Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction. (English) Zbl 1331.70040
Summary: We consider dissipative one-dimensional systems subject to a periodic force. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction, and study numerically how time-varying friction affects the dynamics. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on the final value and the growth rate of the damping coefficient. The relevance of the results for the spin-orbit model is argued and discussed in some detail.{
©2012 American Institute of Physics}
©2012 American Institute of Physics}
MSC:
| 70F40 | Problems involving a system of particles with friction |
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 70M20 | Orbital mechanics |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
References:
| [1] | DOI: 10.1016/S0031-9201(96)03229-3 · doi:10.1016/S0031-9201(96)03229-3 |
| [2] | DOI: 10.1134/S0038094609010079 · doi:10.1134/S0038094609010079 |
| [3] | DOI: 10.1016/0019-1035(87)90033-9 · doi:10.1016/0019-1035(87)90033-9 |
| [4] | DOI: 10.1016/j.nonrwa.2007.06.007 · Zbl 1156.34321 · doi:10.1016/j.nonrwa.2007.06.007 |
| [5] | DOI: 10.1016/S0096-3003(03)00317-5 · Zbl 1068.34053 · doi:10.1016/S0096-3003(03)00317-5 |
| [6] | DOI: 10.1098/rspa.2001.0841 · Zbl 1001.70023 · doi:10.1098/rspa.2001.0841 |
| [7] | DOI: 10.1007/s11214-007-9284-1 · doi:10.1007/s11214-007-9284-1 |
| [8] | Birkhoff G. D., American Mathematical Society Colloquium Publications, in: Dynamical Systems (1966) |
| [9] | DOI: 10.1038/4501169a · doi:10.1038/4501169a |
| [10] | DOI: 10.1016/j.epsl.2009.07.010 · doi:10.1016/j.epsl.2009.07.010 |
| [11] | DOI: 10.1016/0167-2789(93)90182-Z · Zbl 0791.58081 · doi:10.1016/0167-2789(93)90182-Z |
| [12] | DOI: 10.1007/s10569-008-9142-9 · Zbl 1342.70035 · doi:10.1007/s10569-008-9142-9 |
| [13] | Chow S. N., Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften 251 (1982) · Zbl 0487.47039 |
| [14] | DOI: 10.1038/nature02609 · doi:10.1038/nature02609 |
| [15] | DOI: 10.1016/j.icarus.2008.12.034 · doi:10.1016/j.icarus.2008.12.034 |
| [16] | DOI: 10.1038/nature06317 · doi:10.1038/nature06317 |
| [17] | DOI: 10.1016/j.epsl.2008.03.062 · doi:10.1016/j.epsl.2008.03.062 |
| [18] | DOI: 10.1103/PhysRevLett.91.134102 · doi:10.1103/PhysRevLett.91.134102 |
| [19] | DOI: 10.1103/PhysRevE.54.71 · doi:10.1103/PhysRevE.54.71 |
| [20] | DOI: 10.1007/978-94-010-2861-5_34 · doi:10.1007/978-94-010-2861-5_34 |
| [21] | DOI: 10.1142/S0129055X07002985 · Zbl 1202.34078 · doi:10.1142/S0129055X07002985 |
| [22] | DOI: 10.1086/109947 · doi:10.1086/109947 |
| [23] | DOI: 10.1016/0019-1035(66)90051-0 · doi:10.1016/0019-1035(66)90051-0 |
| [24] | Gradshteyn I. S., Table of Integrals, Series, and Products, 6. ed. (2000) · Zbl 0981.65001 |
| [25] | Guckenheimer J., Applied Mathematical Sciences 42, in: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1990) |
| [26] | DOI: 10.1007/s11214-007-9202-6 · doi:10.1007/s11214-007-9202-6 |
| [27] | DOI: 10.1029/2004JE002294 · doi:10.1029/2004JE002294 |
| [28] | DOI: 10.1126/science.1118842 · doi:10.1126/science.1118842 |
| [29] | DOI: 10.1146/annurev.earth.24.1.41 · doi:10.1146/annurev.earth.24.1.41 |
| [30] | Krasovsky N. N., Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay (1963) |
| [31] | Laskar J., Astron. Astrophys. 318 pp 975– (1997) |
| [32] | DOI: 10.1007/978-1-4757-2184-3 · doi:10.1007/978-1-4757-2184-3 |
| [33] | DOI: 10.1029/RG002i003p00467 · doi:10.1029/RG002i003p00467 |
| [34] | Magnus W., Hill’s Equation (1979) · Zbl 0158.09604 |
| [35] | DOI: 10.1007/3-540-44807-1_7 · doi:10.1007/3-540-44807-1_7 |
| [36] | Neishtadt A. I., Sel. Math. Sov. 12 (3) pp 195– (1993) |
| [37] | DOI: 10.1038/ngeo417 · doi:10.1038/ngeo417 |
| [38] | Palis J., Astérisque 261 pp 335– (1995) |
| [39] | DOI: 10.1016/j.anihpc.2005.01.001 · Zbl 1143.37016 · doi:10.1016/j.anihpc.2005.01.001 |
| [40] | DOI: 10.1016/j.icarus.2005.03.017 · doi:10.1016/j.icarus.2005.03.017 |
| [41] | Plummer C. C., Physical Geology (2003) |
| [42] | DOI: 10.1063/1.1779991 · Zbl 1080.82017 · doi:10.1063/1.1779991 |
| [43] | Press W. H., Numerical Recipes in C (1992) · Zbl 0778.65003 |
| [44] | Pritchard M. E., Origin of the Earth and Moon pp 179– (2000) |
| [45] | DOI: 10.1016/j.icarus.2009.12.002 · doi:10.1016/j.icarus.2009.12.002 |
| [46] | Robuchon G., 40th Lunar and Planetary Science Conference (2009) |
| [47] | DOI: 10.1103/PhysRevE.80.026205 · doi:10.1103/PhysRevE.80.026205 |
| [48] | DOI: 10.1007/s11249-006-9071-2 · doi:10.1007/s11249-006-9071-2 |
| [49] | DOI: 10.1023/A:1017585722391 · doi:10.1023/A:1017585722391 |
| [50] | DOI: 10.2138/rmg.2006.60.4 · doi:10.2138/rmg.2006.60.4 |
| [51] | DOI: 10.1016/0016-7037(92)90172-F · doi:10.1016/0016-7037(92)90172-F |
| [52] | DOI: 10.1016/S0032-0633(01)00085-X · doi:10.1016/S0032-0633(01)00085-X |
| [53] | DOI: 10.1016/S0032-0633(01)00093-9 · doi:10.1016/S0032-0633(01)00093-9 |
| [54] | DOI: 10.1016/0012-821X(80)90169-7 · doi:10.1016/0012-821X(80)90169-7 |
| [55] | Walpole R. E., Probability and Statistics for Engineers and Scientists (1998) · Zbl 0245.62006 |
| [56] | DOI: 10.1146/annurev.ea.13.050185.001221 · doi:10.1146/annurev.ea.13.050185.001221 |
| [57] | DOI: 10.1016/0032-0633(95)00154-9 · doi:10.1016/0032-0633(95)00154-9 |
| [58] | DOI: 10.1887/0750304588 · doi:10.1887/0750304588 |
| [59] | Woolfson M. M., On the Origin of Planets (2011) |
| [60] | DOI: 10.1007/BF00116287 · doi:10.1007/BF00116287 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
