
-
Previous Article
On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation
- DCDS-B Home
- This Issue
-
Next Article
Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
Using normal forms to study Oterma's transition in the Planar RTBP
1. | Departament de Matemàtiques i Informàtica, Universitat de Barcelona & Barcelona Graduate School of Mathematics, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain |
2. | Faculty of Applied Mathematics, AGH University of Science and Technology, Aleja Adama Mickiewicza 30, 30-059, Kraków, Poland |
Comet 39P/Oterma is known to make fast transitions between heliocentric orbits outside and inside the orbit of Jupiter. In this note the dynamics of Oterma is quantitatively studied via an explicit computation of high order Birkhoff normal forms at the points $ L_1 $ and $ L_2 $ of the Planar Restricted Three-Body Problem. A previous work [
References:
[1] |
A. Deprit,
Canonical transformations depending on a small parameter, Celestial Mech., 1 (1969/1970), 12-30.
doi: 10.1007/BF01230629. |
[2] |
G. Duarte and À. Jorba, Invariant manifolds of tori near ${L}_1$ and ${L}_2$ in the Planar Elliptic RTBP, In preparation, 2022. |
[3] |
G. Duarte and À. Jorba, Modelling Oterma's transition using the Planar Elliptic RTBP, In preparation, 2022. |
[4] |
E. J. Doedel, R. C. Paenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), 1997. |
[5] |
L. Dieci and J. Rebaza,
Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics, 44 (2004), 41-62.
doi: 10.1023/B:BITN.0000025093.38710.f6. |
[6] |
G. Gómez, J. Llibre, R. Martínez and C. Simó, Dynamics and Mission Design Near Libration Points. Vol. I, Fundamentals: The Case of Collinear Libration Points, World Scientific Monograph Series in Mathematics, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
doi: 10.1142/9789812810632_bmatter. |
[7] |
N. W. Harris and M. E. Bailey,
Dynamical evolution of cometary asteroids, Mon. Not. R. Astron. Soc., 297 (1998), 1227-1236.
doi: 10.1046/j.1365-8711.1998.01683.x. |
[8] |
G. Hori,
Theory of general perturbations with unspecified canonical variables, Publications of the Astronomical Society of Japan, 18 (1966), 287-296.
|
[9] |
À. Jorba,
A methodology for the numerical computation of normal forms, centre manifolds and first integrals of {H}amiltonian systems, Exp. Math., 8 (1999), 155-195.
doi: 10.1080/10586458.1999.10504397. |
[10] |
À. Jorba and J. Masdemont,
Dynamics in the centre manifold of the collinear points of the restricted three body problem,, Phys. D, 132 (1999), 189-213.
doi: 10.1016/S0167-2789(99)00042-1. |
[11] |
À. Jorba and B. Nicolás, Transport and invariant manifolds near ${L}_3$ in the Earth-Moon bicircular model, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105327, 19 pp.
doi: 10.1016/j.cnsns.2020.105327. |
[12] |
À. Jorba and M. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[13] | |
[14] |
W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross,
Resonance and capture of Jupiter comets, Celest. Mech. Dyn. Astron., 81 (2001), 27-38.
doi: 10.1023/A:1013398801813. |
[15] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge,
A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.
doi: 10.1142/S0218127405012533. |
[16] |
K. R. Meyer and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90. Springer, Cham, 2017.
doi: 10.1007/978-3-319-53691-0. |
[17] |
J. Moser,
On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.
doi: 10.1002/cpa.3160110208. |
[18] |
K. Ohtsuka, T. Ito, M. Yoshikawa, D. J. Asher and H. Arakida,
Quasi-Hilda comet 147P/Kushida-Muramatsu - Another long temporary satellite capture by Jupiter, Astron. Astrophys., 489 (2008), 1355-1362.
|
[19] |
D. L. Richardson,
A note on a Lagrangian formulation for motion about the collinear points, Celestial Mech., 22 (1980), 231-236.
doi: 10.1007/BF01229509. |
[20] |
A. Souza and M. Tao,
Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?, European J. Appl. Math., 30 (2019), 830-852.
doi: 10.1017/S0956792518000414. |
[21] |
V. Szebehely, Theory of Orbits, Academic Press, 1967.
![]() |
[22] |
M. Tao,
Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions, Phys. D: Nonlinear Phenomena, 363 (2018), 1-17.
doi: 10.1016/j.physd.2017.10.001. |
show all references
References:
[1] |
A. Deprit,
Canonical transformations depending on a small parameter, Celestial Mech., 1 (1969/1970), 12-30.
doi: 10.1007/BF01230629. |
[2] |
G. Duarte and À. Jorba, Invariant manifolds of tori near ${L}_1$ and ${L}_2$ in the Planar Elliptic RTBP, In preparation, 2022. |
[3] |
G. Duarte and À. Jorba, Modelling Oterma's transition using the Planar Elliptic RTBP, In preparation, 2022. |
[4] |
E. J. Doedel, R. C. Paenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), 1997. |
[5] |
L. Dieci and J. Rebaza,
Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics, 44 (2004), 41-62.
doi: 10.1023/B:BITN.0000025093.38710.f6. |
[6] |
G. Gómez, J. Llibre, R. Martínez and C. Simó, Dynamics and Mission Design Near Libration Points. Vol. I, Fundamentals: The Case of Collinear Libration Points, World Scientific Monograph Series in Mathematics, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
doi: 10.1142/9789812810632_bmatter. |
[7] |
N. W. Harris and M. E. Bailey,
Dynamical evolution of cometary asteroids, Mon. Not. R. Astron. Soc., 297 (1998), 1227-1236.
doi: 10.1046/j.1365-8711.1998.01683.x. |
[8] |
G. Hori,
Theory of general perturbations with unspecified canonical variables, Publications of the Astronomical Society of Japan, 18 (1966), 287-296.
|
[9] |
À. Jorba,
A methodology for the numerical computation of normal forms, centre manifolds and first integrals of {H}amiltonian systems, Exp. Math., 8 (1999), 155-195.
doi: 10.1080/10586458.1999.10504397. |
[10] |
À. Jorba and J. Masdemont,
Dynamics in the centre manifold of the collinear points of the restricted three body problem,, Phys. D, 132 (1999), 189-213.
doi: 10.1016/S0167-2789(99)00042-1. |
[11] |
À. Jorba and B. Nicolás, Transport and invariant manifolds near ${L}_3$ in the Earth-Moon bicircular model, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105327, 19 pp.
doi: 10.1016/j.cnsns.2020.105327. |
[12] |
À. Jorba and M. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[13] | |
[14] |
W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross,
Resonance and capture of Jupiter comets, Celest. Mech. Dyn. Astron., 81 (2001), 27-38.
doi: 10.1023/A:1013398801813. |
[15] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge,
A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.
doi: 10.1142/S0218127405012533. |
[16] |
K. R. Meyer and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90. Springer, Cham, 2017.
doi: 10.1007/978-3-319-53691-0. |
[17] |
J. Moser,
On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.
doi: 10.1002/cpa.3160110208. |
[18] |
K. Ohtsuka, T. Ito, M. Yoshikawa, D. J. Asher and H. Arakida,
Quasi-Hilda comet 147P/Kushida-Muramatsu - Another long temporary satellite capture by Jupiter, Astron. Astrophys., 489 (2008), 1355-1362.
|
[19] |
D. L. Richardson,
A note on a Lagrangian formulation for motion about the collinear points, Celestial Mech., 22 (1980), 231-236.
doi: 10.1007/BF01229509. |
[20] |
A. Souza and M. Tao,
Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?, European J. Appl. Math., 30 (2019), 830-852.
doi: 10.1017/S0956792518000414. |
[21] |
V. Szebehely, Theory of Orbits, Academic Press, 1967.
![]() |
[22] |
M. Tao,
Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions, Phys. D: Nonlinear Phenomena, 363 (2018), 1-17.
doi: 10.1016/j.physd.2017.10.001. |





-1.0952439413131636 |
|
-2.9918455882452549 |
|
-8.0698975794308611 |
|
-1.0358302646539692 |
-1.0952439413131636 |
|
-2.9918455882452549 |
|
-8.0698975794308611 |
|
-1.0358302646539692 |
-23498.68 | 0.23735585604354004 | 0.42878140057480135 | 1.127 |
-23498.675 | 0.23735585604354054 | 0.42878140057480163 | 1.127 |
-23498.67 | 0.23735585604354045 | 0.42878140057480196 | 1.127 |
-23498.665 | 0.23735585604354040 | 0.42878140057480207 | 1.127 |
-23498.66 | 0.23735585604354045 | 0.42878140057480213 | 1.127 |
-23498.655 | 0.23735585604354040 | 0.42878140057480219 | 1.127 |
-23498.65 | 0.23735585604354040 | 0.42878140057480213 | 1.127 |
-23498.645 | 0.23735585604354062 | 0.42878140057480230 | 1.127 |
-23498.64 | 0.23735585604354009 | 0.42878140057480185 | 1.127 |
-23498.635 | 0.23735585604354056 | 0.42878140057480202 | 1.127 |
-23498.63 | 0.23735585604354006 | 0.42878140057480268 | 1.127 |
-23498.625 | 0.23735585604353993 | 0.42878140057480224 | 1.127 |
-23498.62 | 0.23735585604354126 | 0.42878140057480219 | 1.127 |
-23498.615 | 0.23735585604354120 | 0.42878140057480119 | 1.127 |
-23498.61 | 0.23735585604354051 | 0.42878140057480241 | 1.127 |
-23498.605 | 0.23735585604354129 | 0.42878140057480252 | 1.127 |
-23498.6 | 0.23735585604353923 | 0.42878140057480257 | 1.127 |
-23498.595 | 0.23735585604354037 | 0.42878140057480124 | 1.127 |
-23498.59 | 0.23735585604354004 | 0.42878140057480252 | 1.127 |
-23498.585 | 0.23735585604354048 | 0.42878140057480130 | 1.127 |
-23498.58 | 0.23735585604354170 | 0.42878140057480124 | 1.127 |
-23498.68 | 0.23735585604354004 | 0.42878140057480135 | 1.127 |
-23498.675 | 0.23735585604354054 | 0.42878140057480163 | 1.127 |
-23498.67 | 0.23735585604354045 | 0.42878140057480196 | 1.127 |
-23498.665 | 0.23735585604354040 | 0.42878140057480207 | 1.127 |
-23498.66 | 0.23735585604354045 | 0.42878140057480213 | 1.127 |
-23498.655 | 0.23735585604354040 | 0.42878140057480219 | 1.127 |
-23498.65 | 0.23735585604354040 | 0.42878140057480213 | 1.127 |
-23498.645 | 0.23735585604354062 | 0.42878140057480230 | 1.127 |
-23498.64 | 0.23735585604354009 | 0.42878140057480185 | 1.127 |
-23498.635 | 0.23735585604354056 | 0.42878140057480202 | 1.127 |
-23498.63 | 0.23735585604354006 | 0.42878140057480268 | 1.127 |
-23498.625 | 0.23735585604353993 | 0.42878140057480224 | 1.127 |
-23498.62 | 0.23735585604354126 | 0.42878140057480219 | 1.127 |
-23498.615 | 0.23735585604354120 | 0.42878140057480119 | 1.127 |
-23498.61 | 0.23735585604354051 | 0.42878140057480241 | 1.127 |
-23498.605 | 0.23735585604354129 | 0.42878140057480252 | 1.127 |
-23498.6 | 0.23735585604353923 | 0.42878140057480257 | 1.127 |
-23498.595 | 0.23735585604354037 | 0.42878140057480124 | 1.127 |
-23498.59 | 0.23735585604354004 | 0.42878140057480252 | 1.127 |
-23498.585 | 0.23735585604354048 | 0.42878140057480130 | 1.127 |
-23498.58 | 0.23735585604354170 | 0.42878140057480124 | 1.127 |
-23499.67 | 0.19332991772006497 | 0.41263461076678953 | 3.045 |
-23499.665 | 0.19332991772006392 | 0.41263461076678948 | 3.045 |
-23499.66 | 0.19332991772006430 | 0.41263461076678759 | 3.045 |
-23499.655 | 0.19332991772006414 | 0.41263461076678970 | 3.045 |
-23499.65 | 0.19332991772006439 | 0.41263461076678870 | 3.045 |
-23499.645 | 0.19332991772006375 | 0.41263461076678903 | 3.045 |
-23499.64 | 0.19332991772006444 | 0.41263461076678820 | 3.045 |
-23499.635 | 0.19332991772006430 | 0.41263461076678898 | 3.045 |
-23499.63 | 0.19332991772006386 | 0.41263461076678859 | 3.045 |
-23499.625 | 0.19332991772006433 | 0.41263461076678837 | 3.045 |
-23499.62 | 0.19332991772006486 | 0.41263461076678953 | 3.045 |
-23499.615 | 0.19332991772006428 | 0.41263461076678931 | 3.045 |
-23499.61 | 0.19332991772006441 | 0.41263461076678837 | 3.045 |
-23499.605 | 0.19332991772006436 | 0.41263461076678865 | 3.045 |
-23499.60 | 0.19332991772006472 | 0.41263461076678876 | 3.045 |
-23499.595 | 0.19332991772006516 | 0.41263461076678876 | 3.045 |
-23499.59 | 0.19332991772006516 | 0.41263461076678898 | 3.045 |
-23499.585 | 0.19332991772006580 | 0.41263461076678898 | 3.045 |
-23499.58 | 0.19332991772006619 | 0.41263461076678942 | 3.045 |
-23499.575 | 0.19332991772006589 | 0.41263461076678914 | 3.045 |
-23499.57 | 0.19332991772006602 | 0.41263461076678926 | 3.045 |
-23499.67 | 0.19332991772006497 | 0.41263461076678953 | 3.045 |
-23499.665 | 0.19332991772006392 | 0.41263461076678948 | 3.045 |
-23499.66 | 0.19332991772006430 | 0.41263461076678759 | 3.045 |
-23499.655 | 0.19332991772006414 | 0.41263461076678970 | 3.045 |
-23499.65 | 0.19332991772006439 | 0.41263461076678870 | 3.045 |
-23499.645 | 0.19332991772006375 | 0.41263461076678903 | 3.045 |
-23499.64 | 0.19332991772006444 | 0.41263461076678820 | 3.045 |
-23499.635 | 0.19332991772006430 | 0.41263461076678898 | 3.045 |
-23499.63 | 0.19332991772006386 | 0.41263461076678859 | 3.045 |
-23499.625 | 0.19332991772006433 | 0.41263461076678837 | 3.045 |
-23499.62 | 0.19332991772006486 | 0.41263461076678953 | 3.045 |
-23499.615 | 0.19332991772006428 | 0.41263461076678931 | 3.045 |
-23499.61 | 0.19332991772006441 | 0.41263461076678837 | 3.045 |
-23499.605 | 0.19332991772006436 | 0.41263461076678865 | 3.045 |
-23499.60 | 0.19332991772006472 | 0.41263461076678876 | 3.045 |
-23499.595 | 0.19332991772006516 | 0.41263461076678876 | 3.045 |
-23499.59 | 0.19332991772006516 | 0.41263461076678898 | 3.045 |
-23499.585 | 0.19332991772006580 | 0.41263461076678898 | 3.045 |
-23499.58 | 0.19332991772006619 | 0.41263461076678942 | 3.045 |
-23499.575 | 0.19332991772006589 | 0.41263461076678914 | 3.045 |
-23499.57 | 0.19332991772006602 | 0.41263461076678926 | 3.045 |
Order 24 | 7.9760239699283809 |
2.7765943768364664 |
Order 32 | 2.2427370346101517 |
3.9594494889226486 |
Order 40 | 8.1359645807725896 |
1.0198945467998282 |
Order 48 | 2.3883565010998806 |
1.4307930312682510 |
Order 56 | 1.1230484307642622 |
8.3245178575158536 |
Order 60 | 1.1272938164516016 |
3.0458619961877403 |
Order 24 | 7.9760239699283809 |
2.7765943768364664 |
Order 32 | 2.2427370346101517 |
3.9594494889226486 |
Order 40 | 8.1359645807725896 |
1.0198945467998282 |
Order 48 | 2.3883565010998806 |
1.4307930312682510 |
Order 56 | 1.1230484307642622 |
8.3245178575158536 |
Order 60 | 1.1272938164516016 |
3.0458619961877403 |
[1] |
Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 |
[2] |
Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 |
[3] |
Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 |
[4] |
Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 |
[5] |
Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243 |
[6] |
Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 |
[7] |
John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170 |
[8] |
Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022085 |
[9] |
Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769 |
[10] |
Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial and Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693 |
[11] |
Daniel Karrasch, Mohammad Farazmand, George Haller. Attraction-based computation of hyperbolic Lagrangian coherent structures. Journal of Computational Dynamics, 2015, 2 (1) : 83-93. doi: 10.3934/jcd.2015.2.83 |
[12] |
P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677 |
[13] |
Torsten Görner, Ralf Hielscher, Stefan Kunis. Efficient and accurate computation of spherical mean values at scattered center points. Inverse Problems and Imaging, 2012, 6 (4) : 645-661. doi: 10.3934/ipi.2012.6.645 |
[14] |
Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357 |
[15] |
Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022048 |
[16] |
Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 |
[17] |
Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151 |
[18] |
In-Soo Baek, Lars Olsen. Baire category and extremely non-normal points of invariant sets of IFS's. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 935-943. doi: 10.3934/dcds.2010.27.935 |
[19] |
Ling Yun Wang, Wei Hua Gui, Kok Lay Teo, Ryan Loxton, Chun Hua Yang. Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications. Journal of Industrial and Management Optimization, 2009, 5 (4) : 705-718. doi: 10.3934/jimo.2009.5.705 |
[20] |
Marcin Dumnicki, Tomasz Szemberg, Halszka Tutaj-Gasińska. New results on fat points schemes in $\mathbb{P}^2$. Electronic Research Announcements, 2013, 20: 51-54. doi: 10.3934/era.2013.20.51 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]