The infinitesimal order of perturbations preserving the instability of a singular point (Q5942090)

Let \(O\) \((x=0)\) be a singular point of the system \[ \dot x=f(x),\quad x\in \mathbb{R}^n. \] If \(O\) is asymptotically stable, then there exists a continuous function \(\eta (r)\to +0\), \(r\to +0\), such that \(O\) remains asymptotically stable for any system \(\dot x=f(x)+g(t,x)\) with \(|g(t,x)|\leq\eta (|x|)\). Such a function does not necessarily exist for a system with an unstable singular point. Examples are known in which the instability can be destroyed by perturbations arbitrarily rapidly decaying as \(x\to 0\). This kind of instability is referred to as structurally unstable instability. If there exists a continuous function \(\eta (r)>0\), \(r>0\), \(\eta (r)\to 0\) as \(r\to +0\), such that \(O\) remains unstable under any perturbations satisfying \(|g(t,x)|\leq\eta (|x|)\), then this type of instability is referred to as structurally stable instability. The infinitesimal order of instability-preserving perturbations may vary and depends not only on the rate of decay of \(f(x)\) as \(x\to 0\). If the function \(\eta (r)\) can be chosen to have the same infinitesimal order as \(f(x)\) (\(x\to 0\)), i.e. \(f(x)/\eta (x)\) is bounded for \(|x|\leq r_0\), then structurally stable instability is referred to as strongly structurally stable instability. The paper presents the following results: 1. A sufficient condition for the strongly structurally stable instability for homogeneous systems in \(\mathbb{R}^n\); 2. Necessary and sufficient conditions for the strongly structurally stable instability for homogeneous systems in \(\mathbb{R}^2\); 3. Sufficient conditions for the strongly structurally unstable instability for homogeneous systems in \(\mathbb{R}^2\).