^{1}and Elena Lega

^{2}

### Abstract

The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasi–integrable Hamiltonian systems. In particular, long–term instability of the action variables can be observed only in the so–called Arnold web, which is the complement in the phase–space of all KAM invariant tori, and only on the very long times which depend exponentially on an inverse power of the perturbation parameter. Though the structure of the Arnold's web was clearly explained already on Arnold's 1963 article, its numerical detection with a precision sufficient to reveal exponentially slow diffusion of the actions through the web itself has become possible only in the last decade with the extensive computation of dynamical indicators. In this paper, we first review the detection method that allowed us to compute the Arnold web, and then we discuss its use to study the long–term diffusion through the web itself. We also show that the Arnold web of a quasi–integrable Hamiltonian system is useful to track the diffusion of orbits of weakly dissipative perturbations of the same Hamiltonian system.

The dynamics of quasi–integrable Hamiltonian systems is widely studied in the literature. On the one hand, many interesting problems of Physics are in the form of small Hamiltonian perturbations of integrable systems, on the other hand most questions concerning the long-term diffusion are open. Moreover, the application of perturbation theories to the systems of interest for Physics, for example, the systems representing the motions of the asteroids and planets of our Solar System, is complicated by the fact that the small parameters are not extremely small, and their Hamilton functions may be highly degenerate, so that a direct application of the KAM and Nekhoroshev theorems in their original formulation is not possible. One of the most studied problems concerns the orbits which are unstable on suitably long times. The precise definition of unstable orbits depends on the specific system, and is related to a significant change of the action variables. The existence of unstable orbits is a non-trivial problem only depending on the non-degeneracy properties of the integrable approximation (it is quite easy to provide analytic examples of unstable motions even for systems which satisfy the hypotheses of KAM theorem, but not those of Nekhoroshev theorem, see Ref. 40, Sec. 1.7). A paper of fundamental importance was published by Arnold in 1964.

^{2}The paper first provided a quasi–integrable model system with non trivial long-term instability, afterwards called Arnold diffusion, occurring at any small value of the perturbation parameter in the so-called Arnold web, which is the complement in the phase-space of all KAM invariant tori. The use of the term “diffusion” may be here confusing. We remark that all the definitions of Arnold diffusion which have been provided up to now in the literature refer to a phenomenon of

*topological instability*of the action variables. Therefore, what justifies the universal use of the term “diffusion”? The reason can be found in Ref. 2, where the mechanism proposed for proving such instability was based on heteroclinic intersections of a chain of hyperbolic tori. Heteroclinic intersections are well known to be related to some random description of deterministic dynamics. Nevertheless, the idea of random motions behind such heteroclinic chaos is still of topological nature, and up to now, there do not exist in the literature any mathematically rigorous characterizations of the statistical properties of Arnold diffusion. Because of the extremely long times characterizing the Arnold diffusion, Arnold's paper soon motivated a debate about the possibility of numerical detection of Arnold diffusion: few years after the first numerical detections of chaotic motions,

^{30}the problem of the numerical detection of Arnold diffusion was discussed in Ref. 12, and in the following decades, there appeared many studies of numerical diffusion through resonances, such as Refs. 17, 31, 35, 37, and 50, up to our papers.

^{19,25,29,36}Most of these papers are concerned with the detection of the resonances of the system using extensive computations of dynamical indicators on grids of initial conditions of the phase–space. We remark that, specifically in Celestial Mechanics, the numerical detection of the resonances of a system using dynamical indicators is one of the major tools for studying its long–term instability (for recent examples, see Refs. 22, 23, 38, 43–45, and 49). The quasi–integrable structure of many systems gives the resonances the structure of a web, the so called Arnold web. In our papers,

^{19,25,29,36}we provided numerical evidence about the existence of sets of orbits diffusing through the Arnold web with remarkable topological and statistical properties. In particular, by choosing the initial conditions in the single resonances (far from the main multiple resonances), the average over all the orbits of the square of the distance of the action variables from their initial condition (see Sec. III B for precise discussion) is well fitted by a linear time dependence, as in the genuine diffusion processes, in the following two cases: (i) the time interval is suitably long to allow us to measure a slow drift of the orbits along the resonance, but in the same time the orbits do not cross other main multiple resonances (see Ref. 36); (ii) in much longer time intervals the orbits visit several different single and multiple resonances (see Ref. 25). We referred to case (i) as to local diffusion, and to case (ii) as to global diffusion. Therefore, in a regime of moderately small perturbation parameters, it was possible to relate the actions' instability to diffusion coefficients, and we found that the diffusion coefficient scales with the perturbation parameter faster than power laws. In our opinion, the motivation of numerical studies of Arnold diffusion goes farther than providing support or evidence to analytical studies, which nowadays are still far to provide the complete description of Arnold diffusion. An important added value of numerical studies, which cannot handle arbitrarily small perturbations, is to show that the process can be detected even for perturbations which are small, but not extremely small, and on times which are long, but not extremely long, making it relevant also for Physics.

Part of the computations have been done on the “Mesocentre SIGAMM” machine, hosted by the Observatoire de la Côte d'Azur.

I. INTRODUCTION

II. NUMERICAL DETECTION OF THE ARNOLD WEB WITH DYNAMICAL INDICATORS

A. The FLI for different orbits

B. Sensitivity of the FLI

III. DIFFUSION THROUGH THE WEB OF RESONANCES

A. Topological properties of the diffusion

B. Statistical properties of the diffusion

C. Computation of diffusion coefficients

IV. WEAKLY DISSIPATIVE PERTURBATIONS OF QUASI–INTEGRABLE SYSTEMS

### Key Topics

- Diffusion
- 73.0
- Statistical properties
- 6.0
- Topology
- 6.0
- Chaos
- 4.0
- Fourier analysis
- 3.0

## Figures

Detection of the Arnold web of the Hamiltonian system (6), previously studied in Ref. 20 , for obtained with FLI computations. The panel represents, for any initial condition, the average between the FLI computed with initial tangent vectors and , and integration time T = 1000, using a color scale. The average allows to reduce the dependence of the computation on the choice of the initial tangent vector and improves the results presented in Ref. 20 . The initial conditions are chosen on a grid regularly spaced on (the other initial conditions are ); the color scale is set such that light grey (yellow on the electronic version of the paper) corresponds to chaotic motions, motions on KAM tori correspond to grey (orange), while colors from dark grey to black (red to black) identify regular resonant motions.

Detection of the Arnold web of the Hamiltonian system (6), previously studied in Ref. 20 , for obtained with FLI computations. The panel represents, for any initial condition, the average between the FLI computed with initial tangent vectors and , and integration time T = 1000, using a color scale. The average allows to reduce the dependence of the computation on the choice of the initial tangent vector and improves the results presented in Ref. 20 . The initial conditions are chosen on a grid regularly spaced on (the other initial conditions are ); the color scale is set such that light grey (yellow on the electronic version of the paper) corresponds to chaotic motions, motions on KAM tori correspond to grey (orange), while colors from dark grey to black (red to black) identify regular resonant motions.

Detection of the Arnold web of the Hamiltonian system (6), previously studied in Ref. 20 , for obtained with FLI computations. The panel represents, for any initial condition, the average between the FLI computed with initial tangent vectors and , and integration time T = 1000, using a color scale. The average allows to reduce the dependence of the computation on the choice of the initial tangent vector and improves the results presented in Ref. 20 . The initial conditions are chosen on a grid regularly spaced on (the other initial conditions are ); the color scale is set such that light grey (yellow on the electronic version of the paper) corresponds to chaotic motions, motions on KAM tori correspond to grey (orange), while colors from dark grey to black (red to black) identify regular resonant motions.

Detection of the Arnold web of the Hamiltonian system (6) , previously studied in Ref. 20 , in the region of the phase space corresponding to the box of Figure 1 for and T = 3000 (top left), 0.01 and T = 2000 (top right), 0.03 and T = 1000 (bottom left), 0.04 and T = 1000 (bottom right). The panels represent, for any initial condition, the average between the FLI computed withinitial tangent vectors and , and integration times T = 3000 (top left), T = 2000 (top right), and T = 1000 (bottom), using a color scale. The average allows to reduce the dependence of the computation on the choice of the initial tangent vector and improves the results presented in Ref. 20 . The initial conditions are chosen on a grid regularly spaced on (the other initial conditions are ); the color scale is set in any panel such that light grey (yellow on the electronic version of the paper) corresponds to chaotic motions, motions on KAM tori correspond to grey (orange), while colors from dark grey to black (red to black) identify regular resonant motions. In the top left panel, the resonances of the system are clearly organized as a web, the so called Arnold web. When increasing the perturbation parameter the amplitude of the resonances increases (top right panel) and the volume occupied by KAM tori reduces. In the two bottom panels, the resonances of the system overlap in large measure sets of the action space and the volume of KAM tori shrinks to zero.

Detection of the Arnold web of the Hamiltonian system (6) , previously studied in Ref. 20 , in the region of the phase space corresponding to the box of Figure 1 for and T = 3000 (top left), 0.01 and T = 2000 (top right), 0.03 and T = 1000 (bottom left), 0.04 and T = 1000 (bottom right). The panels represent, for any initial condition, the average between the FLI computed withinitial tangent vectors and , and integration times T = 3000 (top left), T = 2000 (top right), and T = 1000 (bottom), using a color scale. The average allows to reduce the dependence of the computation on the choice of the initial tangent vector and improves the results presented in Ref. 20 . The initial conditions are chosen on a grid regularly spaced on (the other initial conditions are ); the color scale is set in any panel such that light grey (yellow on the electronic version of the paper) corresponds to chaotic motions, motions on KAM tori correspond to grey (orange), while colors from dark grey to black (red to black) identify regular resonant motions. In the top left panel, the resonances of the system are clearly organized as a web, the so called Arnold web. When increasing the perturbation parameter the amplitude of the resonances increases (top right panel) and the volume occupied by KAM tori reduces. In the two bottom panels, the resonances of the system overlap in large measure sets of the action space and the volume of KAM tori shrinks to zero.

Time evolution of the FLI numerically computed by integrating the variational equations of the system (6) with : the first orbit (with , the other initial conditions are ) corresponds to the linear law (9) , the second one (with , the other initial conditions are ) to the linear law (10), the third one (with , the other initial conditions are ) to an exponential law. The three different time evolutions are well distinguished already at the time T = 1000.

Time evolution of the FLI numerically computed by integrating the variational equations of the system (6) with : the first orbit (with , the other initial conditions are ) corresponds to the linear law (9) , the second one (with , the other initial conditions are ) to the linear law (10), the third one (with , the other initial conditions are ) to an exponential law. The three different time evolutions are well distinguished already at the time T = 1000.

Test of the sensitivity of the FLI method: 24 plot of d versus for and t = 104 (left panel) and and t = 105 (right panel). In both cases, the relative error of the theoretical result with respect to the numerical one is smaller than 2 × 10−4 for all the computations, thus showing that the Eq. (10) describes indeed the evolution of the FLI with time. Since , the FLI detects the resonance even for very small values of its harmonics amplitudes . Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Physica D 163(1–2), 1–25 (2002). Copyright 2002 Elsevier Science B.V.

Test of the sensitivity of the FLI method: 24 plot of d versus for and t = 104 (left panel) and and t = 105 (right panel). In both cases, the relative error of the theoretical result with respect to the numerical one is smaller than 2 × 10−4 for all the computations, thus showing that the Eq. (10) describes indeed the evolution of the FLI with time. Since , the FLI detects the resonance even for very small values of its harmonics amplitudes . Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Physica D 163(1–2), 1–25 (2002). Copyright 2002 Elsevier Science B.V.

Detection of long–term diffusion along the resonance for the system (6) and (top panels), (middle panels), (bottom panels). Each panel represents the Arnold web computed with the FLI and the black dots corresponding to 100 resonant orbits numerically integrated. First, on each FLI panel, we recognize the two thin parallel yellow lines corresponding to the chaotic borders of the resonance . Then, the diffusion is studied by numerically integrating 100 initial conditions taken in this chaotic border with and , and by representing their orbits on the Arnold web computed with the FLI. Precisely, the black dots on the panels represent the points of the orbits which belong to the small neighbourhood V of the section S, defined by . For each value of ε, the 100 orbits are represented with two different integration times, in order to appreciate the qualitative spread of the orbits along the resonance: the times used for the leftpanels are shorter with respect to the times used for the right panels (t = 107 top-left, 108 top-right; 106 middle-left, middle-right; bottom-left, bottom-right). In all the cases, the orbits spread along the two thin chaotic borders of the resonance, thus providing evidence about the existence of a diffusion along the resonance itself. We also remark that the amplitude of this diffusion is definitely smaller than the distance between the two chaotic borders of the resonance (which is approximately of order ), and that each orbit can transit from one side of the resonance to the otherone in short times of order . Therefore, the possibility of comparing the orbits in the section S with the FLIcomputation is the determining factor in order to appreciate the existence ofthe much slower diffusion along the resonance. Reprinted with permission from E. Lega, M. Guzzo, and C. Froeschlé, Physica D 182, 179–187 (2003) Copyright 2003 Elsevier B.V.

Detection of long–term diffusion along the resonance for the system (6) and (top panels), (middle panels), (bottom panels). Each panel represents the Arnold web computed with the FLI and the black dots corresponding to 100 resonant orbits numerically integrated. First, on each FLI panel, we recognize the two thin parallel yellow lines corresponding to the chaotic borders of the resonance . Then, the diffusion is studied by numerically integrating 100 initial conditions taken in this chaotic border with and , and by representing their orbits on the Arnold web computed with the FLI. Precisely, the black dots on the panels represent the points of the orbits which belong to the small neighbourhood V of the section S, defined by . For each value of ε, the 100 orbits are represented with two different integration times, in order to appreciate the qualitative spread of the orbits along the resonance: the times used for the leftpanels are shorter with respect to the times used for the right panels (t = 107 top-left, 108 top-right; 106 middle-left, middle-right; bottom-left, bottom-right). In all the cases, the orbits spread along the two thin chaotic borders of the resonance, thus providing evidence about the existence of a diffusion along the resonance itself. We also remark that the amplitude of this diffusion is definitely smaller than the distance between the two chaotic borders of the resonance (which is approximately of order ), and that each orbit can transit from one side of the resonance to the otherone in short times of order . Therefore, the possibility of comparing the orbits in the section S with the FLIcomputation is the determining factor in order to appreciate the existence ofthe much slower diffusion along the resonance. Reprinted with permission from E. Lega, M. Guzzo, and C. Froeschlé, Physica D 182, 179–187 (2003) Copyright 2003 Elsevier B.V.

Detection of the long–term diffusion along the Arnold web of the system (6) with . Each panel represents the Arnold web computed with the FLI and the black dots corresponding to 10 resonant orbits numerically integrated. The diffusion is studied by numerically integrating 10 chaotic initial conditions (chosen in the circle marked in the panel a), and by representing on the Arnold their orbits computed up to some time t. Precisely, the black dots on the panels represent the points of the orbits which belong to the small neighbourhood V of the section S, defined by . The panels b, c, and d correspond to the integration times t = 109, , and t = 1.1 × 1011, respectively. With respect to Figure 6 , the much longer integration times allow us to appreciate the qualitative spread of the orbits along the Arnold web: in particular, the orbits visit several resonances and, giving them suitably long times, they spread over large regions of the action space. Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Discrete Contin. Dyn. Syst. B 5(3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Detection of the long–term diffusion along the Arnold web of the system (6) with . Each panel represents the Arnold web computed with the FLI and the black dots corresponding to 10 resonant orbits numerically integrated. The diffusion is studied by numerically integrating 10 chaotic initial conditions (chosen in the circle marked in the panel a), and by representing on the Arnold their orbits computed up to some time t. Precisely, the black dots on the panels represent the points of the orbits which belong to the small neighbourhood V of the section S, defined by . The panels b, c, and d correspond to the integration times t = 109, , and t = 1.1 × 1011, respectively. With respect to Figure 6 , the much longer integration times allow us to appreciate the qualitative spread of the orbits along the Arnold web: in particular, the orbits visit several resonances and, giving them suitably long times, they spread over large regions of the action space. Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Discrete Contin. Dyn. Syst. B 5(3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Top panel: Evolution of with time for a set of 100 orbits of system (6) for previously studied in Ref. 36 . The initial conditions are taken on the resonance and the points of the orbits which belong to a small neighbourhood V of the section S are represented in Fig. 6 , middle panels. Bottom panel: Evolution of the square distance of the actions from the initial datum (Eq. (14) ) for a four dimensional symplectic map previously studied in Ref. 25 . Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Discrete Contin. Dyn. Syst. B 5(3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Top panel: Evolution of with time for a set of 100 orbits of system (6) for previously studied in Ref. 36 . The initial conditions are taken on the resonance and the points of the orbits which belong to a small neighbourhood V of the section S are represented in Fig. 6 , middle panels. Bottom panel: Evolution of the square distance of the actions from the initial datum (Eq. (14) ) for a four dimensional symplectic map previously studied in Ref. 25 . Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Discrete Contin. Dyn. Syst. B 5(3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Diffusion coefficients for the map, Eq. (16) , computed on sets of initial conditions in the resonance . Each dot represents the diffusion coefficient computed on a set of N = 100 initial conditions near , equally spaced on a segment of amplitude 10−10 in the linear unstable space (computed at order 1 in ε) of the hyperbolic torus related to the resonance. The total integration times range from t = 106 up to t = 108. Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Chaos 21(3), 033101 (2011). Copyright 2011, American Institute of Physics.

Diffusion coefficients for the map, Eq. (16) , computed on sets of initial conditions in the resonance . Each dot represents the diffusion coefficient computed on a set of N = 100 initial conditions near , equally spaced on a segment of amplitude 10−10 in the linear unstable space (computed at order 1 in ε) of the hyperbolic torus related to the resonance. The total integration times range from t = 106 up to t = 108. Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Chaos 21(3), 033101 (2011). Copyright 2011, American Institute of Physics.

The panels report the Arnold web of the map Eq. (19) for and different values of ε, obtained by computing the FLI on a grid of 500 × 500 initial conditions regularly spaced in (the other initial conditions are ; the panels report the average between the FLI computed with two different initial tangent vectors). Moreover, on each panel, the black dots represent the projections of two orbits of the map Eq. (19) obtained for , up to some time . The left panels correspond to : the orbits represented on the Arnold web are obtained for , and T = 2 × 107. In the top–left panel, one of the orbits follows closely the evolution obtained from the averaged system (20) . Before converging to the origin of the action space, the orbit crosses several resonances detected by the FLI. These resonances only produce a small scattering (see the zoom in the bottom–left panel), which occurs exactly in correspondence of the resonances detected by the FLI. Instead, the second orbit is characterized by two important resonance captures. The middle panels are obtained for : the orbits represented on the Arnold web are obtained for , and T = 2 × 106. In the top–middle panel, one of the orbits still follows approximately the evolution obtained from the averaged system (20) , but the relevance of the resonance scattering is now more evident (see also the zoom in the bottom-middle panel). The second orbit is characterized by even longer resonance captures, with an interesting change of resonance in correspondence of a resonance crossing (see the bottom–middle panel). The top–right panel is obtained for , and T = 2 × 106, while the bottom–right panel, is obtained for , and T = 2 × 106. As soon as ε increases, the relevance of resonance scatterings and resonance captures increases: for , the orbits follow closely paths obtained by gluing together pieces of resonances.

The panels report the Arnold web of the map Eq. (19) for and different values of ε, obtained by computing the FLI on a grid of 500 × 500 initial conditions regularly spaced in (the other initial conditions are ; the panels report the average between the FLI computed with two different initial tangent vectors). Moreover, on each panel, the black dots represent the projections of two orbits of the map Eq. (19) obtained for , up to some time . The left panels correspond to : the orbits represented on the Arnold web are obtained for , and T = 2 × 107. In the top–left panel, one of the orbits follows closely the evolution obtained from the averaged system (20) . Before converging to the origin of the action space, the orbit crosses several resonances detected by the FLI. These resonances only produce a small scattering (see the zoom in the bottom–left panel), which occurs exactly in correspondence of the resonances detected by the FLI. Instead, the second orbit is characterized by two important resonance captures. The middle panels are obtained for : the orbits represented on the Arnold web are obtained for , and T = 2 × 106. In the top–middle panel, one of the orbits still follows approximately the evolution obtained from the averaged system (20) , but the relevance of the resonance scattering is now more evident (see also the zoom in the bottom-middle panel). The second orbit is characterized by even longer resonance captures, with an interesting change of resonance in correspondence of a resonance crossing (see the bottom–middle panel). The top–right panel is obtained for , and T = 2 × 106, while the bottom–right panel, is obtained for , and T = 2 × 106. As soon as ε increases, the relevance of resonance scatterings and resonance captures increases: for , the orbits follow closely paths obtained by gluing together pieces of resonances.

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