Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

A Dynamical Transition From Confinement To Blow-up in an Oscillatory Differential Equation

Document Type : Original Article

Author
Abbès Laghrour University Khechela
Abstract
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.
Keywords
Subjects

1. P. Hartman, Ordinary Differential Equations,
Classicsics,
in Applied Mathemat
Vol. 38, SIAM, Philadelphia (2002).
https://doi.org/10.1137/1.9780898719222

2. J. K. Hale, Ordinary Differential Equations, Dover
Publications, Mineola (2009).

3. A. Friedman, Foundations of Modern Analysis,
Dover Publications, Mineola (2010).

4. W. F. Osgood, “Beweis der Existenz einer Lösung
der Differentialgleichung dy
dx = f(x,y), ohne Hinzu
nahme der Cauchy–Lipschitz’schen Bedingung,”
Monatshefte für Mathematik und Physik 9 (1898),
331–345. https://doi.org/10.1007/BF01707876

5. H. Amann, Ordinary Differential Equa
tions: An Introduction to Nonlinear Anal
ysis, De Gruyter Studies in Mathematics,
Vol. 13, Walter de Gruyter, Berlin (1990).
https://doi.org/10.1515/9783110853698

6. V. Lakshmikantham and S. Leela, Differential
and Integral Inequalities: Theory and Applications,
Vol. I: Ordinary Differential Equations, Mathemat
ics in Science and Engineering, Vol. 55A, Academic
Press, New York (1969)
Volume 3, Issue 3
Summer 2026
Pages 129-132

  • Receive Date 01 August 2026
  • Accept Date 25 August 2026
  • First Publish Date 25 August 2026
  • Publish Date 01 August 2026