Differential and Integral Equations

Immediate exchange of stabilities in singularly perturbed systems

N. N. Nefedov and K. R. Schneider

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We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to $\varepsilon$ we prove that an immediate exchange of stabilities implies a change of the asymptotic behavior from $0(\varepsilon)$ to $0(\sqrt{\varepsilon})$ near the point of exchange of stabilities.

Article information

Differential Integral Equations, Volume 12, Number 4 (1999), 583-599.

First available in Project Euclid: 29 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34E15: Singular perturbations, general theory
Secondary: 34C60: Qualitative investigation and simulation of models 34C99: None of the above, but in this section 34D99: None of the above, but in this section


Nefedov, N. N.; Schneider, K. R. Immediate exchange of stabilities in singularly perturbed systems. Differential Integral Equations 12 (1999), no. 4, 583--599. https://projecteuclid.org/euclid.die/1367267008

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