Degenerate equilibrium position
of a system of ordinary differential equations
A point for which
and for which the matrix
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 [1], [2], [4] and of M. Frommer [3], [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 [5]. If the mapping
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
which is small in the sense of the
-topology.
References
[1] | I. Bendixson, "Sur les courbes définies par des équations différentielles" Acta Math. , 24 (1901) pp. 1–88 |
[2] | A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian) |
[3] | M. Frommer, "Die Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen" Math. Ann. , 99 (1928) pp. 222–272 |
[4] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) |
[5] | A.A. Bryuno, "Stepwise asymptotic solutions of non-linear systems" Izv. Akad. Nauk SSSR Ser. Mat. , 29 (1965) pp. 329–364 (In Russian) |
Comments
See also Frommer method for the study of the behaviour of trajectories in a neighbourhood of an equilibrium position.
References
[a1] | M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) |
[a2] | J.K. Hale, "Ordinary differential equations" , Wiley (1980) |
Degenerate equilibrium position. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_equilibrium_position&oldid=46609