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The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems
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10.1063/1.4807097
/content/aip/journal/chaos/23/2/10.1063/1.4807097
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4807097
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Figures

Image of FIG. 1.
FIG. 1.

Detection of the Arnold web of the Hamiltonian system (6), previously studied in Ref. , 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  = 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. . 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.

Image of FIG. 2.
FIG. 2.

Detection of the Arnold web of the Hamiltonian system (6), previously studied in Ref. , 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  = 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. . 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.

Image of FIG. 3.
FIG. 3.

Detection of the Arnold web of the Hamiltonian system (6) , previously studied in Ref. , in the region of the phase space corresponding to the box of Figure 1 for and  = 3000 (top left), 0.01 and  = 2000 (top right), 0.03 and  = 1000 (bottom left), 0.04 and  = 1000 (bottom right). The panels represent, for any initial condition, the average between the FLI computed withinitial tangent vectors and , and integration times  = 3000 (top left),  = 2000 (top right), and  = 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. . 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.

Image of FIG. 4.
FIG. 4.

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  = 1000.

Image of FIG. 5.
FIG. 5.

Test of the sensitivity of the FLI method: plot of versus for and  = 10 (left panel) and and  = 10 (right panel). In both cases, the relative error of the theoretical result with respect to the numerical one is smaller than 2 × 10 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 (1–2), 1–25 (2002). Copyright 2002 Elsevier Science B.V.

Image of FIG. 6.
FIG. 6.

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 of the section , defined by . For each value of , 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 ( = 10 top-left, 10 top-right; 10 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 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 , 179–187 (2003) Copyright 2003 Elsevier B.V.

Image of FIG. 7.
FIG. 7.

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 . Precisely, the black dots on the panels represent the points of the orbits which belong to the small neighbourhood of the section , defined by . The panels b, c, and d correspond to the integration times  = 10, , and  = 1.1 × 10, 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 (3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Image of FIG. 8.
FIG. 8.

Top panel: Evolution of with time for a set of 100 orbits of system (6) for previously studied in Ref. . The initial conditions are taken on the resonance and the points of the orbits which belong to a small neighbourhood of the section 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. . Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Discrete Contin. Dyn. Syst. B (3), 687–698 (2005). Copyright 2005, American Institute of Mathematical Sciences.

Image of FIG. 9.
FIG. 9.

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  = 100 initial conditions near , equally spaced on a segment of amplitude 10 in the linear unstable space (computed at order 1 in ) of the hyperbolic torus related to the resonance. The total integration times range from  = 10 up to  = 10. Reprinted with permission from M. Guzzo, E. Lega, and C. Froeschlé, Chaos (3), 033101 (2011). Copyright 2011, American Institute of Physics.

Image of FIG. 10.
FIG. 10.

The panels report the Arnold web of the map Eq. (19) for and different values of , 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  = 2 × 10. 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  = 2 × 10. 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  = 2 × 10, while the bottom–right panel, is obtained for , and  = 2 × 10. As soon as 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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/content/aip/journal/chaos/23/2/10.1063/1.4807097
2013-06-06
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4807097
10.1063/1.4807097
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