Delayed loss of stability in systems with degenerate linear parts (Q1876024)

The paper investigates the delayed loss of stability of the scalar equation \(\varepsilon dx/d\tau =g(\tau ,x)\). The associated autonomous equation is \(dx/dt=g(\tau ,x)\) where \(\tau\) is now considered as a parameter. It is assumed that \(g(\tau ,0)=0\). Under suitable conditions on \(g\), it is shown that the associated autonomous equation undergoes a bifurcation of equilibrium at the point where \(x=0\) loses stability. The asymptotic delay is estimated. In a similar manner the planar system \(\varepsilon dx/d\tau =\sigma (\tau )x-w(\tau )y+f_{1}(\tau ,x,y)\), \(\varepsilon dy/d\tau =w(\tau )x+\sigma (\tau )y+f_{2}(\tau ,x,y)\), loses stability where the associated autonomous system undergoes a Hopf bifurcation at the origin.