On Algebraic-Dynamical Correspondence of Oscillatory Blow-Up Solutions for ODEs

Abstract

We develop an algebraic-dynamical correspondence for type-I oscillatory blow-up solutions of autonomous ODEs with asymptotically quasi-homogeneous vector fields. Periodic solutions of the balance law arising from asymptotic expansions of blow-ups are related to periodic orbits on the horizon for the desingularized vector field. Unlike the stationary case, the correspondence involves a positive periodic scaling function and a nontrivial time reparametrization. We further establish the correspondence of linearized structures: distinguished phase and scaling directions generate invariant subbundles, while the remaining characteristic multipliers are identified through associated quotient bundles. These results yield a criterion for the existence of type-I periodic blow-up solutions based on periodic solutions of the balance law and their characteristic multipliers, without explicitly constructing compactifications or periodic orbits at infinity. The theory also clarifies the difference between stability information in the balance law and in dynamics at infinity. Two examples illustrate the correspondence.