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7Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 1, pp. 59-76. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210106
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 70H33, 37J40, 85A05
Armbruster — Guckenheimer — Kim Hamiltonian System
in 1:1 Resonance
M. Alvarez-Ramirez, A. Garcia, J. Vidarte
This article deals with the autonomous two-degree-of-freedom Hamiltonian system with Armbruster-Guckenheimer-Kim galactic potential in 1:1 resonance depending on two parameters. We detect periodic solutions and KAM 2-tori arising from linearly stable periodic solutions not found in earlier papers. These are established by using reduction, normalization, averaging and KAM techniques.
Keywords: galactic potential, Hamiltonian system, normalization and reduction, KAM tori, reduced space, periodic orbits
Received July 07, 2020 Accepted March 17, 2021
M. Alvarez-Ramirez and A. García are partially supported by Programa Especial de Apoyo a la Investigación (UAM) grant number I5-2019. Jhon Edder Vidarte Olivera was supported by a PRODEP postdoc grant 511-6/18-5782, Mexico.
Martha Alvarez-Ramírez
mar@xanum.uam.mx,malvarez66@gmail.com
Antonio García
agar@xanum.uam.mx
Departamento de Matemáticas, UAM-Iztapalapa 09340 Iztapalapa, Ciudad de México, México
Jhon Vidarte jhon.vidarte@ucsc.cl
Departamento de Matemática y Física Aplicadas
Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile
1. Setting of the problem
In this paper we study the Hamiltonian with Armbruster-Guckenheimer-Kim (shortly, AGK-Hamiltonian) potential:
n(x, y) = + y¡) - + xl) - + xlf - h-x\xl (1.1)
near the fixed point (0,0,0,0), where ¡ < 0, a = 0 and b are real parameters.
In 1989, Simonelli and Gollub [21] carried out an experimental work with surface waves in square and rectangular containers subject to vertical oscillations. Their methods allowed stable and unstable fixed points (sinks, sources or saddles) and bifurcation sequences. Motivated by these experiments, Armbruster et al. [3] derived a dynamical model starting with a normal form given by four first-order differential equations depending on several parameters. Their system provided a general description of a codimension-two bifurcation problem. To simplify the problem, they performed a rescaling of the variables and parameters. When the scaling parameter was set to zero, the differential equations became the AGK-Hamiltonian system, where an important feature is its ^-symmetry. They found a large parameter region of chaotic behavior and argued that the existence of chaos in a given region suggested that the Hamiltonian is nonintegrable. In [2], Andrle proposed a similar potential AGK-Hamiltonian as a dynamical model for a stellar system with an axis and a plane of symmetry.
Many studies have been focused on the study of the dynamics of AGK-Hamiltonian for several values of the parameters. Without being exhaustive, some interesting works are the following. Llibre and Roberto [12] used classical averaging theory of first order to examine the C1 nonintegrability in the sense of Liouville and Arnold, and also studied the existence of periodic orbits. In [7], Elmandou considered some dynamical aspects for the AGK potential in a rotating reference frame and examined the nonintegrability by means of Painlevé analysis. Additionally, he employed the Lyapunov method to seek periodic solutions near the equilibrium positions. Acosta-Humánez et al. [1] applied Morales - Ramis theory to prove that the AGK-Hamiltonian is rationally integrable if and only if b = 0, a = —b or b = 2a. In fact, these are the only Liouville integrable cases. El-Sabaa et al. [8] made a topological study of the integral manifolds for the integrable cases b = 2a and b = —a, finding the structure of the bifurcation of the level sets. As expected, if bifurcations of Liouville tori take place, the level set of the first integrals becomes degenerate, and the existence of families of periodic solutions is possible, and they were given in terms of Jacobi's elliptic functions. In addition, phase portraits associated to integrable systems are shown. However, the complete description of the dynamics of both integrable cases is not given. The gaps in the aforementioned work were completed by Llibre and Valls in [13], who obtained and characterized the full global flow.
The goal of this paper is to provide results on the existence of periodic solutions, their stability and KAM 2-tori for ¡ < 0, a,b G R and positive energy in a neighborhood of the origin. This is achieved by performing an explosion of the origin and a combination of different techniques such as normal forms, symplectic reduction and averaging theories. The key idea of our analysis is that the origin is an elliptic fixed point of the Hamiltonian system associated to (1.1), and that can be seen as a quartic perturbation of two coupled oscillators in 1:1 resonance. We are able to detect periodic solutions and KAM 2-tori which were not reported in previous works. This is done in a set of coordinates that fits better in the problem and allows one to avoid certain technicalities of other approaches related to the 1:1 resonance. We have used Mathematica software to perform all calculations and to make the figures that are essential in the understanding of the results.
We would like to point out that Deprit [5] introduced the so-called Lissajous variables which deal with perturbations of coupled oscillators in 1-1 resonance. Later on, Deprit and Elipe [6] used the Lissajous transformation in the study of perturbed elliptic oscillators shown by an example where the perturbing term is a quartic polynomial. They proved that the reduced phase space is a two-dimensional sphere, where each point represents an orbit. Indeed, the potential in the Hamiltonian (1.1) is a quartic polynomial, however, we have chosen to use a similar, but different, approach, which can be preferable when applying theorems given in the Appendix. Therefore, we will draw the same conclusions as those in [6].
The outline of this paper is as follows. In Section 2 we perform a symplectic scaling of the AGK Hamiltonian system for i < 0 by introducing the small positive parameter e. Then we do several symplectic reductions and normalizations in order to put the Hamiltonian H0 + e2H2 and the Hopf map in the convenient coordinates (L,Q,£,P). Now the case e = 0 is studied and it ends with the averaged Hamiltonian. In Section 3 we consider this in terms of the coordinates (Q,P). We obtain conditions on the parameters a and b for the existence of the nondegenerate critical points, as well as their linear stability. Section 4 is devoted to the use of KAM theory to establish the persistence of KAM 2-tori in the full system, and the elliptic periodic solutions, more precisely, we use the Han, Li and Yi theorem (Theorem 7 in Appendix) that is valid for highly degenerate Hamiltonian systems. Here there is also the application of the Reeb theorem (Theorem 5 in the Appendix) to find that each nondegenerate critical point gives rise to a family of periodic solutions of the full system. In order to make this paper self-contained, we decided to include an appendix. As far as the main theorems are concerned, we present results on averaging theory and KAM theory and stability for Hamiltonian systems.
2. Normalization and reduction
In this section we go through processes of reduction and averaging in order to enlighten the properties of the AGK-Hamiltonian near the origin.
In the AGK-Hamiltonian (1.1), we scale the variables by
x1 — e(-i)-l/4x1, x2 — e(-i)-1/4x2, ^
yi — e(-i)1/4 yi, y2 — e(-i)1/4 y2,
where e is a positive small parameter; this is symplectic with multiplier e2. Applying this change of coordinates to the Hamiltonian (1.1), it assumes the form
H£(x,y)= Ho(x,y)+ e2H2(x), (2.2)
after rescaling time by dt — (-¡)-1/2dt and resetting (-¡)-3/2e2 — e2; where
Uo = \(x2+x2 +y\+yl) (2.3)
Thus, we have a Hamiltonian perturbation of two harmonic oscillators in 1:1 resonance.
H2 = - + • (2.4)
For a moment, we are going to focus on the unperturbed case e = 0. The equations of motion derived from H0 are
x i = yi, yi = -xi, x 2 = y2, V2 = -yi-
We recall that every solution of this system is periodic with period 2n.
In order to describe the orbit space of (2.3), we fix the value H0 = h G R+, such that it describes the 3-sphere of radius V2h, given by
Mh = {(xi,X2,yi,i2): x2 + x2 + y2 + y2 = 2h}.
Thus, we localize around the origin by performing three different changes of symplectic variables. First, we use action-angle variables (1,4) = (I1,I2,41,42)
zl = v^lcos 01, z2 = cos 02,
r~ (2-5)
yi = y/2Ii sin 01, y2 = \J2l2 sin 02,
where 0 ^ I\, I2 ^ h and Ho = h + /2- We see that. I\ = \{x\ + y'\) and I2 = \{x?2 + J/|) are first integrals and not simultaneously equal to zero. As the next step, we introduce the linear symplectic change of variables (L,J2,£,0) through the transformation given by
L = Ii +12, J = I2, £ = 4i, 0 = 02 - 0i,
where 0 ^ £, 0 < 2n and 0 ^ J ^ h. In this symplectic transformation, the unperturbed Hamiltonian (2.3) is converted into H0 = L.
Now we introduce the rectangular symplectic coordinates by (J, 0) through the transformation
Q = V2J cos 9, P = V2J sin 9, (2.6)
where 0 ^ Q2 + P2 = 2 J ^ 2L = 2h. The Hamiltonian function of the harmonic oscillator now reads H0 = L.
In addition, if xi + y2 = 0, then 0 ^ Q2 + P2 < 2L = 2h. Finally, we summarize all the previous successive changes of coordinates as the transformation Ti: Q ^ Mh defined by
xi = \/2L-Q2 -P2 cos £, yx = s/'2L - Q2 - P2 sin^,
(2-7)
x2 = Q cos £ — P sin £, y2 = P cos £ + Q sin £,
where Q = {(L,Q,£,P): Q2 + P2 < 2L,£ G S1}. We stress the fact that the image of Ti excludes the set x"2 + y\ = 0. In this case we have that x2 + y| = 0 and can perform a similar coordinate change, namely, T2: Q ^ Mh defined by
x1 = Q cos £ — P sin £, y1 = P cos £ + Q sin £,
x2 = y/2L - Q2-P2 cos £, y2 = y/2L - Q2 - P2 sin £.
We have that T2 = R o T1, where R(x1,x2,y1,y2) = (x2,x1,y2,y1). Hence, the set
A = {Ti: Q ^ Mh, T2 : Q ^ Mh}
becomes an atlas of symplectic charts, and therefore Mh is a symplectic manifold.
(2.8)
The inverse of the transformations T\ and T2 are expressed, respectively, as
^i „,2 , „,2 i „2 r> ___j-__( yi
L = -(X! +x2 + yl+y2), £ = arctan
2 \xi
_ XlX2 + yiy2 p _ Xiy2 - x2yi
Vxi + y'i ' Vxi + y'l '
and
^f_,2 , 2 , „2 I „ 2\ П ___( y2
(2.9)
L = тт(ж1 + ж2 +У1+У2), Z = arctan _ 2 x2
_ X1X2 + У1У2 p _ Х1У2 - Х2У1
VX2 + y\ ' \/ж2 + y\
(2.10)
-<2 ' У2 V J/2
The Hamiltonian in both charts has the form "Ho = So, the equations of motion derived from it are
L = 0, <¿ = 0, i = -1, ib = 0. (2.11)
Hence, £(t) = — t + and L, Q and P are constant. Since I is an angle variable, these solutions are 2^-periodic.
For our purpose we need to reduce the phase space of the two-dimensional harmonic oscillator by using the symmetry given by the S^action generated by the flow of the Hamiltonian (2.3). Keeping H0 = h fixed, we address this issue for the Hopf fibration given by the map
П: Mh = S3 ^ S2, (2.12)
where the great circles of S3 correspond one-to-one to points on the two-dimensional sphere S2. Thus, the orbit mapping becomes a symplectic reduction of the surface Mh = S3 to the reduced phase space S2. It is given explicitly as follows:
П: S3 ^ S2, (2.13)
(Xl ,Х2,У1, У2) ^ (Wl,W2 ,W3),
where
Wi = 2(xi У2 — Х2У1), W2 = 2(X1X2 + У1У2), W3 = X2 + У2 — X2 — y\, (2.14) and satisfies the relation
(Xl + X2 + У2 + у2 )2 = W2 + W2 + W2 = 4h2. (2.15)
For this, we refer the reader to [16].
Hence, relations (2.9) and (2.10) imply that the coordinates Q and P in terms of (2.14) are transformed to
Q= , W2 , P= , W1 , (2.16)
y/2 (w3 + 2 h) s/2(w3 + 2 h)
or
Q= , W2 , P = - , Wl ■ (2.17)
y/2(2h - w3) \j2(2h - w3)
Let Ui = S2 \ {(0,0, —2h)} and U2 = S2 \ {(0,0,2h)}. It follows that
: Uj R2 j = 1,2
(Wl ,W2,W3) — (Q,P)
are local symplectic charts for the reduced space S2, respectively.
We shall specify an atlas A on the 2-sphere with two charts defined as the composition of the inclusion (L, Q, P) ^ (L, Q, 0, P)
Ti: Q ^ S2, T2 : Q ^ S2,
where T = {(L, Q, P): Q2 + P2 < 2h and L = h}. Then locally the Hopf map is
n: Q ^ Q1, (2.18)
(L,Q,£,P) ^ (L,Q,P).
Indeed, periodic orbits of the system (2.11) are sent to orbits in S2, namely, fixed points.
It is worth noting that the changes of coordinates (2.7) and (2.8) correspond to a particular case of the 1:1 resonance in local symplectic maps for resonant Hamiltonian systems with n degrees of freedom (for more details, see [5] and [17]).
Now we come back to the analysis of the Hamiltonian H£, which is a smooth Hamiltonian perturbation of two harmonic oscillators in 1:1 resonance by a quartic potential. Let us observe that the parameter e measures the smallness of the quartic perturbation. The sets Mh,£ = = {(x1,x2,y1,y2): H£ = h}, where h > 0 and e ^ 0 is small enough, are invariant manifolds.
It follows that there is a homotopy R: Mh x [0,e0] ^ R4 such that for each 0 ^ e ^ e0 the map R£ = R(-,e) is a diffeomorphism between the 3-sphere Mh and Mh,£. Moreover, the set A£ = {01: Q ^ Mh, 02: Q ^ Mh}, where = R£ ◦ Tk, is an atlas of Mh,£, see [9, 11]. In each chart of A£, the Hamiltonian H£ reads as
%£ = L + e2H2(L,Q,£,P). (2.19)
We can eliminate the angle £ from the Hamiltonian to a certain order with the aid of the Lie transformation method. We end up to order four in the small parameter with the following Hamiltonian:
H£ = L + e2H2(L,Q,P) + 0(e'1), (2.20)
where
n2 = [6a,L2 + (2L -Q2- P2) (3bQ2 + (b - 2a)P2)], (2.21)
see [14, 16]. We observe that L = 0(e4) and L is not longer a constant, but its variation is very small. Therefore, the map (2.12), which in the charts of A£ is (2.18), does not send orbits into orbits.
3. The averaged Hamiltonian
Let, H£ = L + e2T~L2 be the Hamiltonian equal to up to terms of order four in t. We notice that, is independent, of £, and thus its conjugate variable L is constant,.
In order to obtain the Hamiltonian H£ on the reduced space S2, we replace L by h > 0 in (2.21), eliminate the factor 1/16 by scaling the time, and remove the constant term —6aL2, arriving at
H = —(2h - Q2 - P2) [36Q2 + (b - 2a)P2]. (3.1)
So the equations of motion become
Q = 4P [(2b - a)Q2 + (2a - b)(h - P2)], P = 4Q [3b(h - Q2) + (a - 2b)P2].
Proposition 1. Let ô = 2a/b. The .system (3.2) has the following critical points on S2 (see Table 1):
1. For ô G R \ {—2,1}7 there are six isolated critical points, Oj, j = 1, . . ., 6.
2. For ô = —2, there are two isolated critical points: Oi and O2. In addition, there is a continuous set of critical points given by Q2 + P2 = h.
3. For ô = 1, there are two isolated critical points: O3 and O4. Additionally, the line Q = 0 is a continuous set of critical points.
4. For b = 0 there are two isolated critical points: O5 and O6. Moreover, the line P = 0 is a continuous set of critical points.
Proof. We recall that the system (3.2) is the flow in local coordinates. A direct computation shows that for ô G R\{—2,1} the system (3.2) has nine critical points, but only five of them satisfy the condition Q2 + P2 < 2h. In addition, the point (0,0) corresponds to different critical points in the flow on S2 and the other four appear in both charts, therefore there are six critical points.
The rest of the statement is obtained in a similar way. □
Table 1. Critical points of H in the variables (Q,P) and in S2
Critical point 2i Ï2 s2
Oi (0,0) -- (0,0,2ft.)
02 -- (0,0) (0, 0, -2ft)
Os (Vft, O) (v/1,0) (0, 2ft, 0)
o4 (-VI, o) (-\/M) (0, -2ft, 0)
o5 (o,Vhj (o ,-Vhj (2ft, 0,0)
Oe (o, -Vh) (o,Vh) (-2/?,0,0)
The number of isolated critical points of the system (3.2) in the plane of parameters a and b is shown in Fig. 1.
Remark 1. Each critical point Oj on S2 is associated with a periodic orbit on S3 parameterized by £. In fact, from (2.12), this curve is n_1(Oj). A direct computation shows the projection of n_1(Oj) on the configuration plane x1 x2, see Table 2.
Table 2. Critical points and their projection onto the configuration plane x1 x2
Critical point projection of II~1(Oj) on the configuration plane x\ xo
Ox segment in the x\ axis
02 segment in the x<± axis
Os segment of line outside the coordinate axes
o4 segment of line outside the coordinate axes
o5 circles with center at the origin and radius \fT%
Oe circles with center at the origin and radius Vft
a
Fig. 1. Number of isolated critical points of the system (3.2) in the plane of parameters a and b. The green and orange lines correspond to b = 0 and 2a — b = 0 (S = 1), respectively. The blue line is a + b = 0 (S = —2).
Now, for convenience, we define the sets of critical points
R = {Oi, O2} , M = {O3, O4} , C = {O5, Ob}.
The next result comprises the information on the stability of the critical points.
Proposition 2. Under the assumptions of Proposition 1, the linear stability of the critical points is as follows:
1. The critical points in R are stable for S < 1 and unstable for S > 1.
2. The critical points in M are stable for S > —2 and unstable for S < —2.
3. The critical points in C are stable for S < —2 or S > 1 and unstable for —2 < S < 1.
4. For b = 0 and a = 0 the critical points in C are stable.
Proof. Let A = JD2HL, where J denotes the standard skew-symmetric matrix of order 2. Its characteristic polynomial is p(X) = A2 + det A. The theorem follows from Table 3. □
The change in the number of critical points of the reduced Hamiltonian (3.1) is related to the emergence of bifurcation curves in terms of the parameters a and b, see Fig. 2.
Remark 2. The reader should be warned that, at S = to (b = 0), there is a degeneracy such that the flow is similar to that shown in the panel (g) of Fig. 2, but rotated through 90 degrees.
Table 3. Eigenvalues associated with the critical points
set of critical points det A eigenvalue: A
n -4852/?2(c5 - 1) ±4\/3|5|/?V<5 - 1
M 4852/?2((5 + 2) ±4v/3|6|/7V-(<5 + 2)
C C with 6 = 0 1652/?2((5 — 1)((5 + 2) 64/?2 a2 ±4\b\hy/-(ö-l)(ö + 2) ±8\a\hi
(a) 6 e (-œ, -2)
(b) 6 « -2"
(c) 6 = -2
(d) 6 « -2+
(e) 6 e (-2,1)
(f) 6 « 1-
0.0 Q
(g) 6 = 1
0.0 Q
(h) 6 « 1+
0.0 Q
(i) 6 e (1, +œ)
Fig. 2. Flow evolution of the reduced Hamiltonian (3.1) in the local chart (U1,^1). The red dots correspond to stable critical points (centers) and the blue points are unstable points (saddles). The thick brown line in panel (g) corresponds to a continuous set of critical points.
4. Reconstruction of the dynamics of the AGK-Hamiltonian from the averaged system
This section is devoted to obtaining conditions for nondegeneracy of the Hamiltonian (2.20) at the critical points that generate families of periodic orbits and KAM tori described in terms of the parameters.
4.1. Periodic orbits
Let p(t,e) = (x1(t,e),x2(t,e),y1(t,e),y2(t,e)) be a solution of the system related to the Hamiltonian (2.2) and let p* = n_1 (Oj) denote the periodic solution associated with the critical point Oj. Now we state our main result on the periodic solutions associated with the Hamiltonian (2.2).
Theorem 1. For the system associated with the Hamiltonian (2.2), there are at most 6 T(e)-periodic solutions p(t, e) such that p(t, 0) = p* and T(0) = 2n. More precisely:
1. The periodic solutions generated by the critical points in R are linearly stable when 5 < 1 and unstable in the Lyapunov sense when 5 > 1.
2. The periodic solutions generated by the critical points in M are linearly stable when 5 > —2 and unstable in the Lyapunov sense when 5 < —2.
3. The periodic solutions generated by the critical points in C are linearly stable when 5 < —2 or 5 > 1 and unstable in the Lyapunov sense when —2 < 5 < 1.
4. If b = 0 and a = 0, there are two linearly stable periodic solutions generated by the critical points in C.
The period of the periodic solutions is given by T(e) = 2n(1 — e2T*) + 0(e4), where the corrections T* are listed in Table 4. Also, the characteristic multipliers of the periodic solutions are
1,1,1 ± t2Ai + 0(t4), with Ai = —,
16
where X is one of the eigenvalues of A = JD2H given in Table 3.
Proof. Reeb's Theorem 5 implies that, each critical point, will rise to a family of periodic solutions provided that. dei{D2/H) / 0. In addition, the kind of stability of the periodic solution is inherited by the corresponding stability of the critical point. Hence, Propositions 1 and 2 imply the first part of the theorem. It remains to prove the last part concerning periods of the periodic orbits.
The Hamiltonian (2.20) yields
dL dL
Let £(0) = 0 and £(T) = 2n, then T is the period of the orbit and
t t _
2tt = £(T) - £(0) = j ¿(t)dt = T + e2 j dt + 0(t-4}_
0 0
A straightforward use of the implicit function theorem shows that T(e) = 2n(1 — e2T*) +0(e4), where T* is shown in Table 4. □
Table 4. Corrections of the periods of the periodic solutions
Set of critical points rp*
n 6bhö
M 6bh(S+l)
C C with 6 = 0 2bh(2S + 1) 8 ah
Remark 3. It is worth noting that using the sets of symplectic coordinates (2.7) and (2.8), we can parameterize each periodic solution of the Hamiltonian (2.2) in terms of the angle I = ¿(t). More precisely, if pj(t,s) = (xl(t,e),x2(t,e),yl(t,e),y2(t,e)) is the periodic solution associated to the critical point Oj, then
x1(£,e) = x^)+ £2xj^)+ O(£3), x2(¿, s) = x0,^) + £2x2^) + O(s3),
yi(^e)= y01^) + £2y2W + O(s3), y2(¿, s) = y°^) + e2ylW + O(s3),
where the expressions for the functions xjk (¿) and y°k (¿) appear in Tables 5-7.
Table 5. Expression for the functions x°k(¿) and yk(¿), k = 1,2, at the critical points
Critical point x'UC) x%(i) yl¡(0 ym
O i \/2h. cos Í 0 V2hsme 0
02 0 \/2h cos Í 0 V^sini
Os A/TJCOS Í — \[T% cos Í a/Tj sin Í — y/h, sin Í
o4 V~h cos Í \/7¿cos Í \fh. sin Í \fh. sin Í
o5 %/TJCOS Í \fh. sin Í \fh. sin Í — \fh. cos Í
Oe %/frcOS Í — \/7% sin Í \fh. sin Í \fh. cos Í
Table 6. Expression for the functions x2k(¿), k = 1, 2, at the critical points
Critical point x2(0 xUn
Ox o2 Os o4 o/?3/2(7 2cos2ñ cosí 8A/2 0 (2a + b)h3/2 v ; (7 2 cos 21) cos Í (2a + b)h3/2 v ; (7 2 cos 21) cos Í 0 o/?3/2(7 2cos2ñ cosí 8a/2 (2a + b)h3/2 v ; (7 2 cos2í) cos Í (2a + b)h3/2 v ; (7 2 cos2í) cos Í
o5 Oe (8a + b + 2b cos 2d) cos (' i-,3/2 /?39 (8a + 6 + 26 cos 2C) cos C h^(8a + b 26 cos 2d) sin (! /,3/2 /?39 (8a+ 6 26 cos 2Í) sin Í
Table 7. Expression for the functions (£), k = 1,2, at the critical points
Critical point ym ym
Ox 02 03 04 3a/r*/2 (3 2 cos 2(') sini 0 3(2a + 6)/?3/2 ----^-(3 + 2cos2í) sin Í 3(2a + 6)/?3/2 ----^-(3 + 2cos2í) sin Í 0 3a/?3/2 /3 2 cos 2d) sin t 8a/2 3(2o + 6)/?3/2 —-^-(3 + 2 cos 2i) sin Í 3(2o + 6)/?3/2 ----^-(3 + 2 cos 2i) sin Í
05 06 ( 8o + 6 + 66cos2í)siní ( 8o + 6 + 66cos2í)siní /?3^(8 a 6 + 66 cos 2d) cosí /?3^(8 a 6 + 66 cos 2d) cosí
4.2. KAM tori
In this section we apply KAM theory to the Hamiltonian (2.20). In order to avoid confusion with the hypothesis or strength of the KAM theorem, we provide a version due to Han, Li and Yi [10]. The theorems displayed below are similar, but they should be separated due to their subtle differences.
We start by rewriting the reduced Hamiltonian (3.1) in terms of 5, which yields
H = -b(2h - Q2 - P2) [3Q2 - (5 - 1)P2]. (4.1)
Remark 4. We have the totally integrable cases for 5 = -2, 1 and b = 0, therefore, their phase space is foliated by invariant tori.
Theorem 2. If 5 £ (-x>, 1), there are invariant KAM tori of dimension two, enclosing the periodic orbits associated with the critical points of the set R. The measure of the complement of their union is of order O(e3).
Proof. Since the points in the set R are elliptic in (Q, P), we use the action-angle coordinates defined by
/1 _ x \ i/4 _ / o \ 1/4
Q=l — J s/2heos0, p=l— J y/2hsmd. (4.2)
In these variables, the Hamiltonian (4.1) becomes
n = —4bh\/3(l - 5)h - 2bif [8 - 4 + (6 + 2) cos 29}. (4.3)
The next step is to reverse the operations that made possible the transformation from H£ to H. This is done by identifying h with the action and adding the dropped term —6aL2. We also introduce a change of time scale in the Hamiltonian by multiplying it by e2/16 and adding the dropped term L. We end up with
H£(L,h,£,e) = L + s2hl{Il,0)+O{sA), (4.4)
where
Uh,e) = -^L2- h _ hi2 [5- 4 + (5 + 2) cos29).
16 4 8
Since hi in (4.4) depends on the angle 0, we cannot directly apply a KAM theorem to obtain 2D-tori. However, since the action Ii close to equilibria is small, we have that Ii is bigger than if. Then, in a small neighborhood of the equilibria, we can replace Ii by eli in the Hamiltonian (4.4). This means that we are approaching Ii = 0, that is, the equilibrium point. So that, hi can be split, into two orders in the parameter t to give
3
He{L, h,£, 6) = L — s2—b5L2 - es^-bVT^Ô LIX + 0(e4)
16
4
(4.5)
This Hamiltonian has the form (A.1) as needed in Theorem 7 of the Appendix, with h0 = L,
hi = — ^rL'2, and ha = —^ Furthermore, with its notation, n = 2, a = 2, mi = 2,
m2 = 3, no = m = 1, n2 = 2, Ino = Ini = L, = (Lji), T° = T1 = L and T2 = while the frequency vector becomes
and the 3x3 matrix with columns Q, dQ/dL and dQ/dli is given by
1 0 0 0
36(5 t
36(5
8
4
4
Since this matrix has rank 2 for all values of 5 G (-x>, 1), the result follows. Moreover, taking into account the Remark 5 in the Appendix, we obtain /3 = Y1 i=i m-i&i — ni-i) = 3 and s = 1. Now, since the perturbation is of order O(e4) and = e3+n > e4 for 0 < n < 1/5, the
measure of the set where we cannot guarantee the existence of invariant tori is of order O(e3).
□
Theorem 3. If 5 G (—2, there are invariant KAM tori of dimension two enclosing
the points in M. These invariant tori form a majority in the sense that the measure of the complement of their union is of order O(e3).
Proof. We proceed in a similar way as in the proof of Theorem 2. So, we just point out the differences. First, we shift the origin to the critical points and scale the variables by
Q = Q=f VTi,
P = P,
where "—" applies for O3 and "+" corresponds to O4. Next,, we normalize the quadratic part, through the symplectic change (Q,P) —> (h,9) given by
h cos 9, P = 2
3
1/4
3 j — - -^5 + 2 ^
Hence, at both critical points the Hamiltonian (4.1) takes the form
sin 0.
H = -36/?2 + 4bhy/S(8 + 2)Ii ±
2 x 31/4 b Vhj'f2 cos 9 [10 -6 + 3(S - 2) cos 26] (Ö + 2)1/4 '
(4.6)
(4.7)
where the upper sign applies for O3 and the lower one for O4.
At this step, we must undo the operations to pass from H£ to H. First, we identify h with the action L and rescale time by multiplying the Hamiltonian (4.7) by e. Next, we add —6aL2, rescale time again by multiplying the Hamiltonian by e2/16 and add L. Then, after some simplifications the resulting Hamiltonian in action-angle coordinates reads as
where
He(L,h,e,0) = L + e2h1(I,0)+0(e4), (4.8)
if2 cos e [10 - 6 + 3{5 - 2) cos 26 ]
=+aMmLh ± 31/4"
16
4
8(5 + 2)1/4
We note that, this function hi depends on angle thus we cannot, directly apply a standard KAM theorem to obtain 2D-tori. However, since the action I1 close to equilibria is small, we have that. Ii is bigger than if. Hence, by doing Ii —>• eli in the Hamiltonian (4.8), we find that., in a small neighborhood of the equilibria, we can split, hi into two orders in the parameter e, namely,
U£(L,h,C,9) = L-e
23&(S + l)r2 , _3 b-\/S(S + 2)
16
-L +t
In order to apply Theorem 7, we set h0 = L, h1 = —
4
36(5-
Lh+0(s4).
(4.9)
—L2, and h2 =
6v^T2)rT
-LI i.
16 ' z 4
Taking n = 2, a = 2, mi = 2, m2 = 3, n0 = = 1, n2 = 2, Ino = Ini = L, Im = (L,Ii)
j.no _ _ ^ j,n2 _ ^^ ^^^ _ _ ^ anc| j^j = we obtain the frequency vector
yn 1
T T"o
T"2 T
Q = Q2, Q3) =
'dho dhi dh2\
'дL,~дL,W¡),
and the 3x3 matrix with columns Q, dQ/dL and dQ/dli, given by
1
0
0
36(5+1) 36(5+1)
6^3(5 + 2) 6^3(5 + 2) -4-^ -4- °
Since the rank of the matrix is 2 for 5 £ (—2, +rc>), the result follows. Moreover, from Remark 5 in the Appendix, we obtain /3 = 2=1 mi(ni — ni-1) = 3 and s = 1. As a consequence, as the perturbation is of order O(e7/2) and es/3+v = e3+v > e1/2, the set where we cannot guarantee the persistence of invariant tori has a measure of order O(e3). □
Theorem 4. If 5 £ (—&>, —2) U (1, to), there are invariant KAM tori of dimension two, enclosing the C critical points. These invariant tori form a majority in the sense that the measure of the complement of their union is of order O(e3).
Proof. As was done before, we apply the following change of coordinates:
Q = Q,
p = p +
(4.10)
where the upper sign applies for O5 and the lower one is used for O6. Now we normalize the quadratic part through the symplectic change
'5- 1 \1/4
q = 2[JT2)
r1 cos d,
1/4
r1 sin d,
(4.11)
so, after some algebraic manipulation, we find that the Hamiltonian (4.1) becomes
H
bh\S-l)-4bhy/(S - 1)(5 + 2)I1^2bVh(§=±) ^I3/2 sin0 [3(5-2)+(5-10) cos 20). (4.12)
<5+2 J
Next, proceeding in a similar way as in the proof of Theorem 3, we incorporate the terms associated to L and undo the time scalings, obtaining the Hamiltonian
H£ = ho(L) + e?hi(I,d) + 0(e*)
(4.13)
where
^j^veMu,,» fiziyVi1"
16 4 8 \d+2 / 1
'sin 6 [3(S-2)+(S-10) cos 20].
Now we proceed as in the previous proofs, scaling the action variable I\ —> sl\ in a small neighborhood of the equilibrium points such that the Hamiltonian (4.12) becomes
H£(L,h,£,Q)=L.
(4.14)
To apply Theorem 7, we take ho = L, hi = --^—j^-^-L2, and h2 = —^"
-Lh.
Here, we have n = 2, a = 2, mi = 2, m2 = 3, no = m = 1, n2 = 2, In° = I'21 = L, J"2 = (L, h),
j.no _ j-ni _ ^ j.n2 _ ^ j-n0 _ j-m _ ^ j.n.2 _ ^ ^^ ^ freqUenCy vector is
Hence, we deduce that the 3 x 3 matrix whose columns are Q, dQ/dL, and dQ/dI\ is
1
b(25 + 1)
L
0
b(25 + 1)
-!)(<? +2) £ by/(5- !)(<* +2) Q
which has rank 2 for 5 £ (-x>, -2) U (1, +rc>). Therefore, by Han, Li, and Yi's theorem (Theorem 7), the first part of the theorem is concluded. Now, by applying Remark 5 in the Appendix, we find that /3 = 3, s = 1 and, since the perturbation is of order O(e7/2) and es/3+v = e3+v > e1/2, we find that the measure of the set where we cannot ensure the persistence of invariant tori is of order O(e3). □
0
8
8
Appendix
Let (M, Q) be a symplectic manifold of dimension 2n and H0: M — R a smooth Hamiltonian which defines a Hamiltonian vector field Y0 = (dH0)# with symplectic flow Let I C R be an interval such that each h £ I is a regular value of H0 and N0(h) = H-1(h) is a compact connected circle bundle over a base space B(h) with projection n: N0(h) — B(h). So, this is the setting of regular reduction theory. Assume that all the solutions of Y0 in N0(h) are periodic and have periods smoothly depending only on the value of the Hamiltonian; i.e., the period is a smooth function T = T(h).
Let e be a small parameter and let Hl: M ^ R be smooth, H£ = H0 + eHl, Ye = Y0 + + eYi = dH#, N£(h) = H- l(h), n: N£(h) ^ B(h) the projection, and be the flow defined by Yi.
Let the average of Hl be
t
H = Hi(4)dt.
o
The next result provides sufficient conditions for characterizing the existence of periodic solutions of the Hamiltonian system associated to H£. For more information on this subject we refer the reader to [14, 15, 20].
Theorem 5 (Reeb). If H has a nondegenerate critical point at, ir(p) = p G £>(/?,) with p € N0(h), then there are smooth functions p(e) and T(e) for e small with p(0) = p, T(0) = T, and p(e) € N£, and the solution of Ys through p(e) is T(e)-periodic. In addition, if the characteristic exponents of the critical point p (that is, the eigenvalues of the matrix A = JD27i{j5)) are Xl, X2,..., X2n—2, then the characteristic multipliers of the periodic solution through p(e) are
1,1,1 + eXiT + 0(e2), 1 + eX2T + 0(e2),..., 1 + eX2n—2T + 0(e2).
Theorem 6. Let. p andp be as in the previous theorem. If one, or more, of the characteristic exponents Xj is real or has nonzero real part, then the periodic solution through p(e) is unstable. If the matrix A is strongly stable, then the periodic solution through p(e) is elliptic, i.e., linearly stable.
The proofs of Theorems 5 and 6 appear in [14].
Consider a Hamiltonian system of the form
He(I,v,e) = ho (In0) + emi hi (Ini) + ... + ema ha (Ina) + ema+l p(I,p,e), (A.1)
where (I, € Rn x Tn are action-angle variables with the standard symplectic structure dI A dp, and e > 0 is a sufficiently small parameter. The Hamiltonian H£ is real analytic, and the parameters a, m,Ui (i = 0,1,..., a) and mj (j = 1,2,..., a) are positive integers satisfying n0 ^ nl ^ ... ^ na = n, ml ^ m2 ^ ... ^ ma = m, Ini = (Il,...,Ini), for i = 1,2,...,a, and p depends on e smoothly.
The Hamiltonian H£(I,p,e) is taken in a bounded closed region Z x Tn c Rn x Tn. For each e the integrable part of H£,
Xi(I) = ho (In0) + emi hl (Ini) + ... + ema ha(Ina),
admits a family of invariant n-tori T£ = {(} x Tn, with linear flows {x0 + w£(()t}, where, for each Z € Z, w£(() = VX£(() is the frequency vector of the n-torus T£ and V is the gradient operator. When u£(() is nonresonant, the n-torus T£ becomes quasi-periodic. We refer to the integrable part X£ and its associated tori {T £} as the intermediate Hamiltonian and intermediate tori, respectively.
Let In' = (Ini_1+1,..., Ini), i = 0,1,..., a (where ??._i = 0, hence = In°), and define
Q = (Vr o h0(Ino),..., VTc hna (Ina)), such that, for each i = 0,1,..., a, Vj«¿ denotes the gradient with respect to
We assume the following high-order degeneracy-removing condition of Bruno-Russmann type (so named by Han, Li, and Yi), giving credit to Bruno and Russmann, who provided weak conditions on the frequencies guaranteeing the persistence of invariant tori, the so-called (A) condition: there is a positive integer s such that
Rank{9rQ(I): 0 < \r\ < s} = n, VI E Z.
For the usual case of a nearly integrable Hamiltonian system of the type
H£(I,p,e) = X(I) + ep(I,p,e), (I,p) e Z x Tra c Mra x Tra, (A.2)
condition (A) given above generalizes the classical Kolmogorov nondegenerate condition that dQ(I) be nonsingular over Z, where Q(I) = VX(I); Bruno's nondegenerate condition that Rank{Q(I),dQ} = n, for all I e Z; and the weakest nondegenerate condition, guaranteeing such persistence provided by Russmann, that w(Z) should not lie in any (n — 1)-dimensional subspace. Russmann's condition is equivalent to condition (A) for systems like (A.2). However, Bruno's or Russmann's conditions do not apply to the Hamiltonian (A.1), as it is too degenerate.
The following theorem gives the setting that makes certain the persistence of KAM tori in Hamiltonians like (A.1).
Theorem 7 (Han, Li and Yi). Assume we are given condition (A) and n with 0 < n < 1/5. Then there exists an e0 > 0 and a family of Cantor sets Z£ c Z, 0 < e < e0, with \Z \ Z£\ = = O(en/s), such that each Z e Z£ corresponds to a real analytic, invariant, quasi-periodic n-torus Tj of the Hamiltonian (A.l), which is slightly deformed from the intermediate n-torus T?. Moreover, the family {T^: ( e Z£, 0 < e < to} varies Whitney smoothly.
See the proof in [10].
Remark 5. In contrast to the case for a usual nearly integrable Hamiltonian system, the excluding measure for the existence of quasi-periodic invariant tori in the properly degenerate case is of a fairly large order of ev/s for a pre-fixed small positive constant n, as shown in the theorem above. This is mainly caused by a normal form reduction which pushes the perturbation to an order higher than esl3+v for
a
¡3 = mi (ni - ni-i), i=i
for which the domain G needs to shrink by order en/s in measure. This is necessary for general properly degenerate Hamiltonian systems like (A.1) in order to apply the standard KAM iterations. However, if the perturbation in (A.1) is already of order O(es^+n), then a normal form reduction will not be necessary, and the excluding measure for the existence of quasi-periodic invariant tori can be improved to order e13.
Acknowledgments
We would like to express our warmest thanks to the anonymous referees for the careful reading of the manuscript and for the comments which allowed us to improve both the quality and the clarity of the paper. Moreover, we wish to thank Jesús Palacián and Patricia Yanguas for helpful comments and suggestions on this work.
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