View source for Degenerate equilibrium position

''of a system of ordinary differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307701.png" />''

A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307702.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307703.png" /> and for which the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307704.png" /> has zero eigen values. The most extensively investigated degenerate equilibrium positions are those of two-dimensional systems, for which several methods for studying the behaviour of trajectories in a neighbourhood of this position are available; these include the methods of I. Bendixson [[#References|[1]]], [[#References|[2]]], [[#References|[4]]] and of M. Frommer [[#References|[3]]], [[#References|[4]]]. Geometric methods of investigation have been recommended in spaces of more than two dimensions; these consist, in essence, in isolating the main terms at the right-hand sides of the equations and demonstrating that the behaviour of the trajectory remains unchanged on passing from the abbreviated to the complete equation [[#References|[5]]]. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307705.png" /> is sufficiently often differentiable or analytic, the degree of degeneracy of the equilibrium position may be considered equal to the number of non-degenerate equilibrium positions into which the given degenerate equilibrium position may be subdivided as a result of a change in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307706.png" /> which is small in the sense of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030770/d0307707.png" />-topology.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bendixson,   "Sur les courbes définies par des équations différentielles"  ''Acta Math.'' , '''24'''  (1901)  pp. 1–88</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Frommer,   "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen"  ''Math. Ann.'' , '''99'''  (1928)  pp. 222–272</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Bryuno,   "Stepwise asymptotic solutions of non-linear systems"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29'''  (1965)  pp. 329–364  (In Russian)</TD></TR></table>



====Comments====
See also [[Frommer method|Frommer method]] for the study of the behaviour of trajectories in a neighbourhood of an [[Equilibrium position|equilibrium position]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,   S. Smale,   "Differential equations, dynamical systems, and linear algebra" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1980)</TD></TR></table>