Blow-up parameter landscapes for polynomial dynamical systems

Finite-time blow-up is one of the ways in which a dynamical model can become singular, often signaling the breakdown of either the modeled physical system or the model itself. Determining whether blow-up occurs, and for which parameter values and initial conditions, is, therefore, a fundamental problem in the analysis of nonlinear dynamical systems. We develop a numerical framework for identifying the regions of parameter space in which a dynamical system governed by a system of first-order ordinary differential equations with polynomial right-hand sides exhibits finite-time blow-up for at least one initial condition. The approach combines the compactification of the phase space with computational-algebraic techniques, producing partitioned parameter landscapes that reveal blow-up and non-blow-up regimes. Through several examples, we show that the method replaces problem-specific hand calculations with an automated computational tool for analyzing blow-up regions in parameter-dependent dynamical systems.