On the existence of periodic solutions to certain classes of systems of differential equations with random pulse influence (Q2784873)

The authors deal with the following stochastic impulsive system NEWLINE\[NEWLINE\dot x=f(t,x,\xi(t)),\;t\neq t_i;\quad \Delta x|_{t=t_i}=I_i(x,\eta_i),NEWLINE\]NEWLINE with \(x\in \mathbb{R}^n\), \(i\in \mathbb{Z}\), \(f:\mathbb{R}^{1+n+k} \mapsto \mathbb{R}^n\), \(I_i:\mathbb{R}^{n+l} \mapsto \mathbb{R}^n\). It is assumed that the function \(f\) is \(T\)-periodic in \(t\), and that there exists a natural number \(p\) such that \(I_{i+p}(x,z)=I_i(x,z)\), \(t_{i+p}=t_i+T\). The \(\mathbb{R}^k\)-valued stochastic process \(\xi(t)\) and the sequence \(\eta_i\) of \(\mathbb{R}^l\)-valued random variables are periodically coupled. The goal of the paper is to obtain conditions for the existence of solutions which are periodic in the sense of finite-dimensional distributions and periodically coupled with \(\xi(t)\). NEWLINENEWLINENEWLINESuch conditions are obtained in the following cases: NEWLINENEWLINENEWLINE(a) \(f=F(x)+\epsilon g(t,x,y), I_i=O(\epsilon)\) \(|\epsilon|\ll 1\), and the system \(\dot x=F(x)\) has an asymptotically stable compact invariant set; NEWLINENEWLINENEWLINE(b) \(f=A(t)x+y, I_i=B_ix+z\), and the spectrum of the monodromy operator of the linear \(T\)-periodic system \(\dot x=A(t)x\), \(t\neq t_i\), \(\Delta x|_{t=t_i}=B_i\), does not intersect with the unit circle; NEWLINENEWLINENEWLINE(c) \(f=A(t)x+O(\epsilon), I_i=B_i+O(\epsilon)\).