A transformation of non-linear dynamical systems with a single singular singularly perturbed differential equation†

Singularly perturbed state differential equations of the form xdot] = f(x, z, t, ?), x(t₀, ?) = x₀(?); μ(?)? = g(x, z, t, ?), z(t₀ ?) = z₀(?) with lim μ(?) = 0; ?, μ > 0 are considered, where the nominal equation 0 = g(x, z, t, 0)^{? → ∞} does not have to be solvable for z. A fairly general transformation of the above system into a form xdot]^* = f ^*(x^*, z, t; z(1),...,z^(d?1), ? ); μ^*(?)z^(d)= g^*(x^*. z(⁰),...z^(d?1), t; ?), with dim x^* = dim x ?(d ? 1), d ? 1 is proposed. The transformed system stands a better chance of being analysed by existing methods (especially by those proposed by Hoppensteadt (1971) and Hoppensteadt and Mi ranker (1976)) than the original singular singularly perturbed form. Informative examples are presented.