Exponentially small oscillation of 2-dimensional stable and unstable manifolds in 4-dimensional symplectic mappings

6 Pages·0.1 MB·English

by  Yoshihiro Hirata

DPNU–99–01
January 1999
Exponentially small oscillation of 2-dimensional
stable and unstable manifolds in 4-dimensional
symplectic mappings
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Yoshihiro Hirata∗, Kazuhiro Nozaki and Tetsuro Konishi
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Department of Physics, Faculty of Science, Nagoya University, Furo-cho,
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Chikusa-ku, Nagoya, 464-8602, Japan
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c Abstract
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X Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptotically
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a studied. We construct the 2-dimensional stable and unstable manifolds near the sub-
manifolds which experience exponentially small splitting, and successfully obtain ex-
ponentially small oscillating terms in the 2-dimensional manifolds.
∗ E-mail address: [email protected]
1 typeset using PTPTEX.sty <ver.0.8>
Homoclinic (or heteroclinic) bifurcation plays an important role to cause chaotic motion
near hyperbolic fixed points. Melnikov’s method (or Melnikov’s integral), which is based on
theregular perturbationmethod, isa powerful toolto detect transversal homoclinicpoints.1)
On the other hand, it is known that in the rapidly forced systems and the standard map,
and so on, splitting of separatrices is exponentially small. Consequently it is impossible to
capture exponentially small oscillation straightforwardly with Melnikov’s method.
In the last decade the difficulty has been overcome by using the method called asymp-
totic expansions beyond all orders. This method enables one to capture exponentially small
oscillation of the stable and unstable manifolds in 2-dimensional symplectic mappings which
are perturbed from linear mappings,2),3),4) and to construct functional approximations of
the manifolds.5),6),7)
For 4-dimensional symplectic mappings, Gelfreigh and Shatarov treated a coupled stan-
dard mapping and computed the crossing angle between the 2-dimensional stable and unsta-
ble manifolds.8) We obtained the functional approximations of particular sub-manifolds of
the stable and unstable manifolds in a 4-dimensional double-well symplectic mapping with a
weak coupling.9) In this letter, we construct the functional approximations of 2-dimensional
stable and unstable manifolds in 4-dimensional double-well symplectic mappings with more
general weak couplings.
We start with the 4-dimensional symplectic map (q ,p ) 7→ (q′,p′)
j j j j
∂J˜
p′ = p +ǫ(q −2p3)−ǫ3
 j j j j ∂qj (1)
 q′ = q +ǫp′
j j j
where j,k ∈ {1,2},j 6= k, 0< ǫ ≪ 1 is a perturbation parameter. To simplify computation
a b
we take the coupling potential J˜ as J˜ = q2q2 + q3q3.† Note that the map (1) possesses
2 1 2 3 1 2
the substitutional invariance between j and k.
The map (1) possesses a hyperbolic × hyperbolic fixed point at the origin (0,0,0,0)
and there exist 2-dimensional stable and unstable manifolds near the origin. In Ref. 9),
we applied the method to the particular 1-dimensional stable and unstable sub-manifolds
defined as q = q and successfully obtained the functional approximations of the sub-
1 2
manifolds when b = 0. In this letter we study the neighborhood of the sub-manifolds as a
perturbation problem.
One can rewrite the map (1) in the second order difference equation
∆2q (t) = q (t)−2q (t)3 −ǫ2(aq (t)q (t)2 +bq (t)2q (t)3), (2)
ǫ j j j j k j k
† InRef.9)weshowthatifJ˜isapolynomialofq ’sofdegreesix,itispossibletocaptureexponentially
j
small oscillation. Hence this coupling term is considerably general.
2
where ∆2q(t) ≡ {q(t+ǫ)−2q(t)+q(t−ǫ)}/ǫ2. The difference equation (2) is equivalent to
ǫ
the infinite order differential equation
d2q ∞ ǫ2l−2d2lq
j = q −2q3 −ǫ2(aq q2 +bq2q3)−2 j, (3)
dt2 j j j k j k (2l)! dt2l
l=2
X
which is called the outer equation. The boundary condition with which we solve the equation
(3) is q±(t;ǫ) → +0(t → ±∞), where the superscript ± stands for the stable and unstable
j
manifolds, respectively.‡
We first construct the solution to the ODE (3) by using the regular perturbation method
withrespect toǫ2, i.e., expanding q (t,ǫ) = s (t)+ǫ2q˜(t)+O(ǫ4). Substituting theexpansion
j j j
into the ODE (3), one obtains the unperturbed equation as
d2s
j = s −2s3,
dt2 j j
and the perturbed equation as
d2q˜
j = (1−6s (t)2)q˜ +F (t), (4)
j j j
dt2
where F (t) is a polynomial of s (t),s (t) and their derivatives. The boundary condition is
j j k
rewritten as s (t),q˜(t) → +0 (t → ±∞).
j j
The unperturbed solution is obtained as s (t,t ) = sech(t+t ), where t ’s are integration
j j j j
constants. Because the system is autonomous, it has translational invariance for t. Hence
the essential parameter is the difference between t and t . We put t = (−1)jα where
1 2 j
α is an arbitrary real constant. To simplify notations we put s(t) ≡ s (t,0) = sech(t).
j
Taking the integration constant α, which was vanished in Ref. 9), as another perturbation
parameter, we successfully obtain exponentially small oscillating terms in the neighborhood
of the sub-manifolds.
The fundamental system of the solution to the homogeneous equation of the ODE (4) is
dt 3 1
given as s˙ (t,t ) and g (t,t ) = g(t+t ), where g(t) = s˙(t) = sech(t)− cosh(t)−
j j j j j
s˙(t)2 2 2
Z
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tsinh(t)sech(t)2. Note that s˙(t) and g(t) are odd and even functions of t respectively, and
2
Wronskian is equal to 1. The general stable and unstable solutions to the inhomogeneous
equation (4) is obtained as
t t
q˜±(t,t ) = c±s˙ (t)−s˙ (t) F (t)g (t)dt+g (t) F (t)s˙ (t)dt, (5)
j j j j j j j j j j
Z0 Z±∞
respectively, where c±’s are integration constants.
j
‡ The same notations will be used without further comment.
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When α = 0, which is equivalent to t = 0,j = 1,2, the solution stands for the 1-
j
dimensionalsub-manifoldsofthe2-dimensionalstableandunstablemanifolds, andthestable
and unstable sub-manifolds coincide, i.e., q+(t) = q−(−t).9) One can therefore restrict the
j j
solution to even fuctions of t. Putting q˜±(t,0) = q˜(t), one obtains
j
b+2 6a+8b+7 t
q˜(t) = sech(t)3 − sech(t)+ sinh(t)sech(t)2.
6 24 24
Next we construct solutions when t 6= 0. Let us suppose |t | ≪ 1, corresponding to the
j j
neighborhood of the sub-maifolds. We expand q˜±(t,t ) in the power of t around q˜(t) as
j j j
2 ∂q˜±(t,t )
q˜±(t,t ) = q˜(t)+ t j j +O(t2)
j j jX=1 j ∂tj (cid:12)(cid:12)tj=0 j
(cid:12)
t ∂q˜±(t,0)(cid:12) t ∂q˜±(t,0)
= q˜(t)+t 1 j (cid:12)+ 2 j +O(t2), (6)
j tj ∂t1 tj ∂t2 ! j
and c± = c0 + t c˜± + O(t2). The condition q˜±(t,t ) → q˜(t) (t → 0) implies c0 = 0. The
j j j j j j j
terms in the bracket of the right hand side of the equation (6) are independent of j because
of the substitutional invariance between j and k of the map (1). Hence we put them q˜±(t),
i.e., q˜±(t,t ) = q˜(t)+t q˜±(t). A straightforward computation gives
j j j
56a+48b
q˜±(t) = c˜±s˙(t)∓ g(t)+q˜˙(t)
j 105
3 28a+24b
+ bsinh(t)sech(t)4 − sinh(t)sech(t)2log(cosh(t))
7 35
8a−6b 28a+24b
+ sinh(t)sech(t)2 − sinh(t).
30 105
Thus the solution to the outer equation (3) is given as
q±(t,t ) = s(t)+ǫ2q˜(t)+t {s˙(t)+ǫ2q˜±(t)} (7)
j j j
up to ǫ4 and t2.
j
The outer solution (7) possesses singular points at t = πi/2 + nπi, n ∈ Z. One must
therefore analyze the neighborhood of the points as the inner problem. Introducing the new
variables as t−πi/2 = ǫz, τ = t /ǫ and Φ (z,τ ) = ǫq (t,t ), the difference equation (2) is
j j j j j j
rewritten as∆2Φ = −2Φ3−bΦ2Φ3+ǫ2(Φ −aΦ Φ2), where ∆2Φ ≡ Φ (z+1,τ )−2Φ (z,τ )+
j j j k j j k j j j j j
Φ (z −1,τ ). In this letter we restrict our attention to the leading term of Φ (z,τ ), hence
j j j j
truncating the terms with respect to ǫ2, we obtain the inner equation as
∆2Φ = −2Φ3 −bΦ2Φ3. (8)
j j j k
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Notethatwhenb 6= 0,thesecondterm,whichisgeneratedfromthecouplingterm,contribute
the inner equation.
When α = 0, putting X(z) = Φ (z,0), the inner equation corresponing to s(t)+ǫ2q˜(t) is
j
obtained as
∆2X(z) = −2X(z)3 −bX(z)5, (9)
which is called the first inner equation. The matching condition is X(z) → ǫ(s(t)+ǫ2q˜(t))
when |z| → ∞. On the other hand, when α 6= 0, expanding Φ (t,τ ) = X(z)+τ X′(z), we
j j j j
obtain
∆2X′(z) = −6X(z)2X′(z)−bX(z)4{2X′(z)−3X′(z)}. (10)
j j j k
Summing up both sides of the equations (10), one obtains the second inner equation
∆2X˜(z) = −6X(z)2X˜(z)+bX(z)4X˜(z), (11)
where X˜(z) = X′(z) +X′(z).§ The matching condition is X˜(z) → ǫ2(s˙(t)+ǫ2q˜±(t)) when
1 2
|z| → ∞.
The inner equations (9) and (11) possess formal solutions when |z| tends to ∞. Although
the formal solutions do not converge, the formal Borel transformations of these formal solu-
tions converge, i.e., these formal solutions are Borel summable. By using Borel transforma-
tion, Stokes phenomena, which are abrupt jump of asymptotic expansions, around t = πi/2,
can be captured, and the analytic continuation from the left half of t-plane to the right one
can be performed. The details of the computation will be reported elsewhere.10)
Finally the stable and unstable solutions which are valid both the left and right half
plane are obtained as
qˆ±(t,t ) = qˆ±(t)+t qˆ±(t), (12)
j j 0 j 1
where
qˆ±(t) = s(t)+ǫ2q˜(t)+S(t,ǫ)M(t,ǫ)B(t,ǫ)
0
qˆ±(t) = s˙(t)+ǫ2q˜±(t)+S(t,ǫ){M(t,ǫ)B˜ (t,ǫ)+M˜(t,ǫ)B˜ (t,ǫ)} (13)
1 1 2
5 c 2
M(t,ǫ) = − e−πǫ
2ǫ4
5 c˜ 2
M˜(t,ǫ) = − e−πǫ
2ǫ5
§ The subtraction of both sides of the equations (10) gives the linearized equation of the equation (9).
In this computation the linearized equation has no information.10)
5
3 2πt 2πt
B(t,ǫ) = πs˙(t)cos +g(t)sin
4 ǫ ǫ
3 2πt 2πt
B˜ (t,ǫ) = πs¨(t)cos +g˙(t)sin
1
4 ǫ ǫ
3 2πt 2πt
B˜ (t,ǫ) = πs˙(t)sin −g(t)cos
2
4 ǫ ǫ
0 (t < 0)
S(t,ǫ) =  1 (t = 0) (for small ǫ).
 2 (t > 0)
The Stokes constants cand c˜, which depend on b, are numerically obtained as c = −763 and
c˜= 1146 when b = 1.0. For other values of b the Stokes constants also converge.10)
By using the approximate solution (12), the splitting angles between the stable and
unstablemanifoldscanbecomputed.10) Onecanshowthattheanglealongthesub-manifolds
(t = t ) is exponentially small7) and the angle orthogonal to the sub-manifolds are O(ǫ2).
1 2
The expression (13) implies that not only the sub-manifolds but also the 2-dimensional
stable and unstable manifolds themselves possess exponentially small oscillating terms. In
the future work, we extend this computation to 2n-dimensional symplectic mappings with
weak coupling terms.
References
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[3] V. G. Gelfreich, V. F. Lazutkin and N. V. Svanidze, Physica 71D (1994), 82.
[4] V. Hakim and K. Mallick, Nonlinearity 6 (1993), 57.
[5] A. Tovbis, M. Tsuchiya and C. Jaff´e, preprint (1994).
[6] A. Tovbis, M. Tsuchiya and C. Jaff´e, Chaos, 8 (1998), 665.
[7] K. Nakamura and M. Hamada, J. of Phys. A29 (1996), 7315
[8] V. G. Gelfreich and D. K. Sharomov, Phys. Lett. A197 (1995), 139
[9] Y. Hirata and T. Konishi, http://xxx.lanl.gov/abs/chao-dyn/9802009.
[10] Y Hirata, K. Nozaki and T. Konishi, In preparation.
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