Chaotic Diffusion in a Triaxial Galactic Model: an Example of Global Stable Chaos

P. M. Cincotta ; C. M. Giordano

Russian journal of nonlinear dynamics, Tome 20 (2024) no. 4, pp. 427-447.

Résumé

In this work we focus on the chaotic diffusion in the phase space of a triaxial potential resembling an elliptical galaxy. The transport process is studied in two different action-like starting spaces in order to cope with circulating and noncirculating orbits. Estimates of the diffusion rate obtained by means of the variance approach are discussed in detail and their limitations are exposed. After revisiting the Shannon-entropy-based method from a conceptual point of view in the framework of simple arguments taken from the information theory, we apply it to measure changes in the unperturbed actions or integrals of motion of the system for different sets of small ensembles of random initial conditions. For such sets of ensembles, estimates of the Lyapunov times are also provided. The results show that, within the chaotic component of the phase space, the Lyapunov times are shorter than any physical time scale as the Hubble time, but the diffusion times are much larger than the latter. Thus, we conclude that stable chaos dominates the dynamics of realistic galactic models.

Détail

Keywords: instabilities, galactic dynamics
Mots-clés : chaos, diffusion

Bibliographie

[1] Hénon, M. and Heiles, C., “The Applicability of the Third Integral of Motion: Some Numerical Experiments”, Astronom. J., 69:1 (1964), 73–79 | MR

[2] Dokl. Akad. Nauk SSSR, 156:1 (1964), 9–12 (Russian) | MR | Zbl

[3] Chirikov, B. V., “A Universal Instability of Many-Dimensional Oscillator Systems”, Phys. Rep., 52:5 (1979), 263–379 | DOI | MR

[4] Contopoulos, G., “Orbits in Highly Perturbed Dynamical Systems: 1. Periodic Orbits”, Astronom. J., 75 (1970), 96–107 | DOI | MR

[5] Contopoulos, G., “Orbits in Highly Perturbed Dynamical Systems: 2. Stability of Periodic Orbits”, Astronom. J., 75 (1970), 108–130 | DOI | MR

[6] Bountis, T., Manos, T., and Antonopoulos, C., “Complex Statistics in Hamiltonian Barred Galaxy Models”, Celest. Mech. Dyn. Astron., 113:1 (2012), 63–80 | DOI

[7] Contopoulos, G., Order and Chaos in Dynamical Astronomy, Astronomy and Astrophysics Library, Springer, Berlin, 2002, xiv, 624 pp. | DOI | MR | Zbl

[8] Katsanikas, M. and Patsis, P. A., “The Phase Space Structure in the Vicinity of Vertical Lyapunov Orbits around $L_{1,\,2}$ in a Barred Galaxy Model”, Mon. Not. R. Astron. Soc., 516 (2022), 5232–5243 | DOI

[9] Katsanikas, M., Patsis, P. A., and Contopoulos, G., “Instabilities and Stickiness in a 3D Rotating Galactic Potential”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23:2 (2013), Art. 1330005, 30 pp. | DOI | MR

[10] Merritt, D., “Chaos and Elliptical Galaxies”, Celest. Mech. Dyn. Astron., 64:1–2 (1996), 55–67 | DOI

[11] Merritt, D., “Elliptical Galaxy Dynamics”, Publ. Astron. Soc. Pac., 111:756 (1999), 129–168 | DOI

[12] Merritt, D. and Fridman, T., “Triaxial Galaxies with Cusps”, Astrophys. J., 460 (1996), 136–162 | DOI

[13] Merritt, D. and Valluri, M., “Chaos and Mixing in Triaxial Stellar Systems”, Astrophys. J., 471:1 (1996), 82–105 | DOI

[14] Muzzio, J. C., Carpintero, D. D., and Wachlin, F. C., “Spatial Structure of Regular and Chaotic Orbits in A Self-Consistent Triaxial Stellar System”, Celest. Mech. Dyn. Astron., 91:1–2 (2005), 173–190 | DOI | MR | Zbl

[15] Muzzio, J. C., “Regular and Chaotic Orbits in a Self-Consistent Triaxial Stellar System with Slow Figure Rotation”, Celest. Mech. Dyn. Astron., 96:2 (2006), 85–97 | DOI | MR | Zbl

[16] Papaphilippou, Y. and Laskar, J., “Global Dynamics of Triaxial Galactic Models through Frequency Map Analysis”, Astron. Astrophys., 329 (1998), 451–481

[17] Pfenniger, D., “Galactic and Stellar Dynamics: Limits and Perspectives”, Celestial Mech. Dynam. Astronom., 72:1–2 (1998/99), 37–67 | DOI | MR

[18] Voglis, N., Kalapotharakos, C., and Stavropoulos, I., “Mass Components in Ordered and in Chaotic Motion in Galactic $N$-Body Models”, Mon. Not. R. Astron. Soc., 337:2 (2002), 619–630 | DOI

[19] Wachlin, F. C. and Ferraz-Mello, S., “Frequency Map Analysis of the Orbital Structure in Elliptical Galaxies”, Mon. Not. R. Astron. Soc., 298:1 (1998), 22–32 | DOI

[20] Milani, A. and Nobili, A. M., “An Example of Stable Chaos in the Solar System”, Nature, 357:6379 (1992), 569–571 | DOI

[21] Maffione, N. P., Darriba, L. A., Cincotta, P. M., and Giordano, C. M., “Chaos Detection Tools: Application to a Self-Consistent Triaxial Model”, Mon. Not. R. Astron. Soc., 429:3 (2013), 2700–2717 | DOI

[22] Maffione, N. P., Darriba, L. A., Cincotta, P. M., and Giordano, C. M., “A Comparison of Different Indicators of Chaos Based on the Deviation Vectors: Application to Symplectic Mappings”, Celestial Mech. Dynam. Astronom., 111:3 (2011), 285–307 | DOI | MR

[23] Darriba, L. A., Maffione, N. P., Cincotta, P. M., and Giordano, C. M., “Comparative Study of Variational Chaos Indicators and ODEs' Numerical Integrators”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22:10 (2012), Art. 1230033, 33 pp. | DOI | MR

[24] Lecar, M., Franklin, F., and Murison, M., “On Predicting Long-Term Orbital Instability: A Relation between the Lyapunov Time and Sudden Orbital Transitions”, Astronom. J., 104:3 (1992), 1230–1236 | DOI

[25] Levison, H. F. and Duncan, M. J., “The Gravitational Sculpting of the Kuiper Belt”, Astrophys. J. Lett., 406 (1993), L35–L38 | DOI

[26] Mikkola, S. and Tanikawa, K., “Correlation of Macroscopic Instability and Lyapunov Times in the General Three-Body Problem”, Mon. Not. R. Astron. Soc., 379:1 (2007), L21–L24 | DOI

[27] Morbidelli, A. and Froeschelé, C., “On the Relationship between the Lyapunov Times and Macroscopial Instability Times”, Celestial Mech. Dynam. Astronom., 63:2 (1996), 227–239 | DOI | Zbl

[28] Murison, M. A., Lecar, M., and Franklin, F. A., “Chaotic Motion in the Outer Asteroid Belt and Its Relation to the Age of the Solar System”, Astron. J., 108:6 (1994), 2323–2329 | DOI

[29] Soper, P., Franklin, F., and Lecar, M., “On the Original Distribution of the Asteroids: 3. Orbits between Jupiter and Saturn”, Icarus, 87:2 (1990), 265–284 | DOI

[30] Shevchenko, I. I., “Hamiltonian Intermittency and Lévy Flights in the Three-Body Problem”, Phys. Rev. E (3), 81:6 (2010), Art. 066216, 11 pp. | DOI | MR

[31] Shevchenko, I. I., “On the Recurrence and Lyapunov Time Scales of the Motion near Chaos Border”, Phys. Lett. A, 241:1–2 (1998), 53–60 | DOI

[32] Cincotta, P. M., Giordano, C. M., and Shevchenko, I. I., “Revisiting the Relation between the Lyapunov Time and the Instability Time”, Phys. D, 430 (2022), Art. 133101, 12 pp. | DOI | MR

[33] Cincotta, P. M., Giordano, C. M., and Shevchenko, I. I., “Diffusion and Lyapunov Timescales in the Arnold Model”, Phys. Rev. E, 106:4 (2022), Art. 044205, 15 pp. | DOI | MR

[34] Mogavero, F., Hoang, N. H., and Laskar, J., “Timescales of Chaos in the Inner Solar System: Lyapunov Spectrum and Quasi-Integrals of Motion”, Phys. Rev. X, 13:2 (2023), Art. 021018, 24 pp.

[35] Contopoulos, G. and Harsoula, M., “Stickiness in Chaos”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18:10 (2008), 2929–2949 | DOI | MR

[36] Contopoulos, G. and Harsoula, M., “Stickiness Effects in Conservative Systems”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20:7 (2010), 2005–2043 | DOI | MR | Zbl

[37] Contopoulos, G. and Harsoula, M., “Stickiness Effects in Chaos”, Celestial Mech. Dynam. Astronom., 107:1–2 (2010), 77–92 | DOI | MR | Zbl

[38] Efthymiopoulos, C., Contopoulos, G., Voglis, N., and Dvorak, R., “Stickiness and Cantori”, J. Phys. A, 30:23 (1997), 8167–8186 | DOI | MR | Zbl

[39] Miguel, N., Simó, C., and Vieiro, A., “Escape Times across the Golden Gantorus of the Standard Map”, Regul. Chaotic Dyn., 27:3 (2022), 281–306 | DOI | MR | Zbl

[40] Cincotta, P. M., Giordano, C. M., Alves Silva, R., and Beaugé, C., “The Shannon Entropy: An Efficient Indicator of Dynamical Stability”, Phys. D, 417 (2021), Paper No. 132816, 10 pp. | DOI | MR

[41] Cincotta, P., Giordano, C. Alves Silva, R., Beaugé, C., “Shannon Entropy Diffusion Estimates: Sensitivity on the Parameters of the Method”, Celestial Mech. Dynam. Astronom., 133:2 (2021), Paper No. 7, 20 pp. | DOI | MR | Zbl

[42] Cincotta, P. M. and Giordano, C. M., “Estimation of Diffusion Time with the Shannon Entropy Approach”, Phys. Rev. E, 107:6 (2023), Paper No. 064101, 9 pp. | DOI | MR

[43] Cincotta, P. M. and Giordano, C. M., “Estimation of the Diffusion Time in a Triaxial Galactic Potential”, Mon. Not. R. Astron. Soc., 526:1 (2023), 895–902 | DOI | MR

[44] Giordano, C. M. and Cincotta, P. M., “The Shannon Entropy As a Measure of Diffusion in Multidimensional Dynamical Systems”, Celestial Mech. Dynam. Astronom., 130:5 (2018), Paper No. 35, 21 pp. | DOI | MR

[45] de Zeeuw, T., Peletier, R., and Franx, N., “Mass Models with Stäckel Potentials”, Mon. Not. R. Astron. Soc., 221:4 (1986), 1001–1022 | DOI | Zbl

[46] Cincotta, P., Giordano, C., and Muzzio, J. C., “Global Dynamics in a Self-Consistent Model of Elliptical Galaxy”, Discrete Contin. Dyn. Syst. Ser. B, 10:2–3 (2008), 439–454 | MR | Zbl

[47] Schwarzschild, M., “Self-Consistent Models for Galactic Halos”, Astrophys. J., 409:2 (1993), 563–577 | DOI | MR

[48] Cincotta, P. M. and Giordano, C. M., “Theory and Applications of the Mean Exponential Growth Factor of Nearby Orbits (MEGNO) Method”, Chaos Detection and Predictability, Lecture Notes in Phys., 915, eds. C. Skokos, G. Gottwald, J. Laskar, Springer, Berlin, 2016, 93–128 | DOI | MR

[49] Cincotta, P. M., Giordano, C. M., and Simó, C., “Phase Space Structure of Multi-Dimensional Systems by Means of the Mean Exponential Growth Factor of Nearby Orbits”, Phys. D, 182:3–4 (2003), 151–178 | DOI | MR | Zbl

[50] Cincotta, P. M. and Simó, C., “Simple Tools to Study Global Dynamics in Non-Axisymmetric Galactic Potentials: 1”, Astron. Astrophys. Suppl. Ser., 147:2 (2000), 205–228 | DOI

[51] Hairer, E., Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations: Vol. 1. Nonstiff Problems, Springer Ser. Comput. Math., 8, 2nd ed., Springer, New York, 1993, xvi, 528 pp. | MR

[52] Prince, P. J. and Dormand, J. R., “High Order Embedded Runge – Kutta Formulae”, J. Comput. Appl. Math., 7:1 (1981), 67–75 | DOI | MR | Zbl

[53] Hurst, H. E., “Long-Term Storage Capacity of Reservoirs”, Trans. Am. Soc. Civil Eng., 116:1 (1951), 770–799 | DOI

[54] Shannon, C. E. and Weaver, W., The Mathematical Theory of Communication, Univ. of Illinois, Urbana, Ill., 1949, vi, 117 pp. | MR | Zbl

[55] Arnol'd, V. I. and Avez, A., Ergodic Problems of Classical Mechanics, translated from the French by A. Avez, Math. Phys.Monogr. Ser., 2nd ed., Addison-Wesley, New York, 1989, ix, 286 pp. | MR | Zbl

[56] Katz, A., Principles of Statistical Mechanics: The Information Theory Approach, Freeman, San Francisco, Calif., 1967, 188 pp.

[57] Lesne, A., “Shannon Entropy: A Rigorous Notion at the Crossroads between Probability, Information Theory, Dynamical Systems and Statistical Physics”, Math. Structures Comput. Sci., 24:3 (2014), e240311, 63 pp. | DOI | MR | Zbl

[58] Cincotta, P. M., Giordano, C. M., and Simó, C., “Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter”, Regul. Chaotic Dyn., 28:3 (2023), 265–294 | DOI | MR | Zbl

[59] Alves Silva, R., Beuagé, C., Ferraz-Mello, S., Cincotta, P. M., and Giordano, C. M., “Instability Times in the HD 181433 Exoplanetary System”, Astron. Astrophys., 652 (2021), A112, 10 pp. | DOI

[60] Beaugé, C. and Cincotta, P. M., “Shannon Entropy Applied to the Planar Restricted Three-Body Problem”, Celestial Mech. Dynam. Astronom., 131:11 (2019), Paper No. 52, 21 pp. | DOI | MR | Zbl

[61] Kövári, E., Érdi, B., and Sándor, Zs., “Application of the Shannon Entropy in the Planar (Non-Restricted) Four-Body Problem: The Long-Term Stability of the Kepler-60 Exoplanetary System”, Mon. Not. R. Astron. Soc., 509:1 (2022), 884–893 | DOI

[62] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 2: Numerical Application”, Meccanica, 15:1 (1980), 21–30 | DOI

[63] Maffione, N. P., Gómez, F. A., Cincotta, P. M., Giordano, C. M., Cooper, A. P., and O'Shea, B. W., “On the Relevance of Chaos for Halo Stars in the Solar Neighbourhood”, Mon. Not. R. Astron. Soc., 453:3 (2015), 2830–2847 | DOI

[64] Maffione, N. P., Gómez, F. A., Cincotta, P. M., Giordano, C. M., Grand, R. J. J., Marinacci, F., Pakmor, R., Simpson, Ch. M., Springel, V., and Frenk, C. S., “On the Relevance of Chaos for Halo Stars in the Solar Neighbourhood: 2”, Mon. Not. R. Astron. Soc., 478:3 (2018), 4052–4067 | DOI

[65] Binney, J., “Orbital Tori for Non-Axisymmetric Galaxies”, Mon. Not. R. Astron. Soc., 474:2 (2018), 2706–2724

[66] Binney, J. and Spergel, D., “Spectral Stellar Dynamics”, Astrophys. J., 252 (1982), 308–321 | DOI

[67] Binney, J. and Spergel, D., “Spectral Stellar Dynamics: 2. The Action Integrals”, Mon. Not. R. Astron. Soc., 206:1 (1984), 159–177 | DOI