Kam theory, Lindstedt series and the stability of the upside-down pendulum

We 
consider the planar pendulum with support point oscillating in the 
vertical direction; the upside-down position of the pendulum 
corresponds to an equilibrium point for the projection of the 
motion on the pendulum phase space. By using the Lindstedt series 
method recently developed in literature starting from the 
pioneering work by Eliasson, we show that such an equilibrium 
point is stable for a full measure subset of the stability region 
of the linearized system inside the two-dimensional space of 
parameters, by proving the persistence of invariant KAM tori for 
the two-dimensional Hamiltonian system describing the model.

1. Introduction 1.1. The state of the art. The upside-down pendulum with the support point oscillating with a frequency ω large enough has been extensively studied in literature as a simple model exhibiting a quite nontrivial behaviour; see [5] and the references quoted therein.
The stability of the upside-down position can be proven by the averaging method (see for instance [19], Ch. 9): the result is that if the support point oscillates fast enough then the upward equilibrium position becomes stable. However such a kind of analysis is not completely rigorous both because no explicit control on the corrections can be obtained and because it can lead to incorrect results, as already pointed out in [8] and [6]. In fact the averaging method approach can be followed also for studying the stability of the downward position, and the result one finds in doing so is that such a position is always stable -provided that ω is large enough to make it possible to apply the averaging, say ω > ω 1 , for some ω 1 -, a result obviously unacceptable as by varying ω above ω 1 one can lose even the linear stability (as it follows from Mathieu's equation theory). A rather complete review on the averaging method can be found in [2].
A rigorous proof of stability has been given for the linearized system in [1], where also the case of several pendula has been considered. In the latter case the linearized system can be written (by a diagonalization procedure) as a system of several uncoupled Mathieu's equations. So both for a single pendulum and for more than one pendulum, the theory of Mathieu's equation applies. In particular for a single pendulum the physical parameters describing the system have to be such that the parameters (a, q) appearing in Mathieu's equationẍ + (a + 2q cos 2τ )x = 0 lay inside the stability region corresponding to negative values of a (see [1]): if the amplitude of the oscillation of the support point is small enough (with respect to the length of the pendulum), then one has stability if the frequency ω is above a threshold value ω 0 (see below), and the same value predicted by the averaging method is found for ω 0 .

Stability and KAM theory.
What is missing in the literature is a rigorous discussion of the full system (not just the linearized one). In this paper we achieve such a task, by studying the full system by means of perturbation theory techniques.
In [2] it is suggested that KAM methods are necessary, but there is no explicit discussion; we think that it can be of interest to discuss in detail the problem by means of the Lindstedt series method recently developed in literature (see [4] for a list of references) both as an application of the general theory presented in [4] and as an occasion to improve the bounds in a case in which additional properties come into play. The method, based on a tree representation of the quasi-periodic solutions describing the KAM invariant tori, was originally introduced in [12], hence revisited in [14] and [11], and adapted to the case of isochronous systems in [4]: it is on the latter that we shall rely.
Since the linearized system can be written as a two-dimensional integrable Hamiltonian system, the full system becomes a perturbation of an integrable one: so KAM theory applies. Then we can prove that a large quantity of invariant tori persists under the perturbations, and the nearer the initial data (θ(0),θ(0)) are to the upside-down position (π, 0) the nearer the curves obtained by projection of the tori on the pendulum phase space (θ,θ) are to such a position: this proves the stability of the upside-down position for the pendulum.
Note that in applying the above argument it is a fundamental fact that the system is a two-dimensional one, so that the existence of the tori yield a topological obstruction for trajectories starting inside a torus to cross it and move far from it. So such a result cannot be used in order to study the stability of the upside-down position in the case of more than one pendulum: as a matter of fact, in the latter case, it is even likely that Arnol'd diffusion can occur and, as a consequence, the position is not stable at all.

Contents.
As far as the proof is concerned, in such a note we simply show that the system can be written in terms of action-angle variables, and gives rise to a perturbed isochronous system; thus we can refer to [4] for the proof of persistence of KAM invariant tori within the framework of the Lindstedt series approach.
Note that, once verified that for our system the conditions under which the KAM theorem for perturbations of isochronous systems apply, one could proceed also by means of the usual KAM techniques. We prefer to use the Lindstedt series method of [4] for the reasons illustrated above, even if the theorem 1.4 of [4], briefly recalled in Section 2, could be proved also through other approaches to KAM theory.
In Section 2 we introduce the model which we are interested in, and we show how the analysis of [4] can be applied to it; this yields some Diophantine-type condition on the oscillation frequency of the support point In Appendix A3 we explicitly check that all the hypotheses under which the results in [4]  Consider a pendulum with support point P = (x P , y P ) oscillating in the vertical direction with the law y P (t) = b cos ωt (and x P (t) ≡ 0). The system is described by the Lagrangian (see, for instance [18] or [19]) which, by the change of variable can be written as The corresponding (non-autonomous) Hamiltonian is where x 1 = x and y 1 is the momentum conjugated to x 1 . We are interested in the stability of the position (θ,θ) = (π, 0), i.e. (x,ẋ) = (0, 0), for ω large enough (see [19] and [1]), so that we can define and write (2.4), divided by δ 2 , as where the last sum is obtained by Taylor expanding the cosine function and disappears for δ = 0.
We can consider the autonomous Hamiltonian with (I 2 , α 2 ) conjugated variables. By a (canonical) rescaling we can put (2.7) into the form where, for notational simplicity, we still denote by (p 1 , q 1 ) the new variables. Let us consider the Hamiltonian obtained from (2.9) by putting δ = 0: one easily realizes that the corresponding equation for q 1 is Mathieu's equation so that the solution is of the form for some ρ (and for a particular choice of the initial phase) and with p 0 (τ ) a periodic function of period π; µ is a real number in (0, 1), for ω large enough (see [3], [17] and [9]; see also Appendix A1 for the notations and for a review of some basic properties of Mathieu's equation which will be used below). In (2.11) we can choose ρ so that ). It is possible to pass to action-angle variables through a canonical transformation see [13] and [10] (see also Appendix A2 below), with α 2 the same in both sides. Then the Hamiltonian (2.9) at δ = 0 becomes which is an isochronous Hamiltonian, while the perturbation (2.14) in (2.9) can be written in terms of the variables (A, α), and gives a function analytic in its arguments (in a suitable domain). By setting δ 2 = ε and defining where the functions F p (α, A 1 ) are analytic in (α, A 1 ) uniformly for all p. Call A the domain of analyticity in A 1 , and, for any A ∈ A, let us denote by B ρ (A) the ball of radius ρ and center A; the analyticity in α is in a strip Σ κ = {α : |Im α j | < κ, j = 1, 2}.
The system (2.16) represents two harmonic oscillators interacting through a potential depending only on the angles and on the action variable A 1 : the latter condition, in particular, implies that one oscillator is simply a clock, i.e. it rotates with fixed frequency.
The corresponding equations of motion are so that one sees that α 2 (t) = t.

2.2.
Stability of the upside-down pendulum. We want to study the persistence of tori near the origin. Note that, for any value of A, the origin is recovered by setting ε = δ 2 = 0 (see (2.5)). On the other hand for ε = 0 the scaled Hamiltonian (2.16) reduces to ω · A, so that it admits invariant tori for any value A 0 = (A 01 , A 02 ) of the action variable A, all run with the same rotation vector ω: such tori are defined by where ω is fixed and A 0 ∈ R 2 . Note that the Hamiltonian (2.16) is of the form considered in [4], so that we can apply the theorem 1.4 of [4]; for simplicity we write here the complete statement (adapted to the notations used above).
In the original variables, along the motion corresponding to the considered torus T , one has in the sense that, for initial data (x 1 (0), y 1 (0)) with x 2 1 (0) + y 2 1 (0) = δ 2 , one has |x 2 1 (t) + y 2 1 (t) − δ 2 | < Cδ 4 , for some constant C; in particular for ε small enough along the motion one has A 1 (t) = 0, so that the trajectory never crosses the origin (which can be outside the analyticity domain of the conjugating functions). So we have a closed curve surrounding the origin and which can be made arbitrarily near to it. As all trajectories starting from initial data contained inside the torus described by (2.26) have to remain inside, we have that their projections onto the plane (x 1 , y 1 ) have to remain inside the closed curve C. Therefore we obtain the stability of the origin for the motion of the variables (x 1 , y 1 ).

Extension to vectors satisfying only a non-resonance condition.
Note that the theorem 1.4 of [4] requires ω to be Diophantine. In the study of stability of elliptic equilibrium points it is well known (see for instance [16]) that a Diophantine condition on the unperturbed frequencies is not necessary, and in fact it can be relaxed (by using that the perturbation depends on A and ε in a precise way). A mechanism of this kind works also in the present case, and, by using the special form of the perturbation (see (2.15)÷(2.18)), one can show that also for the vertically driven pendulum the conditions on the rotation vectors can be weakened. This could certainly be done within the formalism of the classical approaches to KAM theory, but here we prefer to discuss such an extension of the proof within the tree formalism introduced in [4]. The analysis is performed in Appendix A4, and gives the following result.

2.4.
Remark. Note that the Diophantine condition imposed on ω in the theorem 1.4 of [4] (which allows a full measure set of rotation vectors) can in fact be improved: for weaker conditions holding in general we refer to [20], while a case in which optimal conditions can be explicitly worked out can be found in [7]. Nevertheless in general, that is without making any assumptions on the perturbation, no condition of the kind of (2.28), or whatever else, can be obtained: the possibility of imposing only a non-resonance condition like (2.28) can arise only if the perturbation is of some special form, as it is the case for the elliptic equilibrium points and for the vertically driven pendulum studied in the present paper. In this sense, the results formulated in [4] are as general as in any other KAM approach to the study of isochronous systems. where a < 0 and q ∈ R. The solutions of (A1.1) are of the form where µ 0 ∈ C and p 0 is π-periodic. The regions of the plane (a, q) such that µ 0 ∈ R are called stability regions: the corresponding solutions are quasi-periodic, hence (in particular) bounded.

A2.2. Second change of coordinates. Consider the change of coordinates
Then (A2.4) is an analytic change of coordinates, such that the change of coordinates C 2 • C 1 is analytic and canonical, and, in the new variables, the Hamiltonian (2.9) for δ = 0 becomes ω · A ≡ µA 1 + A 2 . (A2.5) For the proof see [10] and [13], in particular the theorem 2 in Section 1.3 of [10]. Moreover the second line of (2.9) can be written as a function which does not depend on A 2 , and such that the dependence of Φ p (α, A 1 ) on α 1 involves only harmonics with |ν 1 | ≤ 2p (simply combine the definitions of the two changes of coordinates C 1 and C 2 ).
A4. Non-resonance conditions on the rotation vectors A4.1. Non-resonant condition on the rotation vector. The result in Section 2 can be improved in the following way.
Let ω be such that Then it is still possible to perform the analysis of [4] with some minor changes. We refer to the formalism introduced in [4]: we use the same notations used there, by assuming a full knowledge of that paper by the reader, with no further comments henceforth.
Assume that the rotation vector ω satisfies the non-resonance condition (A1.1). This means that there exists a constant α such that so that, for ε small enough then one can prove the following bound.
In general note that k = 1 implies V = 1 so that the above argument yields N α (θ) = 0 for θ of order 1.
Given V > 1, assuming (A4.6) to hold for V < V we can show that it holds also for V . Let θ be a tree of order k with V nodes and let v 0 be the node which the root line 0 ≡ v0 of θ exits from. Call θ 1 , . . . , θ m the subtrees of order ≥ 2 entering the node v 0 and denote by k 1 , . . . , k m and V 1 , . . . , V m , respectively, the orders and the numbers of nodes of such subtrees. Call also k and V the sum of the orders and of the numbers of nodes, respectively of the subtrees of order 1 entering v 0 (of course k = V ). One has m ≥ 0, k 1 +. . .+k m +k v0 +k = k and V 1 +. . .+V m +1+V = V .
A4.4. Improvement of the bound on the radius of convergence. With respect to [4] we can modify the multiscale decomposition by defining the scales starting from n such that C 0 2 n < α/8 ≤ C 0 2 n+1 .
Then one can reason as in [15] in order to show that, by using the lemma A4.2, instead of a factor C −k 0 one has a factor (8α −1 ) k C −3k/5 0 (here the exponent k for 8α −1 is a bound on the number of lines which do not contribute to N α (θ)).
This implies that the radius of convergence ε 0 of the series defining the functions h, H, η can be bounded by ε 0 = E C 3/5 0 (instead of ε 0 = EC 0 as in Section 2), for some constant E depending on α but not on C 0 . A4.5. Persistence of KAM tori. Under the condition (A4.1) on ω the analysis of [4] can be repeated for the Hamiltonian (2.16) with f as in (2.17). By the discussion in Section A4.4 one finds ε 0 ≡ E C 3/5 0 , so that one can choose ε * ∈ [0, ε 0 ] and repeat the same argument as in Section 5 of [4]: the main difference is that one can fix the interval I of size |I| = aC 3/5 0 and, as C 3/5 0 C 0 for C 0 small enough (if C is not small one can choose the constant b in the theorem 1.4 of [4] so that the same conclusions hold), one can easily prove that there are infinitely many µ 0 ∈ I such that the corresponding ω 0 verify the Diophantine condition (1.11) of [4].