Abstract
Period-2 badly ordered orbits (non-Birkhoff orbits) are studied in the standard mapping. Points of symmetric non-Birkhoff orbits appear on symmetry axes due to the saddle-node bifurcation, and non-symmetric non-Birkhoff orbits appear due to the equi-period bifurcation of symmetric non-Birkhoff orbits. The braids of symmetric non-Birkhoff orbits are constructed, and the topological entropy is estimated.
Let F : (x n , y n ) → (x n+1 , y n+1 ) be a monotone twist mapping of the cylinder (0 ≤ x < 2π, −∞ < y < ∞), where the condition for monotonicity is ∂x n+1 /∂y n > 0. 1) There exist two types of periodic orbits for this mapping. Suppose that u, v ∈ O q , where O q is the set of points belonging to a periodic orbit of period q in the universal cover of the cylinder. Let π x (u) be the x coordinate of u and letF be a lift of F . If the relation
holds, the periodic orbit is said to be Birkhoff. If this relation does not hold, such orbits are said to be non-Birkhoff. 2)−5) If, in addition, the mapping possesses reversibility, then the orbits can be classified differently, as symmetric and nonsymmetric ones. Correspondingly, there are symmetric and nonsymmetric periodic orbits. Thus, given a reversible monotone twist mapping, we have four kinds of periodic orbits: symmetric Birkhoff, nonsymmetric Birkhoff, symmetric non-Birkhoff, and non-symmetric non-Birkhoff orbits. In a previous paper, 5) we studied non-Birkhoff orbits of period-3 in standard-like mappings. A system containing such orbits is pseudo-Anosov and possesses chaotic motion with positive topological entropy.
In this paper, we consider the standard mapping as a typical example of reversible monotone twist mappings and study the properties of the mapping associated with symmetric and non-symmetric non-Birkhoff period-2 orbits. It is well-known that period-2 Birkhoff orbits exist for any parameter value, including the integrable situation (a = 0). Non-Birkhoff periodic points, which exist for non-integrable parameters, do not exist in the integrable situation. They appear through a tangent bifurcation. Interestingly, as we show in this paper, non-Birkhoff period-2 orbits, which have no points on the symmetry axes, are born from symmetric non-Birkhoff orbits through an equi-period bifurcation. In other words, non-symmetric period-2 orbits are never born from non-symmetric ones. Initially they are all symmetric. We confirm numerically that this applies to period-3 orbits. We suspect that all

Let F : (x n , y n ) → (x n+1 , y n+1 ) be a monotone twist mapping of the cylinder (0 ≤ x < 2π, −∞ < y < ∞), where the condition for monotonicity is ∂x n+1 /∂y n > 0. 1) There exist two types of periodic orbits for this mapping. Suppose that u, v ∈ O q , where O q is the set of points belonging to a periodic orbit of period q in the universal cover of the cylinder. Let π x (u) be the x coordinate of u and letF be a lift of F . If the relation
holds, the periodic orbit is said to be Birkhoff. If this relation does not hold, such orbits are said to be non-Birkhoff. 2)−5) If, in addition, the mapping possesses reversibility, then the orbits can be classified differently, as symmetric and nonsymmetric ones. Correspondingly, there are symmetric and nonsymmetric periodic orbits. Thus, given a reversible monotone twist mapping, we have four kinds of periodic orbits: symmetric Birkhoff, nonsymmetric Birkhoff, symmetric non-Birkhoff, and non-symmetric non-Birkhoff orbits. In a previous paper, 5) we studied non-Birkhoff orbits of period-3 in standard-like mappings. A system containing such orbits is pseudo-Anosov and possesses chaotic motion with positive topological entropy.
In this paper, we consider the standard mapping as a typical example of reversible monotone twist mappings and study the properties of the mapping associated with symmetric and non-symmetric non-Birkhoff period-2 orbits. It is well-known that period-2 Birkhoff orbits exist for any parameter value, including the integrable situation (a = 0). Non-Birkhoff periodic points, which exist for non-integrable parameters, do not exist in the integrable situation. They appear through a tangent bifurcation. Interestingly, as we show in this paper, non-Birkhoff period-2 orbits, which have no points on the symmetry axes, are born from symmetric non-Birkhoff orbits through an equi-period bifurcation. In other words, non-symmetric period-2 orbits are never born from non-symmetric ones. Initially they are all symmetric. We confirm numerically that this applies to period-3 orbits. We suspect that all non-symmetric periodic orbits bifurcate from symmetric periodic points. We derive a dynamical order relation for non-Birkhoff period-2 orbits of different rotation numbers. We construct braids for these orbits and calculate a lower bound of the topological entropy. The topological entropy of the standard mapping tends to infinity with parameter a.
The standard mapping T (x, y) is given by
where f (x) = a sin x and a ≥ 0. The mapping T is expressed as a product of two involutions G and H: 6),7), * )
The sets of fixed points of G and H are called the 'symmetry axes'. The involution G has symmetry axes S 1 and S 2 , and H has symmetry axes S 3 and S 4 :
These axes correspond to the left-right symmetry of the orbit. * * ) In the following, S + i (resp. S − i ) denotes the line segment located in the region satisfying y > 0 (reps. < 0). There exist infinitely many symmetry axes in the universal cover.
In what follows, we use x and y arbitrarily for the coordinates of the universal cover. We characterize period-2 orbits by their rotation numbers ν = p/2 with p ≥ 0. We denote a period-2 orbit by {p 1 , p 2 }, where p 1 = (x 1 , y 1 ) and p 2 = (x 2 , y 2 ). Without loss of generality, we assume y 1 > 0.
The following equations are the general conditions for an orbit to be of period-2:
Hereafter we only consider non-Birkhoff periodic orbits. The condition that a period-2 orbit is non-Birkhoff is x 2 < x 1 . This is equivalent to
According to Eqs. (10) and (11), no point of a non-Birkhoff orbit of period-2 is on S 1 or S 2 . Then, obviously, we have
Thus we have two solutions for period-2 orbits,
where m ≥ 0. Equation (13) gives symmetric non-Birkhoff orbits (SNBO). Indeed, we obtain Gp 1 = p 2 = T p 1 = HGp 1 . Then p 2 is on a symmetry axis of H, i.e., on S 3 or S 4 . We have p 3 = T p 2 = HGp 2 = Hp 1 . Since p 3 = p 1 on the cylinder, p 1 is also on the symmetry axis of H, i.e., on S 3 or S 4 . Hence, this is a symmetric non-Birkhoff orbit (SNBO) on S 3 or S 4 . Equation (14) gives non-symmetric non-Birkhoff orbits (NSNBO).
The two symmetry axes S 3 and S 4 of the universal cover are expressed as
where even integers give S 3 and odd integers give S 4 . Below we comment on the solutions for k = −1. There are two types of solutions. The first type is the period-2 orbit with p = 0 bifurcated from Q, due to the period-doubling bifurcation at a = 4. The second type with p > 0 does not satisfy Eq. (11). The solutions for k ≤ −2 do not satisfy the assumptions. Using the condition p 1 ∈ S 3 or p 1 ∈ S 4 , Eqs. (10) and (11) can be rewritten
Combining Eqs. (16) and (17), we obtain the maximum value of p, and then we have the relation
For example, NBOs with ν = 1/2(k = 2), 3/2(k = 3) and 5/2(k = 4) appear at the same value of a. This fact results from Eqs. (16) 
Using Eq. (19), we can determine the dynamical order relation for SNBOs with ν = p/2 (see Table I ).
Next we study NSNBOs determined by Eq. (14). We have the equation to
We have the critical value a ns c at which non-symmetrical non-Birkhoff period-2 orbits appear, and have x i and y i (i = 1, 2) at a = a ns c as follows: 
These solutions also satisfy Eq. (13). This means that NSNBO appears due to the equi-period bifurcation of the SNBO. Since p 1 is located on the symmetry axis expressed by Eq. (15), m is obtained as
Here, we study the stability of NSNBOs using a linear analysis. The trace of the linearized matrix is R(a) = 2 − a 2 cos 2 x 1 , where Eq. (14) has been used. Increasing a from a ns c , one solution of x 1 increases and the other solution decreases from 3π/2. Thus R(a) decreases from 2. This implies that NSNBOs are elliptic and thus that the mother point on the symmetry axis changes from a stable elliptic point to an unstable saddle.
The geometrical properties of NSNBOs are the same as those of SNBOs, since NSNBOs appear due to the equi-period bifurcation of SNBOs on the symmetry axis. The bifurcated NSNBO gives the same braid as the mother SNBO. Thus it is sufficient to construct the braid of SNBOs to estimate the topological entropy.
The turning back occurs during the transition from p 1 to p 2 . The rotation number is approximately equal to −α−(k+1−p), where 0 < α < 0.5 for a s c < a < a ns c , α = 0.5 for a = a ns c and 0.5 < α < 1 for a > a ns c . The rotation number for the transition from p 2 to p 3 [= p 1 (Mod2π)] is p + α + (k + 1 − p). Using these facts, we can construct the braid. The first step is the deformation of the cylinder to an annulus. Next, we shrink an inner circle to a point (termed c). The point c is a fixed freynolds@example.net.
We construct the braid of three strings connecting two annuli. 5) Let the string from p 1 to p 2 be A, that from p 2 to p 1 be B, and that from c to c be C. The two strings A and B revolve round c. This determines how two strings intersect (A and C, B and C). If A and B intersect each other, we draw the string of the forward rotation behind the string of the backward rotation. This intersection corresponds to the twist property of T . Figure 1 shows how to construct the braid with p = 1 and k = 2. This braid is separated into three blocks. The 
We have the braid β p,k given by
The component σ
is expressed as θ 
The maximum of the absolute values of the eigenvalues gives the topological entropy. 10),11) It is obtained in the case t = −1. This can be proved by repeating the calculation given in the Appendix of Ref. 5). The topological entropy h is obtained as h ≥ ln (n + 1 + n 2 + 2n).
The fact that the topological entropy is positive implies that the system with nonBirkhoff period-2 orbits is pseudo-Anosov. Their minimum entropy is larger than ln 3, and thus these orbits cannot be embedded in a two-fold horseshoe. This implies the existence of strong chaotic motion in the system and the existence of an n-fold horseshoe with n ≥ 4. Formulating an interpretation of chaos in terms of n-fold horseshoes is a future problem.

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