Compact invariant sets of the static spherically symmetric Einstein-Yang-Mills equations. (English) Zbl 1236.83001
Summary: In this Letter we obtain results concerning compact invariant sets of the static spherically symmetric Einstein-Yang-Mills (EYM) equations with help of studies of its localization. Let \(a\) be a cosmological constant and \(s\) be another parameter entering into these equations which is used for considering the physical time as a temporal variable, with \(s=1\), while \(s= - 1\) is used for considering the physical time as a spatial variable. We show that in case \(s=1\); \(a<0\) the location of any compact invariant set is described by some system of linear inequalities. Then we prove that in case \(s=1\); \(a>0\) the set of all compact invariant sets consists of two equilibrium points only. Further, we state that in cases \(s= - 1\); \(a<0\) and \(s= - 1\); \(a>0\) there are only two equilibrium points and there are no periodic orbits. In addition, we prove that in the last two cases there are neither homoclinic orbits nor heteroclinic orbits as well.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C45 | Quantization of the gravitational field |
| 83F05 | Relativistic cosmology |
| 15A39 | Linear inequalities of matrices |
| 37C29 | Homoclinic and heteroclinic orbits for dynamical systems |
References:
| [1] | Breitenlohner, P.; Forgács, P.; Maison, D., Commun. Math. Phys., 261 (2006), 569 · Zbl 1104.58004 |
| [2] | Buzzi, C. A.; Llibre, J.; da Silva, P. R., J. Phys. A: Math. Theor., 40, 7187 (2007) · Zbl 1189.82077 |
| [3] | Llibre, J.; Valls, C., J. Math. Anal. Appl., 336, 2, 1203 (2007) · Zbl 1124.83005 |
| [4] | Linden, A. N., Commun. Math. Phys., 221, 525 (2001) · Zbl 0992.83014 |
| [5] | Linden, A. N., Classical Quantum Gravity, 18, 695 (2001) · Zbl 0973.83043 |
| [6] | Linden, A. N., Michigan Math. J., 50, 201 (2002) · Zbl 1027.83010 |
| [7] | Linden, A. N., TAMS, 359, 11, 5125 (2007) · Zbl 1206.83047 |
| [8] | Breitenlohner, P.; Forgács, P.; Maison, D., Commun. Math. Phys., 163, 141 (1994) · Zbl 0809.53081 |
| [9] | Smoller, J.; Wasserman, A.; Yau, S. T.; McLeod, J. B., Commun. Math. Phys., 143, 115 (1991) · Zbl 0755.53061 |
| [10] | Winstanley, E., Classical Quantum Gravity, 16, 1963 (1999) · Zbl 0947.83030 |
| [11] | Llibre, J.; Valls, C., J. Phys. A: Math. Gen., 38, 8155 (2005) · Zbl 1098.83012 |
| [12] | Llibre, J.; Yu, J., Commun. Math. Phys., 286, 277 (2009) · Zbl 1173.58301 |
| [13] | Krishchenko, A. P.; Starkov, K. E., Phys. Lett. A, 353, 383 (2006) · Zbl 1181.37044 |
| [14] | Starkov, K. E., Phys. Lett. A, 372, 6269 (2008) · Zbl 1225.70012 |
| [15] | Boichenko, V. A.; Leonov, G. A.; Reitmann, V., Dimension Theory for Ordinary Differential Equations (2005), Teubner Verlag: Teubner Verlag Wiesbaden · Zbl 1094.34002 |
| [16] | Pogromski, A. Y.; Nijmeijer, H.; Rooda, J. E., IEEE Trans. Automat. Control, 52, 4, 586 (2007) · Zbl 1366.93281 |
| [17] | K.E. Starkov, Int. J. Bifurc. Chaos, in press; K.E. Starkov, Int. J. Bifurc. Chaos, in press |
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