In this paper, we investigate the nonlinear ordinary differential equation.
y′=y^{p}(sin(y)+ε), p>1,
with positive initial data. We show that the qualitative behavior of solutions strongly depends on the perturbation parameter ε. For 0≤ε<1, the nonlinearity possesses infinitely many zeros, which generate invariant intervals trapping the trajectories. As a consequence, all solutions are global and converge to equilibrium points. In contrast, when ε>1, all zeros disappear and the equation becomes strictly positive, leading to finite-time blow-up for every positive solution via an Osgood-type argument. The results provide a simple example illustrating how a small structural perturbation may completely change the global dynamics of an oscillatory differential equation.
Said, K. (2026). A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation. Transactions in Theoretical and Mathematical Physics, 3(3), 129-132. https://doi.org/10.30511/ttmp.2026.2096322.1092
MLA
Said, K. "A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation", Transactions in Theoretical and Mathematical Physics, 3, 3, 2026, 129-132. doi: 10.30511/ttmp.2026.2096322.1092
HARVARD
Said K. (2026). 'A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation', Transactions in Theoretical and Mathematical Physics, 3(3), pp. 129-132. doi: 10.30511/ttmp.2026.2096322.1092
CHICAGO
K. Said, "A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation," Transactions in Theoretical and Mathematical Physics, 3 3 (2026): 129-132, doi: 10.30511/ttmp.2026.2096322.1092
VANCOUVER
Said K. A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation. TTMP. 2026;3(3):129-132. doi: 10.30511/ttmp.2026.2096322.1092
