On one scenario for changing the stability of invariant manifolds of singularly perturbed systems (Q6610442)

Consider the singularly perturbed system\N\[\N\begin{aligned} \N\frac{dx}{dt} &= f(x,y,\mu,\varepsilon), \\\N\varepsilon \frac{dy}{dt} &=g(x,y,\mu,\varepsilon) \N\end{aligned}\tag{1}\N\]\Nwith \(x \in \mathbb{R}^n, y \in \mathbb{R}^m\), \(\varepsilon\) and \(\mu\) are real parameters, where \(\varepsilon\) is sufficiently small. Under some conditions system \((1)\) possesses a slow invariant manifold \(M_\varepsilon\) satisfying for \(\varepsilon =0\) the relation \(g(x,y,\mu,0)=0\). The stability of \(M_\varepsilon\) can be determined by the eigenvalues of the Jacobian of its linearization. The problem of exchange of stability of \(M_\varepsilon\) has been studied in the literature using different approachess. The authors investigate the exchange of stability of \(M_\varepsilon\) under degenerate conditions on the eigenvalues of the Jacobian. The result is illustrated by several examples.