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Another look at the saddle-centre bifurcation: Vanishing twist
Physica D 211 (2005) 47–56
Another look at the saddle-centre bifurcation: Vanishing twist H.R. Dullin∗ , A.V. Ivanov Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK Received 13 February 2005; received in revised form 25 July 2005; accepted 28 July 2005 Available online 22 August 2005 Communicated by C.K.R.T. Jones
Abstract In the saddle-centre bifurcation a pair of periodic orbits is created “out of nothing” in a Hamiltonian system with two degrees of freedom. It is the generic bifurcation with multiplier one. We show that “out of nothing” should be replaced by “out of a twistless torus”. More precisely, we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits. Vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes. We explicitly derive the position of the twistless torus in phase and in parameter space near the saddle-centre bifurcation. The theory is applied to the area preserving H´enon map. © 2005 Elsevier B.V. All rights reserved. Keywords: Twist maps; Hamiltonian systems; Saddle-centre bifurcation; KAM; Normal forms; Elliptic integrals
1. Introduction
there exist symplectic angle-action coordinates (ϕ, I) on this plane such that the mapping reads
The dynamics near a periodic orbit in a Hamiltonian system with two degrees of freedom is described in linear approximation by its Floquet multipliers µi , i = 1, 2 where µ1 = µ ¯ 2 ∈ C. For an elliptic orbit the multiplier is on the unit circle and can be written as µ = exp(2πiω), where the rotation number ω satisfies 0 < ω < 1/2. Higher order approximations are constructed using the Birkhoff normal form. This can be done by consideration of the Poincar´e map on a local transverse plane in the energy surface. When the rotation number ω of the periodic orbit is irrational
(ϕ, I) → (ϕ + 2πΩ(I), I)
∗
Corresponding author. E-mail address: [email protected] (H.R. Dullin).
0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2005.07.019
(1)
up to arbitrary high order. Iterating the map preserves the invariant circles I = constant and the dynamics on each circle is given by a rotation by 2πΩ(I). The rotation number (or winding number) Ω(I) can be expanded near the periodic orbit I = 0 as Ω(I) = ω + τ0 I + 21 τ1 I 2 + · · · The twist (or torsion) τ(I) is the derivative of the rotation number with respect to the action, τ(I) =
dΩ (I) = τ0 + τ1 I + · · · dI
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A mapping of the form (1) is called a monotone twist map if the twist is non-zero in the region of interest. In that case Moser’s KAM theorem [16] shows that the invariant circle I = I0 with diophantine rotation number Ω(I0 ) and non-vanishing twist τ(I0 ) persists under perturbations. In particular such perturbations can be the neglected higher order terms in the normal form. The analogous statement for flows is Arnold’s KAM theorem [1] in Hamiltonian systems with two degrees of freedom, in which the isoenergetic non-degeneracy condition is the equivalent of the twist condition. In this paper we will show that the twist condition is violated near the saddle-centre bifurcation of periodic orbits (or fixed/periodic points of an area preserving map). The dynamical consequences of violating the twist condition have first been described by Howard and Hohs [11], and the resulting effects have been observed in many examples, see, e.g. [18,15]. Probably the most spectacular effect is the appearance of so-called meandering curves [11,10,17], which are invariant tori that are not graphs over any unperturbed torus. A renormalisation theory for twistless tori has been worked out in [2,3] and more recently it has been shown [4] that an extension of the KAM theorem can also be proved in this context. Even though the phenomenon had been observed in many examples it was not clear how typical it is. In [9,13] it was shown that the twist of a fixed point in a family of area preserving maps vanishes for some rotation number ω near 1/3. At this point a twistless torus is born in a twistless bifurcation [9]. Whenenver this torus passes through a rational winding number meandering curves appear. A similar connection between resonances and vanishing twist holds true in four-dimensional symplectic maps [8]. The techniques of [9] can also be applied to higher order resonances, in particular for ω = 1/4, where a similar result is conceivable. For flows there is the interesting case of the 1:−1 resonance or Hamiltonian Hopf bifurcation, which was shown to produce a family of twistless tori in [7]. In [6] we announced that a similar result holds for the 1:1 resonance in maps, while for the 1:2 resonance in maps the twist does not vanish. Here we are going to give the proofs for the more interesting case of the 1:1 resonance. The main result is encapsulated in a new bifurcation diagram for the saddle-centre bifurcation in area-preserving maps, see Fig. 4. In addition to the stable and unstable branch meeting at the bifurcation it has an additional branch on the other
side of the bifurcation that gives the position of the twistless torus in phase space. This curve has universal shape relative to the other two. The universal frequency of the corresponding twistless curve is also derived. The observation that a twistless curve is destroyed/created in a saddle-centre bifurcation of periodic orbits was first made in [2]. In this work the saddle-centre bifurcation of symmetric periodic orbits (called hyperbolic–elliptic collision in [2]) in the standard nontwist map was used to give bounds for the existence of twistless curves. Other types of orbit collision, e.g. hyperbolic–hyperbolic, were also found in [2] and more recently [19] found so called non-standard reconnection and collision scenarios. These bifurcations are not generic and appear because of the discrete symmetries of the standard nontwist map [14]. We restrict ourselves to the generic saddle-centre bifurcation and obtain analytic results for position and frequency of the nearby twistless curve. The fact that this is a generic bifurcation implies that the resulting twistless curves are not only found in the standard nontwist maps, but in any generic one-parameter family of maps. Even though our main statement is about the saddlecentre bifurcation of periodic orbits in two degrees of freedom conservative systems, most of the analysis is done in one degree of freedom. The conclusions are specific to two degrees of freedom, however, because the concept of twist does not exist in one degree of freedom. Results about the period of the corresponding one degree of freedom system have been obtained by [5], in the context of an adiabatic perturbation of the one degree of freedom saddle-centre bifurcation. But as far as we are aware our exact results are even new for the case of one degree of freedom. In this setting we determine the slowest orbit near the bifurcation point but before the bifurcation occurs. In panel (c) in Fig. 1 it is the one shown with the closest approach to the origin. The paper is organised as follows. We introduce the normal form for the saddle centre bifurcation in the next section. The main theorem on the vanishing of the twist is presented in Section 3. In the following section we show that higher order terms in the normal form do not change our results, and thus there are certain universal constants that can be computed in terms of complete elliptic integrals associated with this bifurcation. In the final section we treat the non-integrable area preserving H´enon map as an example.
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Fig. 1. Lines of constant energy on the phase space (u, v) for the saddle-centre bifurcation, h = 0.1. (a) ε = −0.1, (b) ε = 0, (c) ε = 0.1. For ε = 0.1 the invariant curve closest to the origin has vanishing twist.
2. Saddle-centre bifurcation The normal form of an area preserving map of the plane near a generic bifurcation is integrable [12]. The in general non-integrable mapping of the plane is approximated near the bifurcation by the time one map of an integrable Hamiltonian flow with one degree of freedom. By studying this one degree of freedom system we thus understand the behaviour near the bifurcation in the original non-integrable map of the plane. This in turn can be viewed as the Poincar´e map of a Hamiltonian system with two degrees of freedom on a local transversal section of a periodic orbit of the flow. The Hamiltonian whose time one map generates the map is given by [12] H(u, v) =
v2 2
+
u3 3
+ εu.
(2)
The bifurcation parameter is ε. We must keep in mind that this one degree of freedom system gives an approximation to the area preserving map we are interested in.
The level lines of H are shown in Fig. 1. These are the invariant curves of the area preserving map. The critical points of the energy map H : R2 → R and their dependence on ε give the main structure of √ the bifurcation. The critical points are (u, v) = (± −ε, 0) with critical values h = ∓2(−ε)3/2 /3. The upper sign corresponds to a local minimum of H, while the lower sign gives a saddle. The critical values and their dependence on ε are shown in Fig. 2. Proposition 1. The rotation number Ω(h, ε) of the invariant curve H(u, v; ε) = h near a saddle-centre bifurcation in an area preserving map of the plane is given by the complete elliptic integral √ 1 3/2 1 = 1/4 dz Ω(h, ε) |ε| β w
(3)
along the real cycle β of the elliptic curve E : w2 = P3 (z)= 2γ − z3 − 3σz,
σ = sign(ε).
(4)
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In the original variables the curve ∆ = 0 of colliding roots is given by 9h2 = −4ε3 ,
Fig. 2. Schematic sketch of the (ε, h) bifurcation diagram for the saddle-centre bifurcation. Graphs of P3 are shown together with a horizontal line indicating the value of h. The bold lines marks the critical values of the saddle, the thin lines those of the centre.
The single continuous parameter γ of E is given by γ=
3h . 2|ε|3/2
(5)
Proof. The period of the reduced system is obtained by separating variables in u˙ = v(h, u), giving du T (h, ε) = . (6) 2h − (2/3)u3 − 2εu Since the time one map of the flow of the Hamiltonian gives the map we are studying, the rotation number of the invariant curve H = h of this map is given by Ω = 1/T . By scaling u = z(σε)1/2 , where σ = sign(ε) and introducing the one essential parameter γ, the period T becomes an elliptic integral on the curve E. The essential integral now reads 1 S(γ) = dz (7) w β and combining the formulas gives the result for Ω.
The polynomial P3 has one or three real roots, depending on the sign of the discriminant ∆ = −108(σ + γ 2 ).
(8)
and it is shown in Fig. 2. The collision of roots occurs at the critical values of H. In the cuspoidal region to the left there are three real roots, otherwise there is only one. This is the bifurcation diagram of the saddle-centre bifurcation. When considering two degrees of freedom integrable flows this would be the image of the energymomentum map. As such it is the natural domain on which the rotation number should be studied. The correspondence between the phase portraits Fig. 1 and the bifurcation diagram Fig. 2 is as follows. For ε > 0 there is only one family of invariant curves (or tori) and there is always only one root in P3 . The torus corresponds to the interval where P3 > 0, and this therefore also defines the real cycle β. For ε < 0 there are three families of tori in the phase portrait, inside the loop, or outside to the left or to the right of the separatrix. The only tricky bit in the mapping to the bifurcation diagram is that the tori to the left of the separatrix are below the bold line in the bifurcation diagram and continue smoothly outside the cuspoidal region. Stated otherwise, for each point in the cuspoidal region there are two tori in the preimage, while otherwise there is only one. A contour plot of the rotation number Ω(ε, h) is shown in Fig. 3. As expected the rotation number has a singularity along the curve of the saddle critical values (bold in Fig. 2), while it is smooth across the centre critical values (thin in Fig. 2). The reason for this latter fact is that the curve is elliptic. Thus it only has one real cycle. Since additionally the differential of T is of first kind the real integrals over the two disjoint real intervals for ∆ > 0 are identical.
3. Vanishing twist Vanishing twist occurs where the rotation number has a critical point within a family of invariant curves of an area preserving map of the plane. In Fig. 3 this condition means that the contour lines have a vertical tangent, as indicated by the dashed line. Our main result gives the precise location of this curve. Theorem 2. In the saddle-centre bifurcation of an area preserving map the twist is vanishing on a curve
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has no fixed point, ε > 0. This implies ∆ < 0. All of the following is valid for ∆ < 0 only. The integrand w is positive for z ∈ (−∞, z0 ), where z0 is the single root of P3 (z). Let the factorized polynomial be given by P3 (z) = −(z − z0 )P2 (z),
(11)
P2 (z) = (z − ζ1 )2 + ζ22 ,
so that the complex roots are ζ1 ± iζ2 . Denote the distance between the real and complex roots by r, hence r2 = P2 (z0 ). In this notation the discriminant of P3 is given by ∆ = −4r4 ζ22 . The modulus k of the elliptic integrals is given by z 0 − ζ1 1 1 sign(γ) 2 k = 1+ , (12) = 1+ √ 2 r 2 1 + α2
Fig. 3. Lines of constant rotation number Ω = 1/T equidistant with Ω = 0.2 on the parameter plane (ε, h) for the saddle-centre bifurcation. The vanishing twist is indicated by the dashed curve of vertical tangents ∂Ω/∂h = 0.
in the parameter plane (ε, h) given by 3h = 2γ0 ε3/2 .
(9)
for positive ε where γ0 ≈ 0.9152203 is determined from the root of a transcendental equation involving complete elliptic integrals. Proof. By definition the twist is given by the derivative of the rotation number Ω with respect to the action 2πJ = vdu. Since Ω = 1/T and 2πT = ∂J/∂h the J derivative of the rotation number is ∂Ω ∂Ω/∂h ∂T/∂h = =− . ∂J ∂J/∂h 2πT 3
(8k4 − 9k2 + 1)K(k) = (16k4 − 16k2 + 1)E(k). This equation has a single non-trivial root k = k0 where k02 ≈ 0.7097215. The proof of the Lemma is given in the Appendix A. In order to calculate the parameter γ0 in Theorem 2 corresponding to k0 we observe that α as introduced in (12) is related to γ by ζ2 = − 3ζ = √1 ΓΓ 2 +σ , −σ 1 3 Γ = (γ + σ + γ 2 )1/3 .
α=
Hence the twist vanishes when ∂T (3/2)3/2 ∂S = =0 ∂h ε7/4 ∂γ and this is only possible for finite ε when ∂S 1 =− dz = 0. 3 ∂γ w β
and in the last equality the parameter α = ζ2 /(z0 − ζ1 ) has been introduced. In the second equality, in addition sign(z0 − ζ1 ) = sign(γ) is used, which is true because z0 = ζ1 in (11) together with the vanishing of the quadratic coefficient in (4) implies z0 = ζ1 = 0, and therefore the polynomial has no constant term and h = γ = 0. The solution of (10) is given by our main technical Lemma 3. The integral β (1/w3 )dz vanishes when
ζ2 z0 −ζ1
2
(13)
Now (12) relates α0 to k0 as given in Lemma 3. The corresponding value of α is α0 ≈ 2.164255. Finally inverting (13) we find γ0 ≈ 0.9152203. (10)
This is an elliptic integral of the second kind on the curve E. The relevant case is where the phase portrait
Since γ0 < 1 the curve of vanishing twist is bent downward as compared to the bifurcation curve (8) for ε < 0 and h > 0. See Fig. 3 for a graph of this curve together with the numerically computed lines of constant rotation number. The lines of constant rotation
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number have vertical slope at their intersection with the twistless curve, as must be the case. The scaling of u reduces the number of parameters to one. The essential parameter γ, in its dependence on h and ε, organizes the bifurcation. It allows us to compute explicitly all the important characteristics of the twistless torus. Combining (12) and (13) shows that k is a function of γ. The curves in the parameter plane that have the same value of γ are given by (5). The most prominent ones are the curve of critical values γ = ±1, as shown in Fig. 2, and γ = γ0 , the curve of twistless tori, see Fig. 3. They all have the same shape of a semicubical parabola, except when γ = 0, hence h = 0, or γ = ±∞, hence ε = 0 with ±h > 0. k2
γ
α
Curve
1
+1 +∞ 0.91522 0 −∞ −1
0
3h = −2(−ε)3/2 ε = 0, h > 0 3h = 2γ0 (−ε)3/2 h = 0, ε > 0 ε = 0, h < 0 3h = 2(−ε)3/2
√ + 3) 0.709721497 1 4 (2 1 2 1 4 (2
−
√ 3)
0
√1 3
2.164255 ∞ − √1 3 0
Anywhere along the curve (5) the modulus k(γ) is constant. The function k(h, ε) is therefore not continuous at the origin. Moving on (h, ε)-curves given by (5) for any γ = 1 the change in the period T (h, ε) is elementary. From (3) it follows that T is proportional to |ε|−1/4 , and the constant of proportionality is determined from (3) and (7). The divergence of the period upon approaching the bifurcation from ε > 0 is therefore not caused by the elliptic integral, but merely by the algebraic dependence |ε|−1/4 . Similarly the rotation number of the stable fixed point of the map is given by 1 Ω(γ = −1, ε) = √ (−ε)1/4 ≈ 0.225079|ε|1/4 . 2π The integral S(−1) contained in this expression can be easily calculated using residue calculus because for γ = −1 the curve has a double root, P3 (z) = −(z + 2)(z − 1)2 . For general values of γ and in particular for the twistless torus, the elliptic integral S(γ) (7) needs to be evaluated. For γ = γ0 the single real root is given by z0 = Γ −
σ , Γ
hence z0 (γ0 ) ≈ 0.5535942.
(14)
Fig. 4. Bifurcation diagram of the position in phase space u vs. the bifurcation parameter ε. For ε < 0 the location of the fixed points at √ ± −ε are shown, while for ε > 0 the maximal u of the twistless √ torus at z0 (γ0 ) ε is shown. Compare Fig. 1.
Finally S(γ) is obtained in terms of the standard elliptic integral of first kind K as S(γ) =
4K(k) . √ 3|z0 | 1 + α2 /2
Using the above value of z0 , the position of the rightmost point of the√twistless torus in phase space is located at u0 = z0 ε. This means that for ε = ε0 this point is on the same side as the stable periodic orbit was for ε = −ε0 before the bifurcation, but by a factor of z0 (γ0 ) ≈ 1/2 closer to the origin. See Fig. 4 for an illustration in the form of a standard bifurcation diagram showing position in phase space u versus bifurcation parameter ε. Thus we have proved Proposition 4. The rotation number of the twistless curve is z0 (γ0 ) 1 + α20 Ω(γ = γ0 , ε) = ε1/4 4K(k0 ) ≈ 0.1374244ε1/4 .
(15)
The position of the twistless curve in phase space as determined by its intersection with v = 0 is given by √ √ u0 = z0 ε ≈ 0.5535942 ε (16) The formulae (15) and (16) will be illustrated by the area preserving H´enon map in Section 5.
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4. Universality There are two questions to be sorted out. First of all, the invariant curves of the normal form are not compact, so that it is not clear what the rotation number (or period) means. Secondly, the influence of higher order terms needs to be studied. The two problems are related: Adding higher order terms in the normal form can compactify the invariant curves. What we will show is that in the limit ε → 0 from above, the leading order diverging contribution to the period T comes from the term that we analysed in the previous sections. In particular this means that the constant γ0 that determines the shape of the curve of twistless tori in relation to the curve of critical values of the unstable orbits has the universal value γ0 ≈ 0.91522. Quantities derived from γ0 , like z0 and the coefficients in (15), are accordingly also universal. Let the high-order truncated normal form Hamiltonian (whose time one map generates the Poincar´e map) be ˜ H(u, v; ε) = 21 v2 + 13 u3 + εu + G(u, ε) where G(0, ε) = 0 and G(u, 0) = 0 and G is an analytic function. We assume that G(u, ε) is such that the invariant curves near the origin for ε > 0 are compact. As ˜ usual the period is obtained by solving H(u, v; ε) = h for v = v(u; h, ε) and then by integrating du T˜ (h, ε) = β v(u; h, ε) along β, which projects to the values of u for which v is real. The integral is now split in two ways. First of all the integration path β is decomposed into a part β0 that is close to zero, and in all the rest β1 . Moreover, the expression under the square root is factored as ˜ h, ε) v(u; h, ε)2 = Q(u; h, ε)Q(u;
53
The first integral T1 is an incomplete elliptic integral. Specifying for β0 that it be contained in the interval [−d, d] it can be integrated explicitly to √ 4 3/2 d+C T1 (h, ε) = 1/2 F (φ, k), φ = 2 arctan , r r where r = 2d 2 + |ζ|2 ,
2k2 = 1 +
3d . 2r
The main observation is that the integral T1 does not diverge because the modulus k2 tends to one, but because in the prefactor r tends to zero, and this independently of the size of the neighbourhood d. In fact, we already observed that inside the first √ quadrant ε ≥ 0, h > 0 the modulus satisfies k2 ≤ ( 3 + 2)/4 < 1. The behaviour of r for small ε is √ r 2 = 23 d 1 + α2 = D(γ)ε1/4 . This shows that φ → π, and the incomplete integral T1 approaches the complete integral T with the same modulus k(γ) as before. So even though T1 is an incomplete integral, in the limit of small ε it approaches the complete integral T (h, ε), which diverges. The other integrals T2 and T3 are finite. Since the rotation number Ω is determined by the period T a similar conclusion holds for Ω. Thus we have proved Proposition 5. The results obtained from the analysis of the normal form with non-compact invariant curves (G = 0) correctly describes the behaviour of the rotation number of the compact invariant curves (G = 0).
5. Example: H´enon map
The H´enon map in the area preserving case, where Q(u; h, ε) is the old expression (6). As a result (x , y ) = (y − κ + x2 , −x), we find du du √ √ T˜ (h, ε) = + Q(u, ε, h) Q(u, ε, h) β0 β1 ˜ −1/2 du −1 Q(u, h, ε) du √ √ √ = + du + = T1 + T 2 + T 3 . Q(u, ε, h) Q(u, ε, h) Q(u, ε, h) β0 β0 β1
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illustrates the above, in particular the predictions made in Proposition 4. There are saddle-centre bifurcations in the H´enon map for which the invariant tori before the bifurcation are not compact, and hence the vanishing twist cannot be observed. This applies to the initial bifurcation at κ = −1 that creates the pair of fixed points, and also to many saddle-centre bifurcations that occur for κ > 4. However for κ = 1 a pair of period 3 orbits is created in a saddle-centre bifurcation of the third iterate of the map. One point of this orbit is at the origin for κ = 1. The invariant tori for ε < 0 (before the bifurcation) are compact near the bifurcation point. Hence there will be a torus with vanishing twist. The family of twistless tori for κ < 1 continues down to κ = 9/16, where the central elliptic fixed point of the H´enon map has vanishing twist. At this end the family of twistless curves has been studied in [9]. The present work confirms a conjecture [9] that the family of twistless curves created when the twist of the fixed point vanishes for κ = 9/16 ends in the saddlecentre bifurcation of the period 3 orbit for κ = 1. Since the H´enon map is non-integrable the family of twistless tori is not continuously parametrized by κ, but instead it is a cantorset determined by the condition that the frequency of the twistless curve is sufficiently irrational. One of the period three points is located on the symmetry line x = −y. In new coordinates (u, v) = (x, y + x) the third iterate of the map expanded near the origin with parameter κ = 1 − ε is (u , v ) = ((u − v)(1 + 4v), v + ε +(u2 − 2uv + 3v2 )) + O(u3 , v3 , |ε|3/2 ). The (exact) location of the bifurcating fixed points √ is (u,√v) = (± −ε, 0), with trace of the Jacobian 2 ∓ 2 −ε + 8ε ∓ 8(−ε)3/2 for the elliptic (−) and hyperbolic√(+) orbit. The (approximate) multiplier is µ ≈ 1 + i 2(−ε)1/4 and the corresponding rotation number ω of the elliptic fixed point is obtained from √ µ = exp(2πiω), so that ω ≈ (−ε)1/4 /( 2π). For small positive ε the H´enon map possesses compact invariant curves near the origin. A higher order normal form would give a G that describes these invariant curves. The H´enon map is non-integrable, so that only sufficiently irrational invariant curves of the (high order) normal form will exist in the H´enon map. For the situation under consideration numerical experiments
Table 1 Numerically measured position u and rotation number Ω of the twistless curve for the three times iterated H´enon map with parameter κ =1−ε ε
uε−1/2
Ωε−1/4
0.1 0.01 0.001 0.0001 0.00001 0.000001
0.027 0.328 0.478 0.528 0.545 0.550
0.2831 0.1728 0.1466 0.1401 0.1382 0.1377
The values converge to the numbers given in Proposition 4.
show that many of these invariant curves do exist. Considering the third iterate of the H´enon map turns the pair of period three orbits into three pairs of fixed points with heteroclinic connections. In the normal form there is only one pair of fixed points, and the unstable fixed point has a homoclinic connection. To match the prediction in the case of more than one unstable fixed point the period and hence rotation number must be calculated for the heteroclinic connection. For ε > 0 there are invariant tori on which the dynamics becomes slow near the three points that are close to the three bifurcation points of the third iterate of the map. Hence the rotation number for the third iterate of the map between two successive such points gives the correct rotation number. The results together with the position of the twistless curve are shown in Table 1. The values shown converge to the predicted values given in Proposition 4, however, fairly small ε are needed to see this. The convergence to the true value 0.1374244 . . . occurs approximately as (1/2) ε5/4 .
Acknowledgements This research is funded by the EPSRC under contract GR/R44911/01. Partial support by the European Research Training Network Mechanics and Symmetry in Europe (MASIE), HPRN-CT2000-00113, is also gratefully acknowledged. AVI was also supported in part by INTAS grant 00221, RFBI grant 01-01-00335 and RFME grant E00-1-120.
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Appendix A Proof (Lemma 3). The complete elliptic integral ∂S =− ∂γ
1 dz = 0 w3
can be written as a linear combination of integrals of the first and second kind. Legendre’s standard integral of the first kind K(k) has differential ω1 = √
1 dz P3 (z)
(17)
up to a constant factor. A non-standard form of the differential of Legendre’s standard integral of the second kind E(k) is ω2 =
P2 (z)dz ω1 , (z − (z0 ± r))2
up to the same constant factor as in (17). The differential dz/w3 we are interested in is of the second kind, and can therefore be written as a linear combination of ω1 and ω2 with constant coefficients, up to a total differential: dz = Aω1 + Bω2 + dF (18) w3 where F = Q2 (z)/((z − (z0 ± r))w). Together with the undetermined coefficients of the quadratic polynomial Q2 this gives a system of five linear equations for the five unknown coefficients. Solving these equations gives 2r 4r ((z0 − ζ1 )r − r 2 + 4ζ22 ), B = (r 2 − 4ζ22 ) ∆ ∆ Since the quadratic coefficient of P3 is zero, the roots of P3 add up to zero. Therefore, the real parts satisfy z0 + 2ζ1 = 0, hence
A=
2k2 = 1 −
3ζ1 r
and
r 2 = 9ζ12 + ζ22 .
With these equations the coefficients A and B can be expressed in terms of k alone, up to the factor (r 2 ∆)−1 . Integrating (18) gives the condition of vanishing twist, ∂S/∂γ = 0 in terms of A and B as (8k4 − 9k2 + 1)K(k)= (16k4 − 16k2 + 1)E(k),
(19)
55
where K(k) and E(k) stand for elliptic integrals of the first and the second kind, respectively. Now we prove the existence of a real positive root in the interval (0, 1). For k = 0 we have equality since both elliptic integrals equal π/2. The first derivatives on either side vanish, but the second derivatives are −35π/4 and −65π/4, respectively, so that the left hand side of (19) is larger for small k. For k = 1 the prefactor of K vanishes, while that of E gives 1 and E(1) = 1. Hence for k → 1 the right hand side dominates. This proves that there exists a solution of this equation for k ∈ (0, 1). Numerically we find k02 ≈ 0.7097215. √ Expansion of the equation defining k0 at k2 = 1/ 2 gives the approximation k02
√ √ 1 2 (19 + 14 2)K − (34 − 25 2)E √ √ ≈√ + 2 5 (18 − 13 2)K − 8(4 − 3 2)E
which is correct to 6 decimal places. Here the √ modulus for K and E is the expansion point k2 = 1/ 2.
References [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978. [2] D. del Castillo-Negrette, J.M. Greene, P.J. Morrison, Area preserving nontwist maps: periodic orbits and transition to chaos, Physica D 91 (1) (1996) 1–23. [3] D. del Castillo-Negrette, J.M. Greene, P.J. Morrison, Renormalization and transition to chaos in area preserving nontwist maps, Physica D 100 (3–4) (1997) 311–329. [4] A. Delshams, R. de la Llave, KAM theory and a partial justification of Greene’s criterion for nontwist maps, SIAM J. Math. Anal. 31 (6) 1235–1269 (2000) (electronic). [5] D.C. Diminnie, R. Haberman, Slow passage through homoclinic orbits for the unfolding of a saddle-center bifurcation and the change in the adiabatic invariant, Physica D 162 (1–2) (2002) 34–52. [6] H.R. Dullin, A.V. Ivanov, Twistless tori near low order resonance, in: V.F. Lazutkin, Workshop on Differential Equations, St. Petersburg, 2002, pp. 135–144. Petersburg Department of Steklov Institute, 2003; Zapiski Nauchnyh Seminarov POMI, vol. 300; J. Math. Sci. 128 (2005) 2754–2760. [7] H.R. Dullin, A.V. Ivanov, Vanishing twist in the Hamiltonian Hopf bifurcation, Physica D 201 (2005) 27–44. [8] H.R. Dullin, J.D. Meiss, Twist singularities for symplectic maps, Chaos 13 (1) (2003) 1–16. [9] H.R. Dullin, J.D. Meiss, D.G. Sterling, Generic twistless bifurcations, Nonlinearity 13 (2000) 203–224. [10] J.E. Howard, J. Humpherys, Nonmonotonic twist maps, Physica D 80 (3) (1995) 256–276.
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[11] J.E. Howard, S.M. Hohs, Stochasticity and reconnection in hamiltonian systems, Phys. Rev. A 29 (1984) 418. [12] K.R. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, Berlin, 1992. [13] R. Moeckel, Generic bifurcations of the twist coefficient, Ergodic Theory Dynam. Syst. 10 (1) (1990) 185–195. [14] E. Petrisor, Nontwist area preserving maps with reversing symmetry group, Int. J. Bifur. Chaos 11 (2) (2001) 497–511. [15] D.A. Sadovskii, J.B. Delos, Bifurcation of the periodic orbits of hamiltonian systems: an analysis using normal form theory, Phys. Rev. E 54 (1996) 2033–2070.
[16] C.L. Siegel, J.K. Moser, Lectures on Celestial Mechanics, Springer, Berlin, 1971. [17] C. Sim´o, Invariant curves of analytic perturbed nontwist area preserving maps, Regular Chaotic Dynam. 3 (1998) 180– 195. [18] J.P. Van der Weele, T.P. Valkering, The birth process of periodic orbits in nontwist maps, Physica A 169 (1) (1990) 42– 72. [19] A. Wurm, A. Apte, K. Fuchss, P.J. Morrison, Meanders and reconnection-collision sequences in the standard nontwist map. Chaos 15 (2005) 023108 (13 pp.).
Another look at the saddle-centre bifurcation: Vanishing twist H.R. Dullin∗ , A.V. Ivanov Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK Received 13 February 2005; received in revised form 25 July 2005; accepted 28 July 2005 Available online 22 August 2005 Communicated by C.K.R.T. Jones
Abstract In the saddle-centre bifurcation a pair of periodic orbits is created “out of nothing” in a Hamiltonian system with two degrees of freedom. It is the generic bifurcation with multiplier one. We show that “out of nothing” should be replaced by “out of a twistless torus”. More precisely, we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits. Vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes. We explicitly derive the position of the twistless torus in phase and in parameter space near the saddle-centre bifurcation. The theory is applied to the area preserving H´enon map. © 2005 Elsevier B.V. All rights reserved. Keywords: Twist maps; Hamiltonian systems; Saddle-centre bifurcation; KAM; Normal forms; Elliptic integrals
1. Introduction
there exist symplectic angle-action coordinates (ϕ, I) on this plane such that the mapping reads
The dynamics near a periodic orbit in a Hamiltonian system with two degrees of freedom is described in linear approximation by its Floquet multipliers µi , i = 1, 2 where µ1 = µ ¯ 2 ∈ C. For an elliptic orbit the multiplier is on the unit circle and can be written as µ = exp(2πiω), where the rotation number ω satisfies 0 < ω < 1/2. Higher order approximations are constructed using the Birkhoff normal form. This can be done by consideration of the Poincar´e map on a local transverse plane in the energy surface. When the rotation number ω of the periodic orbit is irrational
(ϕ, I) → (ϕ + 2πΩ(I), I)
∗
Corresponding author. E-mail address: [email protected] (H.R. Dullin).
0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2005.07.019
(1)
up to arbitrary high order. Iterating the map preserves the invariant circles I = constant and the dynamics on each circle is given by a rotation by 2πΩ(I). The rotation number (or winding number) Ω(I) can be expanded near the periodic orbit I = 0 as Ω(I) = ω + τ0 I + 21 τ1 I 2 + · · · The twist (or torsion) τ(I) is the derivative of the rotation number with respect to the action, τ(I) =
dΩ (I) = τ0 + τ1 I + · · · dI
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A mapping of the form (1) is called a monotone twist map if the twist is non-zero in the region of interest. In that case Moser’s KAM theorem [16] shows that the invariant circle I = I0 with diophantine rotation number Ω(I0 ) and non-vanishing twist τ(I0 ) persists under perturbations. In particular such perturbations can be the neglected higher order terms in the normal form. The analogous statement for flows is Arnold’s KAM theorem [1] in Hamiltonian systems with two degrees of freedom, in which the isoenergetic non-degeneracy condition is the equivalent of the twist condition. In this paper we will show that the twist condition is violated near the saddle-centre bifurcation of periodic orbits (or fixed/periodic points of an area preserving map). The dynamical consequences of violating the twist condition have first been described by Howard and Hohs [11], and the resulting effects have been observed in many examples, see, e.g. [18,15]. Probably the most spectacular effect is the appearance of so-called meandering curves [11,10,17], which are invariant tori that are not graphs over any unperturbed torus. A renormalisation theory for twistless tori has been worked out in [2,3] and more recently it has been shown [4] that an extension of the KAM theorem can also be proved in this context. Even though the phenomenon had been observed in many examples it was not clear how typical it is. In [9,13] it was shown that the twist of a fixed point in a family of area preserving maps vanishes for some rotation number ω near 1/3. At this point a twistless torus is born in a twistless bifurcation [9]. Whenenver this torus passes through a rational winding number meandering curves appear. A similar connection between resonances and vanishing twist holds true in four-dimensional symplectic maps [8]. The techniques of [9] can also be applied to higher order resonances, in particular for ω = 1/4, where a similar result is conceivable. For flows there is the interesting case of the 1:−1 resonance or Hamiltonian Hopf bifurcation, which was shown to produce a family of twistless tori in [7]. In [6] we announced that a similar result holds for the 1:1 resonance in maps, while for the 1:2 resonance in maps the twist does not vanish. Here we are going to give the proofs for the more interesting case of the 1:1 resonance. The main result is encapsulated in a new bifurcation diagram for the saddle-centre bifurcation in area-preserving maps, see Fig. 4. In addition to the stable and unstable branch meeting at the bifurcation it has an additional branch on the other
side of the bifurcation that gives the position of the twistless torus in phase space. This curve has universal shape relative to the other two. The universal frequency of the corresponding twistless curve is also derived. The observation that a twistless curve is destroyed/created in a saddle-centre bifurcation of periodic orbits was first made in [2]. In this work the saddle-centre bifurcation of symmetric periodic orbits (called hyperbolic–elliptic collision in [2]) in the standard nontwist map was used to give bounds for the existence of twistless curves. Other types of orbit collision, e.g. hyperbolic–hyperbolic, were also found in [2] and more recently [19] found so called non-standard reconnection and collision scenarios. These bifurcations are not generic and appear because of the discrete symmetries of the standard nontwist map [14]. We restrict ourselves to the generic saddle-centre bifurcation and obtain analytic results for position and frequency of the nearby twistless curve. The fact that this is a generic bifurcation implies that the resulting twistless curves are not only found in the standard nontwist maps, but in any generic one-parameter family of maps. Even though our main statement is about the saddlecentre bifurcation of periodic orbits in two degrees of freedom conservative systems, most of the analysis is done in one degree of freedom. The conclusions are specific to two degrees of freedom, however, because the concept of twist does not exist in one degree of freedom. Results about the period of the corresponding one degree of freedom system have been obtained by [5], in the context of an adiabatic perturbation of the one degree of freedom saddle-centre bifurcation. But as far as we are aware our exact results are even new for the case of one degree of freedom. In this setting we determine the slowest orbit near the bifurcation point but before the bifurcation occurs. In panel (c) in Fig. 1 it is the one shown with the closest approach to the origin. The paper is organised as follows. We introduce the normal form for the saddle centre bifurcation in the next section. The main theorem on the vanishing of the twist is presented in Section 3. In the following section we show that higher order terms in the normal form do not change our results, and thus there are certain universal constants that can be computed in terms of complete elliptic integrals associated with this bifurcation. In the final section we treat the non-integrable area preserving H´enon map as an example.
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Fig. 1. Lines of constant energy on the phase space (u, v) for the saddle-centre bifurcation, h = 0.1. (a) ε = −0.1, (b) ε = 0, (c) ε = 0.1. For ε = 0.1 the invariant curve closest to the origin has vanishing twist.
2. Saddle-centre bifurcation The normal form of an area preserving map of the plane near a generic bifurcation is integrable [12]. The in general non-integrable mapping of the plane is approximated near the bifurcation by the time one map of an integrable Hamiltonian flow with one degree of freedom. By studying this one degree of freedom system we thus understand the behaviour near the bifurcation in the original non-integrable map of the plane. This in turn can be viewed as the Poincar´e map of a Hamiltonian system with two degrees of freedom on a local transversal section of a periodic orbit of the flow. The Hamiltonian whose time one map generates the map is given by [12] H(u, v) =
v2 2
+
u3 3
+ εu.
(2)
The bifurcation parameter is ε. We must keep in mind that this one degree of freedom system gives an approximation to the area preserving map we are interested in.
The level lines of H are shown in Fig. 1. These are the invariant curves of the area preserving map. The critical points of the energy map H : R2 → R and their dependence on ε give the main structure of √ the bifurcation. The critical points are (u, v) = (± −ε, 0) with critical values h = ∓2(−ε)3/2 /3. The upper sign corresponds to a local minimum of H, while the lower sign gives a saddle. The critical values and their dependence on ε are shown in Fig. 2. Proposition 1. The rotation number Ω(h, ε) of the invariant curve H(u, v; ε) = h near a saddle-centre bifurcation in an area preserving map of the plane is given by the complete elliptic integral √ 1 3/2 1 = 1/4 dz Ω(h, ε) |ε| β w
(3)
along the real cycle β of the elliptic curve E : w2 = P3 (z)= 2γ − z3 − 3σz,
σ = sign(ε).
(4)
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In the original variables the curve ∆ = 0 of colliding roots is given by 9h2 = −4ε3 ,
Fig. 2. Schematic sketch of the (ε, h) bifurcation diagram for the saddle-centre bifurcation. Graphs of P3 are shown together with a horizontal line indicating the value of h. The bold lines marks the critical values of the saddle, the thin lines those of the centre.
The single continuous parameter γ of E is given by γ=
3h . 2|ε|3/2
(5)
Proof. The period of the reduced system is obtained by separating variables in u˙ = v(h, u), giving du T (h, ε) = . (6) 2h − (2/3)u3 − 2εu Since the time one map of the flow of the Hamiltonian gives the map we are studying, the rotation number of the invariant curve H = h of this map is given by Ω = 1/T . By scaling u = z(σε)1/2 , where σ = sign(ε) and introducing the one essential parameter γ, the period T becomes an elliptic integral on the curve E. The essential integral now reads 1 S(γ) = dz (7) w β and combining the formulas gives the result for Ω.
The polynomial P3 has one or three real roots, depending on the sign of the discriminant ∆ = −108(σ + γ 2 ).
(8)
and it is shown in Fig. 2. The collision of roots occurs at the critical values of H. In the cuspoidal region to the left there are three real roots, otherwise there is only one. This is the bifurcation diagram of the saddle-centre bifurcation. When considering two degrees of freedom integrable flows this would be the image of the energymomentum map. As such it is the natural domain on which the rotation number should be studied. The correspondence between the phase portraits Fig. 1 and the bifurcation diagram Fig. 2 is as follows. For ε > 0 there is only one family of invariant curves (or tori) and there is always only one root in P3 . The torus corresponds to the interval where P3 > 0, and this therefore also defines the real cycle β. For ε < 0 there are three families of tori in the phase portrait, inside the loop, or outside to the left or to the right of the separatrix. The only tricky bit in the mapping to the bifurcation diagram is that the tori to the left of the separatrix are below the bold line in the bifurcation diagram and continue smoothly outside the cuspoidal region. Stated otherwise, for each point in the cuspoidal region there are two tori in the preimage, while otherwise there is only one. A contour plot of the rotation number Ω(ε, h) is shown in Fig. 3. As expected the rotation number has a singularity along the curve of the saddle critical values (bold in Fig. 2), while it is smooth across the centre critical values (thin in Fig. 2). The reason for this latter fact is that the curve is elliptic. Thus it only has one real cycle. Since additionally the differential of T is of first kind the real integrals over the two disjoint real intervals for ∆ > 0 are identical.
3. Vanishing twist Vanishing twist occurs where the rotation number has a critical point within a family of invariant curves of an area preserving map of the plane. In Fig. 3 this condition means that the contour lines have a vertical tangent, as indicated by the dashed line. Our main result gives the precise location of this curve. Theorem 2. In the saddle-centre bifurcation of an area preserving map the twist is vanishing on a curve
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51
has no fixed point, ε > 0. This implies ∆ < 0. All of the following is valid for ∆ < 0 only. The integrand w is positive for z ∈ (−∞, z0 ), where z0 is the single root of P3 (z). Let the factorized polynomial be given by P3 (z) = −(z − z0 )P2 (z),
(11)
P2 (z) = (z − ζ1 )2 + ζ22 ,
so that the complex roots are ζ1 ± iζ2 . Denote the distance between the real and complex roots by r, hence r2 = P2 (z0 ). In this notation the discriminant of P3 is given by ∆ = −4r4 ζ22 . The modulus k of the elliptic integrals is given by z 0 − ζ1 1 1 sign(γ) 2 k = 1+ , (12) = 1+ √ 2 r 2 1 + α2
Fig. 3. Lines of constant rotation number Ω = 1/T equidistant with Ω = 0.2 on the parameter plane (ε, h) for the saddle-centre bifurcation. The vanishing twist is indicated by the dashed curve of vertical tangents ∂Ω/∂h = 0.
in the parameter plane (ε, h) given by 3h = 2γ0 ε3/2 .
(9)
for positive ε where γ0 ≈ 0.9152203 is determined from the root of a transcendental equation involving complete elliptic integrals. Proof. By definition the twist is given by the derivative of the rotation number Ω with respect to the action 2πJ = vdu. Since Ω = 1/T and 2πT = ∂J/∂h the J derivative of the rotation number is ∂Ω ∂Ω/∂h ∂T/∂h = =− . ∂J ∂J/∂h 2πT 3
(8k4 − 9k2 + 1)K(k) = (16k4 − 16k2 + 1)E(k). This equation has a single non-trivial root k = k0 where k02 ≈ 0.7097215. The proof of the Lemma is given in the Appendix A. In order to calculate the parameter γ0 in Theorem 2 corresponding to k0 we observe that α as introduced in (12) is related to γ by ζ2 = − 3ζ = √1 ΓΓ 2 +σ , −σ 1 3 Γ = (γ + σ + γ 2 )1/3 .
α=
Hence the twist vanishes when ∂T (3/2)3/2 ∂S = =0 ∂h ε7/4 ∂γ and this is only possible for finite ε when ∂S 1 =− dz = 0. 3 ∂γ w β
and in the last equality the parameter α = ζ2 /(z0 − ζ1 ) has been introduced. In the second equality, in addition sign(z0 − ζ1 ) = sign(γ) is used, which is true because z0 = ζ1 in (11) together with the vanishing of the quadratic coefficient in (4) implies z0 = ζ1 = 0, and therefore the polynomial has no constant term and h = γ = 0. The solution of (10) is given by our main technical Lemma 3. The integral β (1/w3 )dz vanishes when
ζ2 z0 −ζ1
2
(13)
Now (12) relates α0 to k0 as given in Lemma 3. The corresponding value of α is α0 ≈ 2.164255. Finally inverting (13) we find γ0 ≈ 0.9152203. (10)
This is an elliptic integral of the second kind on the curve E. The relevant case is where the phase portrait
Since γ0 < 1 the curve of vanishing twist is bent downward as compared to the bifurcation curve (8) for ε < 0 and h > 0. See Fig. 3 for a graph of this curve together with the numerically computed lines of constant rotation number. The lines of constant rotation
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number have vertical slope at their intersection with the twistless curve, as must be the case. The scaling of u reduces the number of parameters to one. The essential parameter γ, in its dependence on h and ε, organizes the bifurcation. It allows us to compute explicitly all the important characteristics of the twistless torus. Combining (12) and (13) shows that k is a function of γ. The curves in the parameter plane that have the same value of γ are given by (5). The most prominent ones are the curve of critical values γ = ±1, as shown in Fig. 2, and γ = γ0 , the curve of twistless tori, see Fig. 3. They all have the same shape of a semicubical parabola, except when γ = 0, hence h = 0, or γ = ±∞, hence ε = 0 with ±h > 0. k2
γ
α
Curve
1
+1 +∞ 0.91522 0 −∞ −1
0
3h = −2(−ε)3/2 ε = 0, h > 0 3h = 2γ0 (−ε)3/2 h = 0, ε > 0 ε = 0, h < 0 3h = 2(−ε)3/2
√ + 3) 0.709721497 1 4 (2 1 2 1 4 (2
−
√ 3)
0
√1 3
2.164255 ∞ − √1 3 0
Anywhere along the curve (5) the modulus k(γ) is constant. The function k(h, ε) is therefore not continuous at the origin. Moving on (h, ε)-curves given by (5) for any γ = 1 the change in the period T (h, ε) is elementary. From (3) it follows that T is proportional to |ε|−1/4 , and the constant of proportionality is determined from (3) and (7). The divergence of the period upon approaching the bifurcation from ε > 0 is therefore not caused by the elliptic integral, but merely by the algebraic dependence |ε|−1/4 . Similarly the rotation number of the stable fixed point of the map is given by 1 Ω(γ = −1, ε) = √ (−ε)1/4 ≈ 0.225079|ε|1/4 . 2π The integral S(−1) contained in this expression can be easily calculated using residue calculus because for γ = −1 the curve has a double root, P3 (z) = −(z + 2)(z − 1)2 . For general values of γ and in particular for the twistless torus, the elliptic integral S(γ) (7) needs to be evaluated. For γ = γ0 the single real root is given by z0 = Γ −
σ , Γ
hence z0 (γ0 ) ≈ 0.5535942.
(14)
Fig. 4. Bifurcation diagram of the position in phase space u vs. the bifurcation parameter ε. For ε < 0 the location of the fixed points at √ ± −ε are shown, while for ε > 0 the maximal u of the twistless √ torus at z0 (γ0 ) ε is shown. Compare Fig. 1.
Finally S(γ) is obtained in terms of the standard elliptic integral of first kind K as S(γ) =
4K(k) . √ 3|z0 | 1 + α2 /2
Using the above value of z0 , the position of the rightmost point of the√twistless torus in phase space is located at u0 = z0 ε. This means that for ε = ε0 this point is on the same side as the stable periodic orbit was for ε = −ε0 before the bifurcation, but by a factor of z0 (γ0 ) ≈ 1/2 closer to the origin. See Fig. 4 for an illustration in the form of a standard bifurcation diagram showing position in phase space u versus bifurcation parameter ε. Thus we have proved Proposition 4. The rotation number of the twistless curve is z0 (γ0 ) 1 + α20 Ω(γ = γ0 , ε) = ε1/4 4K(k0 ) ≈ 0.1374244ε1/4 .
(15)
The position of the twistless curve in phase space as determined by its intersection with v = 0 is given by √ √ u0 = z0 ε ≈ 0.5535942 ε (16) The formulae (15) and (16) will be illustrated by the area preserving H´enon map in Section 5.
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4. Universality There are two questions to be sorted out. First of all, the invariant curves of the normal form are not compact, so that it is not clear what the rotation number (or period) means. Secondly, the influence of higher order terms needs to be studied. The two problems are related: Adding higher order terms in the normal form can compactify the invariant curves. What we will show is that in the limit ε → 0 from above, the leading order diverging contribution to the period T comes from the term that we analysed in the previous sections. In particular this means that the constant γ0 that determines the shape of the curve of twistless tori in relation to the curve of critical values of the unstable orbits has the universal value γ0 ≈ 0.91522. Quantities derived from γ0 , like z0 and the coefficients in (15), are accordingly also universal. Let the high-order truncated normal form Hamiltonian (whose time one map generates the Poincar´e map) be ˜ H(u, v; ε) = 21 v2 + 13 u3 + εu + G(u, ε) where G(0, ε) = 0 and G(u, 0) = 0 and G is an analytic function. We assume that G(u, ε) is such that the invariant curves near the origin for ε > 0 are compact. As ˜ usual the period is obtained by solving H(u, v; ε) = h for v = v(u; h, ε) and then by integrating du T˜ (h, ε) = β v(u; h, ε) along β, which projects to the values of u for which v is real. The integral is now split in two ways. First of all the integration path β is decomposed into a part β0 that is close to zero, and in all the rest β1 . Moreover, the expression under the square root is factored as ˜ h, ε) v(u; h, ε)2 = Q(u; h, ε)Q(u;
53
The first integral T1 is an incomplete elliptic integral. Specifying for β0 that it be contained in the interval [−d, d] it can be integrated explicitly to √ 4 3/2 d+C T1 (h, ε) = 1/2 F (φ, k), φ = 2 arctan , r r where r = 2d 2 + |ζ|2 ,
2k2 = 1 +
3d . 2r
The main observation is that the integral T1 does not diverge because the modulus k2 tends to one, but because in the prefactor r tends to zero, and this independently of the size of the neighbourhood d. In fact, we already observed that inside the first √ quadrant ε ≥ 0, h > 0 the modulus satisfies k2 ≤ ( 3 + 2)/4 < 1. The behaviour of r for small ε is √ r 2 = 23 d 1 + α2 = D(γ)ε1/4 . This shows that φ → π, and the incomplete integral T1 approaches the complete integral T with the same modulus k(γ) as before. So even though T1 is an incomplete integral, in the limit of small ε it approaches the complete integral T (h, ε), which diverges. The other integrals T2 and T3 are finite. Since the rotation number Ω is determined by the period T a similar conclusion holds for Ω. Thus we have proved Proposition 5. The results obtained from the analysis of the normal form with non-compact invariant curves (G = 0) correctly describes the behaviour of the rotation number of the compact invariant curves (G = 0).
5. Example: H´enon map
The H´enon map in the area preserving case, where Q(u; h, ε) is the old expression (6). As a result (x , y ) = (y − κ + x2 , −x), we find du du √ √ T˜ (h, ε) = + Q(u, ε, h) Q(u, ε, h) β0 β1 ˜ −1/2 du −1 Q(u, h, ε) du √ √ √ = + du + = T1 + T 2 + T 3 . Q(u, ε, h) Q(u, ε, h) Q(u, ε, h) β0 β0 β1
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illustrates the above, in particular the predictions made in Proposition 4. There are saddle-centre bifurcations in the H´enon map for which the invariant tori before the bifurcation are not compact, and hence the vanishing twist cannot be observed. This applies to the initial bifurcation at κ = −1 that creates the pair of fixed points, and also to many saddle-centre bifurcations that occur for κ > 4. However for κ = 1 a pair of period 3 orbits is created in a saddle-centre bifurcation of the third iterate of the map. One point of this orbit is at the origin for κ = 1. The invariant tori for ε < 0 (before the bifurcation) are compact near the bifurcation point. Hence there will be a torus with vanishing twist. The family of twistless tori for κ < 1 continues down to κ = 9/16, where the central elliptic fixed point of the H´enon map has vanishing twist. At this end the family of twistless curves has been studied in [9]. The present work confirms a conjecture [9] that the family of twistless curves created when the twist of the fixed point vanishes for κ = 9/16 ends in the saddlecentre bifurcation of the period 3 orbit for κ = 1. Since the H´enon map is non-integrable the family of twistless tori is not continuously parametrized by κ, but instead it is a cantorset determined by the condition that the frequency of the twistless curve is sufficiently irrational. One of the period three points is located on the symmetry line x = −y. In new coordinates (u, v) = (x, y + x) the third iterate of the map expanded near the origin with parameter κ = 1 − ε is (u , v ) = ((u − v)(1 + 4v), v + ε +(u2 − 2uv + 3v2 )) + O(u3 , v3 , |ε|3/2 ). The (exact) location of the bifurcating fixed points √ is (u,√v) = (± −ε, 0), with trace of the Jacobian 2 ∓ 2 −ε + 8ε ∓ 8(−ε)3/2 for the elliptic (−) and hyperbolic√(+) orbit. The (approximate) multiplier is µ ≈ 1 + i 2(−ε)1/4 and the corresponding rotation number ω of the elliptic fixed point is obtained from √ µ = exp(2πiω), so that ω ≈ (−ε)1/4 /( 2π). For small positive ε the H´enon map possesses compact invariant curves near the origin. A higher order normal form would give a G that describes these invariant curves. The H´enon map is non-integrable, so that only sufficiently irrational invariant curves of the (high order) normal form will exist in the H´enon map. For the situation under consideration numerical experiments
Table 1 Numerically measured position u and rotation number Ω of the twistless curve for the three times iterated H´enon map with parameter κ =1−ε ε
uε−1/2
Ωε−1/4
0.1 0.01 0.001 0.0001 0.00001 0.000001
0.027 0.328 0.478 0.528 0.545 0.550
0.2831 0.1728 0.1466 0.1401 0.1382 0.1377
The values converge to the numbers given in Proposition 4.
show that many of these invariant curves do exist. Considering the third iterate of the H´enon map turns the pair of period three orbits into three pairs of fixed points with heteroclinic connections. In the normal form there is only one pair of fixed points, and the unstable fixed point has a homoclinic connection. To match the prediction in the case of more than one unstable fixed point the period and hence rotation number must be calculated for the heteroclinic connection. For ε > 0 there are invariant tori on which the dynamics becomes slow near the three points that are close to the three bifurcation points of the third iterate of the map. Hence the rotation number for the third iterate of the map between two successive such points gives the correct rotation number. The results together with the position of the twistless curve are shown in Table 1. The values shown converge to the predicted values given in Proposition 4, however, fairly small ε are needed to see this. The convergence to the true value 0.1374244 . . . occurs approximately as (1/2) ε5/4 .
Acknowledgements This research is funded by the EPSRC under contract GR/R44911/01. Partial support by the European Research Training Network Mechanics and Symmetry in Europe (MASIE), HPRN-CT2000-00113, is also gratefully acknowledged. AVI was also supported in part by INTAS grant 00221, RFBI grant 01-01-00335 and RFME grant E00-1-120.
H.R. Dullin, A.V. Ivanov / Physica D 211 (2005) 47–56
Appendix A Proof (Lemma 3). The complete elliptic integral ∂S =− ∂γ
1 dz = 0 w3
can be written as a linear combination of integrals of the first and second kind. Legendre’s standard integral of the first kind K(k) has differential ω1 = √
1 dz P3 (z)
(17)
up to a constant factor. A non-standard form of the differential of Legendre’s standard integral of the second kind E(k) is ω2 =
P2 (z)dz ω1 , (z − (z0 ± r))2
up to the same constant factor as in (17). The differential dz/w3 we are interested in is of the second kind, and can therefore be written as a linear combination of ω1 and ω2 with constant coefficients, up to a total differential: dz = Aω1 + Bω2 + dF (18) w3 where F = Q2 (z)/((z − (z0 ± r))w). Together with the undetermined coefficients of the quadratic polynomial Q2 this gives a system of five linear equations for the five unknown coefficients. Solving these equations gives 2r 4r ((z0 − ζ1 )r − r 2 + 4ζ22 ), B = (r 2 − 4ζ22 ) ∆ ∆ Since the quadratic coefficient of P3 is zero, the roots of P3 add up to zero. Therefore, the real parts satisfy z0 + 2ζ1 = 0, hence
A=
2k2 = 1 −
3ζ1 r
and
r 2 = 9ζ12 + ζ22 .
With these equations the coefficients A and B can be expressed in terms of k alone, up to the factor (r 2 ∆)−1 . Integrating (18) gives the condition of vanishing twist, ∂S/∂γ = 0 in terms of A and B as (8k4 − 9k2 + 1)K(k)= (16k4 − 16k2 + 1)E(k),
(19)
55
where K(k) and E(k) stand for elliptic integrals of the first and the second kind, respectively. Now we prove the existence of a real positive root in the interval (0, 1). For k = 0 we have equality since both elliptic integrals equal π/2. The first derivatives on either side vanish, but the second derivatives are −35π/4 and −65π/4, respectively, so that the left hand side of (19) is larger for small k. For k = 1 the prefactor of K vanishes, while that of E gives 1 and E(1) = 1. Hence for k → 1 the right hand side dominates. This proves that there exists a solution of this equation for k ∈ (0, 1). Numerically we find k02 ≈ 0.7097215. √ Expansion of the equation defining k0 at k2 = 1/ 2 gives the approximation k02
√ √ 1 2 (19 + 14 2)K − (34 − 25 2)E √ √ ≈√ + 2 5 (18 − 13 2)K − 8(4 − 3 2)E
which is correct to 6 decimal places. Here the √ modulus for K and E is the expansion point k2 = 1/ 2.
References [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978. [2] D. del Castillo-Negrette, J.M. Greene, P.J. Morrison, Area preserving nontwist maps: periodic orbits and transition to chaos, Physica D 91 (1) (1996) 1–23. [3] D. del Castillo-Negrette, J.M. Greene, P.J. Morrison, Renormalization and transition to chaos in area preserving nontwist maps, Physica D 100 (3–4) (1997) 311–329. [4] A. Delshams, R. de la Llave, KAM theory and a partial justification of Greene’s criterion for nontwist maps, SIAM J. Math. Anal. 31 (6) 1235–1269 (2000) (electronic). [5] D.C. Diminnie, R. Haberman, Slow passage through homoclinic orbits for the unfolding of a saddle-center bifurcation and the change in the adiabatic invariant, Physica D 162 (1–2) (2002) 34–52. [6] H.R. Dullin, A.V. Ivanov, Twistless tori near low order resonance, in: V.F. Lazutkin, Workshop on Differential Equations, St. Petersburg, 2002, pp. 135–144. Petersburg Department of Steklov Institute, 2003; Zapiski Nauchnyh Seminarov POMI, vol. 300; J. Math. Sci. 128 (2005) 2754–2760. [7] H.R. Dullin, A.V. Ivanov, Vanishing twist in the Hamiltonian Hopf bifurcation, Physica D 201 (2005) 27–44. [8] H.R. Dullin, J.D. Meiss, Twist singularities for symplectic maps, Chaos 13 (1) (2003) 1–16. [9] H.R. Dullin, J.D. Meiss, D.G. Sterling, Generic twistless bifurcations, Nonlinearity 13 (2000) 203–224. [10] J.E. Howard, J. Humpherys, Nonmonotonic twist maps, Physica D 80 (3) (1995) 256–276.
56
H.R. Dullin, A.V. Ivanov / Physica D 211 (2005) 47–56
[11] J.E. Howard, S.M. Hohs, Stochasticity and reconnection in hamiltonian systems, Phys. Rev. A 29 (1984) 418. [12] K.R. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, Berlin, 1992. [13] R. Moeckel, Generic bifurcations of the twist coefficient, Ergodic Theory Dynam. Syst. 10 (1) (1990) 185–195. [14] E. Petrisor, Nontwist area preserving maps with reversing symmetry group, Int. J. Bifur. Chaos 11 (2) (2001) 497–511. [15] D.A. Sadovskii, J.B. Delos, Bifurcation of the periodic orbits of hamiltonian systems: an analysis using normal form theory, Phys. Rev. E 54 (1996) 2033–2070.
[16] C.L. Siegel, J.K. Moser, Lectures on Celestial Mechanics, Springer, Berlin, 1971. [17] C. Sim´o, Invariant curves of analytic perturbed nontwist area preserving maps, Regular Chaotic Dynam. 3 (1998) 180– 195. [18] J.P. Van der Weele, T.P. Valkering, The birth process of periodic orbits in nontwist maps, Physica A 169 (1) (1990) 42– 72. [19] A. Wurm, A. Apte, K. Fuchss, P.J. Morrison, Meanders and reconnection-collision sequences in the standard nontwist map. Chaos 15 (2005) 023108 (13 pp.).