aperiodic solutions remain aperiodic for all the values 40 < r2 < 45. As r2 is further increased. The chaotic solutions destabilize into asymmetric 2-tori, one of these is shown at r2 = 46 in Fig. 1. A knot or link in S3 or in R3 is called a double torus knot or link (DTK), if L is contained in a Heegaard surface on genus two (an embedded surface in a closed 3-manifold M 3, which divides M 3 into two handlebodies [15]). The Heegaard surface of genus two is the standard double torus. Now let us give a formal definition of DTK as it is given in [15,16,17]. h Definition 1. Let K be a knot embedded in a double torus H = TL ] TR, which is the gluing of two once-punctured tori TL,TR along a circle C. On each tori KnC consists of at most three parallel classes, ðn1 ; n2 ; n3 ; n01 ; n02 ; n03 Þ for the numbers of the constituent arcs, then n1 þ n2 þ n3 ¼ n01 þ n02 þ n03 ¼ n. Let (p, q), (r, s) the slopes of the first and the second parallel classes of arcs in TL, where the slope of the third is (p + r, q + s). Also denote by (p 0 , q 0 ), (r 0 , s 0 ) the slopes of the
Fig. 2. A topological transformation of the asymmetric 2-torus in Fig. 1 to the standard double torus.
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two of the parallel classes in TR. Finally, in gluing the arcs along C, we have a choice, which is denoted by n < q 6 n. Then by arranging the above numbers as in K ¼ fðn1 ; n2 ; n3 ; n01 ; n02 ; n03 jqÞðp; q; r; sÞðp0 ; q0 ; r0 ; s0 Þg, we can express a double torus knot K, as in Fig. 3. Let K be a DTK with only one parallel classes of arcs on TL and TR, then K can be denoted by K = {(n, 0, 0; n, 0, 0jq)(p, q, ,)(p 0 , q 0 , ,)} and we say that K is of type (1, 1), [22]. Then from lemma 1 and definition 1, we have directly the following result. Corollary 1. Closed periodic orbits of the convective solutions in the rapidly rotating system, contain double torus knots and links. In fact double torus knots and links contains torus knots, 2-bridge knots, knots with genus one and bridge one, (1, 1) – decomposition tunnel number one knot, Berge’s doubly positive knots which are fibred, Dean’s twisted torus knots and Pretzel knots. Double torus knots is either of period 2 or strongly invertible and their tunnel number at most 2, [15,16]. As it is clear in Fig. 3, DTK can be obtained from two torus knots (each of which is embedded in a genus one surface) by gluing together along discs. But torus knots are fibred knots, so it is quite likely to fined fibred DTK. It is also known that a nonlinear system can make a transition from quasiperiodic motion directly to chaos, which is known as the quasiperiodic route to chaos. That was one of the first proposed mechanisms leading to the formation of a strange attractor. There are many routes to chaos, one of them is the formulation of a double torus, Fig. 4. In the torus doubling route to chaos, the original torus (which is a closed curve in a cross section) appears to split into two circles at the torus doubling bifurcation point [18]. Proposition 1. Torus knots and links do occur as closed orbits of the convective solutions in the rapidly rotating system. Proof. Numerically, it is got a symmetric 2-torus, T2, for u at r2 = 42, As in Fig. 5.
h
Fig. 3. A double torus knot, K{(3, 3, 3; 3, 3, 3j4) (1, 0, 1, 1)(0, 1, 1, 1)}.
Fig. 4. Schematic of a torus doubling bifurcation.
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Fig. 5. The symmetric 2-torus for u at r2 = 42.
Fig. 6. (a) A recurrent flow around a hyperbolic fixed point. (b) A close recurrence of a chaotic trajectory and gently adjusting a segment of the chaotic trajectory we can locate a nearby periodic orbit.
Remark 1. There are knots which are not DTK for example knots 816 and 817 are not DTK, where this enumeration as in Rolfson’s book [19]. But here we show that Lorenz knots are non-DTK. In fact the following result is an answer of question 6 in the open problems which are given in [16] by Hill and Murasugi. Theorem 1. There is no double torus Lorenz knots. Proof. It is known that the only possible satellites of a Lorenz knot are parallels with possible twists [7]. It is also known that Lorenz knots are prime [20]. But in [21] Osawa answered question 12 (a) in [16], where he classified satellite DTK and showed that if a satellite double torus knot is not a cable knot, then it has torus knot companion. On the other hand Lorenz knots are fibred, while fibred DTKs [17] are either band sum of two torus knots, which are composite knots, or a closure of some positive (or negative) braids, which are the Murasugi sum of several generalized Pretzel knots. Therefore there is no double torus Lorenz knots. h
3. Calculations Remark 2. Consider a three-dimensional flow in the vicinity of a hyperbolic orbit, Fig. 6a. Since the flow is recurrent, we can choose a surface of section in the vicinity of the fixed point. This section gives a compact map of the disk onto itself with at least one fixed point. In the vicinity of this fixed point a chaotic limit set, a horseshoe, containing an infinite number of unstable periodic points can exist. A single chaotic trajectory meanders around this chaotic limit set in an ergodic manner, passing arbitrarily close to every point in the set including its starting point and each periodic point. If we consider a small segment of the chaotic trajectory that returns close to some periodic points. Intuitively, we expect to be able to gently adjust the starting point of the segment so that the segment precisely returns to its initial starting point, thereby creating a periodic orbit, Fig. 6b, [23]. Now applying the algorithm in the above remark on the trajectories in Fig. 6, we have the knots in Figs. 7–9.
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Fig. 7. The symmetric 2-torus, Ts, for u at one selection of time frames showing its progression to final stage for the value r2 = 42. The horizontal is u (0.25, t) and the vertical is u (0.75, t).
Fig. 8. Knots as periodic orbits in the solution of rapidly rotating system.
Fig. 9. The frequency in the 2-torus is now locked at r2 = 45.15.
4. Speculations In this work we have find a torus doubling as a chaotic attractor. A central problem of topological analysis is to determine a template describing the structure of this attractor. Where the template is specified by the branched manifold and the semi flow that result from the Birman–Williams work [3,6]. In fact their work provides a bridge between phase space where chaotic attractors live and abstract space of equivalent classes where templates live. This can be down as
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resulting from smooth deformation which preserves periodic orbits and does induce crossings between them. Therefore knot and braid invariants of any set of periodic orbits in the real phase space and of its counterpart in the abstract space are identical. In a revolutionary and very surprising work of Ghrist [24], he proved that any knot or link can be realized as a periodic orbit on any one of three kinds of templates. As a future work, we hope to study double torus knots and links via theory of template.
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