Ermakov Systems.

(See also: resources and, in particular, research topics.)

Athough named after a Ukrainian mathematician of the nineteenth century who persued some of Lie's ideas for ordinary differential equations using classical geometry, these systems have become important only during the last twenty years. Initially concieved as a way of creating an integral for the time dependent harmonic oscillator, the first fully Ermakov system was written down by Ray and Reid. It is a pair of time-dependent harmonic oscillators with homogeneous nonlinearity of degree -3 with (at least) one invariant, the Lewis-Ray-Reid invariant, I. Such systems have application in many fields including shallow water wave theory, nonlinear optics and elastic media.

The work in the references cited below concerns questions of linearization, generalization and classification.

1. Athorne, C., Projective lifts and generalized Bernoulli and Ermakov systems, J. Math. Analysis and its Applications 223 (1999) 552-563, MathSciNet Review
2. Govinder, K.S.; Athorne, C. and Leach, P.G.L., The algebraic structure of generalized Ermakov systems in three dimensions, J. Phys. A26 (1993) 4035-4046, MathSciNet Review
3. Athorne, C. Stability and periodicity in coupled Pinney equations, J. Diff. Equ. 100 (1992) 82-94.MathSciNet Review
4. Athorne, C. On generalized Ermakov systems, Phys. Lett. A 159 (1991) 375-378,MathSciNet Review
5. Athorne, C., Polyhedral Ermakov systems, Phys. Lett. A 151 (1990) 407-411. MathSciNet Review
6. Athorne, C. Rational Ermakov systems of Fuchsian type, J. Phys. A 24 (1991) 945-961, MathSciNet Review
7. Athorne, C, Kepler-Ermakov problems, J. Phys. A 24 (1991) L1385-1389, MathSciNet Review
8. Athorne, C. On a subclass of Ince equations, J. Phys. A 23 (1990) L137-139, MathSciNet Review
9. Athorne,C; Rogers, C., Ramgulam, U. and Osbaldestin, A. On linearization of the Ermakov system, Phys. Lett. A 143 (1990) 207-212,MathSciNet Review