Periodic solutions to systems of differential equations with a random impulse action. (English. Ukrainian original) Zbl 0999.34057

Perestyuk, M. O.; Stanzhyts’kyj, O. M.

Theory Probab. Math. Stat. 63, 131-136 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 119-124 (2000).

The authors consider \(T\)-periodic systems of differential equations \[ {dx\over dt}=f(t,x,\xi(t)),\;t\neq t_{i},\quad \Delta x|_{t=t_{i}}=x(t_{i}+0)-x(t_{i}-0)=I_{i}(x,\eta_{i}), \] with \(t\in \mathbb{R}\), \(x\in \mathbb{R}^{n}\), \(i\in \mathbb{Z}\). The function \(f(t,x,y)\), \(y\in \mathbb{R}^{k},\) is periodic in \(t\) with the period \(T\); the functions \(I_{i}(x,z)\), \(z\in \mathbb{R}^{l}\), and time moments \(t_{i}\) are such that for some positive integer \(p\): \(I_{i+p}(x,z)=I_{i}(x,z),\;t_{i+p}=t_{i}+T\). Let \(\xi(t)\) be a stochastically continuous random process and let \(\eta_{i}\) be a sequence of random variables which are periodically connected. The authors obtain necessary and sufficient conditions for the existence of periodic solutions to the considered system. Also, for the system \[ dx/dt=F(x)+\varepsilon g(t,x,\xi(t)), \;t\neq t_{i}, \quad \Delta x|_{t=t_{i}}=\varepsilon I_{i}(x,\eta_{i}), \] where \(\varepsilon>0\) is a small parameter, sufficient conditions for the existence of periodic solutions are derived.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

34F05	Ordinary differential equations and systems with randomness
34C25	Periodic solutions to ordinary differential equations
34K45	Functional-differential equations with impulses
34K50	Stochastic functional-differential equations

Keywords:

periodic solutions; system of differential equations; impulse action