Associated with the original Hamiltonian5 His the real analytic vector field X
H=f(u),
u> = (ˆθ>, x>,Iˆ>, y>). When a symplectic integrator with step size h is applied to the vector fieldf it induces a symplectic map Φh,f. In order to study whether periodic orbits
of f(u) persist in the numerical solution given by the integrator we want to embed Φh,f
in a modified vector field close to f(u) and ask whether the modified vector field still contains a periodic orbit. We assume that the symplectic integrator is given by a one-step method, Φh,f, analytic in bothhand u. Iterating the numerical method gives a numerical
trajectory {un} by generating the sequence of vectorsun:
un+1 = Φh,f(un), n= 0,1, . . . u0 =u(0).
The problem of embedding the map Φh,f in a flow means finding an analytic, modified
vector field ˜f which exactly interpolates un.
It is well known that it is possible to find an autonomous vector field whose flow is close to the numerical trajectory — we already proved in section 1.4.2 (theorem 1.4) that a local modified Hamiltonian exists for a symplectic integrator applied to a Hamiltonian vector field. The following theorem, presented in [98] states that there is always a local modified vector field which comes exponentially close to interpolating the numerical solution.
Theorem 4.3. [98] Let f be an analytic vector field in some open domainD ⊂Cdaround
the trajectory, and with a corresponding bound kfkD and let ϕh,f be the time-h flow of
5
We use the notationHfor the unperturbed Hamiltonian both before and after any change of coordi- nates and/or transformation to the semi-normal/Floquet form (4.2) or (4.7).
4.4. Embedding a map in the flow of a modified vector field 83
f. Let u1= Φh,f(u0) =u0+hf(u0) +O(h2) be an approximation produced by a one-step
method. Then there exists an autonomous modified vector field f ,˜ bounded on the smaller domainD ⊂ D˜ such that
ku1−ϕh,f˜(u0)k=O(hexp(−h0/hkfkD))
for a sufficiently small step size h and where the positive constant h0 depends on the
difference betweenD and D˜ and on the method.
Proofs of theorem 4.3 are given by Hairer and Lubich [51] for any B-series methods (e.g. Runge-Kutta methods), by Benettin and Giorgilli [11] for any symplectic map, and, in a general setting, by Reich6 [117]. Iterating the bound in theorem 4.3 for each step of a numerical integrator, one sees that the numerical trajectory stays exponentially close to
ϕnh,f˜, the flow of the modified vector field, for some finite time. Unfortunately this result is too weak for our intended use since a trajectory which is close to an invariant curve may diverge from it and may do so after only a short time in the case of exponentially divergent systems. For an arbitrary numerical trajectory, an autonomous flow interpolating the trajectory need not exist, and, in general, rarely does [110]. The following proposition from [98] is an example of the failure of maps, or diffeomorphisms, to embed into flows. Further discussion and examples of this point can be found in the book by Banyaga [8, 1.3.6].
Proposition 4.4. [98] There exist vector fields f and one-step methods Φh,f for which
no time-independent vector fieldf˜exists with time-h flow ϕh,f˜equal to Φh,f.
Proof. Consider the following special case of a diffeomorphism defined on the circle T.
First, note that arbitrarily close to the identity there is a diffeomorphism Φ : T → T
with unstable period−s points. That is, Φs(u0) = Φs−1◦Φ(u0) = u0 and |DΦs(u0)| > 1. Therefore, the Jacobian DΦns(u0) = (DΦs(u0))n can be made arbitrarily large for sufficiently largen.
On the other hand, if the map Φ = ϕh,f˜ is without fixed points, then so too is the vector field ˜f which generates ϕh,f˜. Then t0 =
R
T1/|
˜
f(s)|dsis finite and ϕt
0,f˜= Id. For n∈N, the decompositionn=t0bn/t0c+rn, 0≤rn≤t0implies Φn=ϕnf˜=ϕrn,f˜, hence,
ϕn,f˜is uniformly bounded in C1 and so DΦn is continuous and is also bounded. This is in contrast to the case for maps.
The proof above is an example of the sort of topological mismatch that occurs between flows and diffeomorphisms. Most diffeomorphisms have periodic points which are isolated from other periodic points with the same period, while a periodic point of a flow is isolated only if it is a stationary point.
6
The results concerning preservation of structural properties in [117] must be treated with caution as counter-examples are known for some cases [59].
84 Chapter 4. Preservation of Periodic Orbits
If, however, we allow the modified vector field to be time dependent then the inclusion of an (analytic) symplectic map in an (analytic) Hamiltonian flow is possible. The first publications to this effect are by Douady [34, 35] (however, these are not easily accessible, the first being a Ph.D. thesis, and both being in French).
Kuksin and P¨oschel [77] prove that inclusion in an analytic Hamiltonian flow with pe- riodic time dependence is possible for analytic symplectic maps which are perturbations of integrable systems. They also suggest that such a result can be used to show preservation of lower dimensional tori by symplectic maps — by combining the non-autonomous inclu- sion result with a result from P¨oschel [113] on preservation of elliptic lower dimensional tori of Hamiltonian flows. Compared with that in [113], the theorem by Jorba and Villa- neuva in [73], which we make use of, has the advantage that it holds for lower dimensional tori with any combination of elliptic and hyperbolic eigenvalues for the normal directions, and is hence more general.
Pronin and Treschev [114] use a time-averaging procedure to construct an analytic, non-autonomous, periodic flow which exactly interpolates analytic maps isotopic to the identity7 If the original map is symplectic, then the flow is Hamiltonian, (similarly for volume preserving or reversible maps). The theorem also holds for maps without such geometric properties.
If, in the symplectic case, the original map is close to integrable then so too is the Hamiltonian flow associated with it and the orders of the closeness are the same for both. More generally, if the map Φ is O(ε) close to a map ˆΦ, where ˆΦ is already included in the flow of a periodic analytic vector field ˆX, then Φ can be included in a vector field X
which is O(ε) close to ˆX. In the case of a symplectic integrator, of order p, applied to a Hamiltonian vector field XH, which contains a periodic orbit or lower dimensional torus,
we can take the time-h flow of the original vector field to be ˆX, that is ϕh,XH = ˆΦ. The map Φh,XH generated by the symplectic integrator is O(h
p) close to the exact flow map
ˆ
Φ, and, hence, Φh,XH can be included in a the flow of a Hamiltonian vector fieldX=XH˜
which is O(hp) close to the original flow which contained the periodic orbit or invariant
torus.
In [96, 98] Moan presents the following theorem which gives estimates on the size of the non-autonomous component of the modified vector field ˜f = XH˜. The proof of this theorem remains unpublished; though it is available as a preprint [97].
Theorem 4.5. [96, 97, 98] LetΦh,f be a one-step method and assume thatf(u)is analytic
for u∈ D ⊂Cd. Then there exists a modified vector field
˜
f(u, t, h) =f(u) +εr1(u) +εr2(u, t;h) (4.8)
7Two smooth maps Φ
i :M → M0, i = 0,1 of manifolds M, M0 are called isotopic if there exists a
family of maps ˆΦs :M →M0 of the same smoothness class and continuous in the parameter s∈ [0,1],
4.4. Embedding a map in the flow of a modified vector field 85
analytic inD ⊂ D˜ , analytic and h-periodic in t and with a flow that exactly interpolates the numerical trajectory {un} for all time. Additionally, if the step size is sufficiently
small then the time-dependent term is exponentially small inh. More precisely, for
hkfkδ1+δ2 < 2πδ2 e
the size of the non-autonomous term is bounded by
kεr2kδ1 ≤C·exp − 2πδ2 ehkfkδ1+δ2 , where kfkδ= sup z∈Dδ(x) |f(z)|∞, and Dδ(x) ={z∈Cd:|z i−xi| ≤δ, i= 1, . . . , d}, x∈Rd.
One can now see theO(hexp(−h0/hkfkD)) term in theorem 4.3 as being a consequence
of the non-autonomous termεr2 in theorem 4.5.
By theorem 4.5, and the results of Pronin and Treschevet al. we know that there exists an analytic vector field ˜f(u, t;h) =f(u) +εr1(u) +εr2(u, t;h) which is also analytic and
h-periodic in t. This modified vector field exactly interpolates the numerical trajectory
{un}and is symplectic. Since the non-autonomous perturbation is periodic in the time-like
variable, we can, by Floquet’s theorem write the associated Hamiltonian as
Hpert = ˜ω(0)I˜+εH˜(θ, x, I, y, ε),
whereε(r1+r2) =J−1∇Hpert, and where we have extended the phase space of the Hamil- tonian system to include the periodic non-autonomous component as a new time/angle variable ˜θ, withθ>= (ˆθ>,θ˜),and with ˜I conjugate to ˜θ, andI>= ( ˆI>,I˜). The frequency of the new angle variable is ˜ω(0) = 2hπ, ω(0)>= (ˆω(0)>,ω˜(0)).
The perturbed Hamiltonian associated with the modified vector field ˜f =f+ε(r1+
r2) =XH =XH+Hpert can be written as H(θ, x, I, y, ε) = ˆω(0)>Iˆ+ ˜ω(0)I˜ | {z } =ω(0)>I +1 2z >B z+1 2Iˆ >Cˆ I+H∗(ˆθ, x,I, yˆ ) +εH˜(θ, x, I, y, ε). (4.9)
The dependence of ˜H, and henceH, onεis due to the dependence of the perturbation sizekε(r1+r2)k of the modified vector field on the step size h. However, for the result which follows, we don’t make any use ofεas a parameter.
86 Chapter 4. Preservation of Periodic Orbits