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The ratio λ r as given by equation (38), for r even and odd. (a) ω 1 = 1, ω 2 = ( √ 5−1)/2; (b) ω 1 = 1, ω 2 = 12 1/4 − 1.
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We investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 DOF Hamiltonian systems. In particular, we study the dependence of the optimal order of truncation Nopt on the distance ρ from the elliptic equilibrium, by numerical and analytical means. The function Nopt(ρ) determines...
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Context 1
... any rate, λ r can be found immediately once ω is given. Figure 5(a) shows the factor λ r as a function of r for the case ω 2 /ω 1 = ( √ 5 − 1)/2, for r even (left panel) and for r odd (right panel). Figure 5(b) shows the same function for the case ω 2 /ω 1 = 12 1/4 − 1. ...
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... 5(a) shows the factor λ r as a function of r for the case ω 2 /ω 1 = ( √ 5 − 1)/2, for r even (left panel) and for r odd (right panel). Figure 5(b) shows the same function for the case ω 2 /ω 1 = 12 1/4 − 1. In both cases, the function λ r increases by abrupt steps. ...
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... secondary steps may also appear at orders corresponding to the appearance of low-order multiples of a diophantine divisor. The same is true in the case of figure 5(b), as seen from a comparison with figure 4(a). ...
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... figure 5(a) the plateaux appear in quite regular intervals on a logarithmic scale so that an average power law fit can be found. This is shown as a straight line in figure 5(a). ...
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... figure 5(a) the plateaux appear in quite regular intervals on a logarithmic scale so that an average power law fit can be found. This is shown as a straight line in figure 5(a). The best fit power law is λ r 0.036r 1.67 r even, λ r 0.020r 2.03 r odd. ...
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... it should be emphasized that this average power law applies only because the successive steps are at quite regular intervals, a fact due to the nature of the golden mean number. On the other hand, in the case of the less noble number 12 1/4 − 1 ( figure 5(b)) the steps and subsequent plateaux appear at irregular intervals of values of r, and one must go to orders much higher than 120 in order to obtain a good number of plateaux that allow us to draw a meaningful power law. ...
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... 9(a) shows the ratio R r+2 /R r , where again we see the existence of plateaux corresponding to a piecewise constant geometrical growth. These plateaux correspond to the plateaux of figure 5(a). The variations of λ r are replotted in figure 9(a) as a dashed line going alternatively upwards and downwards. ...
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Citations
... for some constant coefficients C i,1 , C i,2 computable from the series (3). Thus, with all frequencies of the problem fixed in advance, establishing the convergence of the inverse series (5) suffices to answer the question posed at (ii). The question of the convergence of the series is, of course, crucial, and related to the kind, and pattern of accumulation in the series terms, of small divisors appearing at successive perturbative steps. ...
... Note that since the divisors depend on the new frequencies ω 1 , ω 2 , choosing non-resonant values for the latter permits the formal construction to proceed; this, even when the unperturbed frequencies ω 0,1 , ω 0,2 are, instead, resonant. As regards convergence, in the case (i) Poincaré ([19], Ch.IX) already emphasizes that the Lindstedt series with divisors depending on the original harmonic frequencies ω 0,1 , ω 0,2 are divergent, exhibiting the well known asymptotic character associated with the series computed via a Birkhoff normal form (see [5] for a review). Indeed, as shown by example in section 2 below, it possible to construct Birkhoff series yielding the same individual solutions as those of the Lindstedt series of scheme (i). ...
... As regards the series convergence, this is guaranteed in an open domain in the 1DOF case. However, when N > 1 the series are in general only asymptotic (see [5] for reviews). ...
We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in `isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $\mathcal{H}=\mathcal{H}_0+\varepsilon \mathcal{H}_1$ where $\mathcal{H}_0$ is the Hamiltonian of $N$ linear oscillators, and $\mathcal{H}_1$ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one producing series analogous to those of the Birkhoff normal form scheme, and another, analogous to the Kolomogorov normal form scheme in which we fix in advance the frequency of the torus.
... This method of computation of halo orbits (as well as the method used for the CRTBP) is based on series expansions of the Hamiltonian, truncated at a large order. The dependence of the error on this truncation order is influenced both by the singularity of the Hamiltonian corresponding to a collision with P 2 (see [21]) and by the well known problem of accumulation of small divisors (see for example [22] and references therein). Since the family of halo orbits forms with a minimum libration amplitude, it is necessary to perform a test on the error introduced with the truncation of series. ...
The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1−L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1−L2 from the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem.
... truncated at a large order. The dependence of the error on this truncation order is influenced both by the singularity of the Hamiltonian corresponding to a collision with P 2 (see [27]) and by the well known problem of accumulation of small divisors (see for example [5] and references therein). Since the family of halo orbits forms with a minimum libration amplitude, it is necessary to perform a test on the error introduced with the truncation of series. ...
The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1-L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1-L2 to the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem.
... Note that since the divisors depend on the new frequencies ω 1 , ω 2 , choosing non-resonant values for the latter permits the formal construction to proceed; this, even when the unperturbed frequencies ω 0,1 , ω 0,2 are, instead, resonant. As regards convergence, in the case (i) Poincaré ( [19], Ch.IX) already emphasizes that the Lindstedt series with divisors depending on the original harmonic frequencies ω 0,1 , ω 0,2 are divergent, exhibiting the well known asymptotic character associated with the series computed via a Birkhoff normal form (see [5] for a review). Indeed, as shown by example in section 2 below, it possible to construct Birkhoff series yielding the same individual solutions as those of the Lindstedt series of scheme (i). ...
... As regards the series convergence, this is guaranteed in an open domain in the 1DOF case. However, when N > 1 the series are in general only asymptotic (see [5] for reviews). ...
We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in 'isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $ {\mathcal{H}} = {\mathcal{H}}_0+\varepsilon {\mathcal{H}}_1 $ where $ {\mathcal{H}}_0 $ is the Hamiltonian of $ N $ linear oscillators, and $ {\mathcal{H}}_1 $ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one producing series analogous to those of the Birkhoff normal form scheme, and another, analogous to the Kolomogorov normal form scheme in which we fix in advance the frequency of the torus.
... The obtained times of stability are of order R (r opt ) −1 ρ (ropt) ,σ (ropt) , where r opt is the normalization order yielding the smallest possible remainder norm. The value of r opt can be obtained via theoretical estimates (see Efthymiopoulos et al. 2004), but in practice, it is also limited by the maximum order in which our computer-algebra normal form calculations can proceed. Theoretical estimates imply that the size of the remainder norm is exponentially small in the inverse of the size of the perturbation H 1 ρ,σ in Eq. (25). ...
... Similar estimates hold in the case of the resonant normal form constructions (see Efthymiopoulos et al. 2004). The behavior of the size of the remainder as a function of the normalization order r will be examined in detail in our semi-analytical computations in Sects. ...
Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. (i) We demonstrate the long-term stability of the semimajor axis within the framework of the $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ H gls ), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ H gls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.
... From definitions (11) and (13), it is evident that both χ r and Z r belong to P r+2 . In order to conclude the normalization step, we have to perform the canonical change of coordinates induced by exp L χr and to update accordingly all the terms that compose the Hamiltonian. ...
... On the other hand, far away from the origin the Birkhoff normal form starts to diverge earlier. Let us mention that in [13] the mechanism of divergence is carefully investigated in a numerical way and it is shown to be much more subtle with respect to what has been discussed just above. Moreover, this has allowed those authors to conclude that the analytical upper bounds should largely overestimate the effective size of the remainders. ...
... , log Z min{r−1 , RII} disappear in the list of elements making part of the set S (r−1) . 13 It is not easy to code with X̺óνoζ, which is the algebraic manipulator we actually used for all the applications discussed in the present paper, because its syntax looks quite difficult for new users. Therefore, we think that an implementation of the algorithm constructing the Birkhoff normal form in the framework provided by Mathematica can be more helpful for a reader that is interested in reproducing our results. ...
Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the H\'enon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.
... The obtained times of stability are of order R (ropt) −1 ρ (ropt) ,σ (ropt) , where r opt is the normalization order yielding the smallest possible remainder norm. The value of r opt can be obtained via theoretical estimates (see [11]), but in practice, it is also limited by the maximum order in which our computer-algebra normal form calculations can proceed. Theoretical estimates imply that the size of the remainder norm is exponentially small in the inverse of the size of the perturbation H 1 ρ,σ in Eq. (25). ...
... Similar estimates hold in the case of the resonant normal form constructions (see [11]). The behavior of the size of the remainder as a function of the normalization order r will be examined in detail in our semi-analytical computations in Sections 4 and 5 below. ...
Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the 'geolunisolar' Hamiltonian $H_{gls}$), after a suitable reduction of the Hamiltonian to the Laplace plane. iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the holding of Nekhoroshev's theorem on the exponential stability of the orbits. We find that the $H_{J_2}$ model is non-convex, but satisfies a 'three-jet' condition, while the $H_{gls}$ model restores quasi-convexity by adding lunisolar terms in the Hamiltonian's integrable part.
... From definitions (11) and (13), it is evident that both χ r and Z r belong to P r+2 . In order to conclude the normalization step, we have to perform the canonical change of coordinates induced by exp L χr and to update accordingly all the terms that compose the Hamiltonian. ...
... On the other hand, far away from the origin the Birkhoff normal form starts to diverge earlier. Let us mention that in [13] the mechanism of divergence is carefully investigated in a numerical way and it is shown to be much more subtle with respect to what has been discussed just above. Moreover, this has allowed those authors to conclude that the analytical upper bounds should largely overestimate the effective size of the remainders. ...
... , log Z min{r−1 , RII} disappear in the list of elements making part of the set S (r−1) . 13 It is not easy to code with X̺óνoζ, which is the algebraic manipulator we actually used for all the applications discussed in the present paper, because its syntax looks quite difficult for new users. Therefore, we think that an implementation of the algorithm constructing the Birkhoff normal form in the framework provided by Mathematica can be more helpful for a reader that is interested in reproducing our results. ...
... Different approaches to the problem of stability of perturbed, near to integrable systems have been developed. The basic question is expressed by (16). The dependence of T (ε) on the size of the perturbation makes the difference. ...
A concise, not too technical account of the main results of perturbation theory is presented, paying particular attention to the mathematical development of the last 60 years, with the work of Kolmogorov on one hand and of Nekhoroshev on the other hand. The main theorems are recalled with the aim of providing some insight on the guiding ideas, but omitting most details of the proofs that can be found in the existing literature.
... The lack of upper limit in the optimal order simply reflects the integrability of the model when e = 0 (a fact which implies that the series are convergent in this case for appropriate bounds in ξ and δ). On the other hand, the lower bound is close to the power law n opt,min ∝ ρ −1 , a relation which is characteristic of resonant normal forms (see [7] for more details). This power-law behavior breaks, however, at ρ ≈ ρ c . ...
Accepted for publications in Communications in Nonlinear Science and Numerical Simulation, 2019
