Constructing Countably Many Distinct Suslin Sets of Characteristic Exponents in the Perron Effect of Change of Their Values

Abstract

For any parameters $m>1$, $\lambda _1\le \lambda _2<0$, and $\varepsilon >0$ and for two sequences $\{S_{in}\}$ of uniformly bounded arbitrary Suslin sets $S_{1n}\subset [\lambda _1+\varepsilon ,b_1]$ and $S_{2n}\subset [\max \{\lambda _2+\varepsilon ,b_1\},b_2]$, we prove the existence of a two-dimensional nonlinear differential system with a linear approximation that has characteristic exponents $\lambda _1$ and $\lambda _2$ and with a disturbance of the $m$th order of smallness in a neighborhood of the origin and possible growth outside it such that all nontrivial solutions of this system are infinitely extendible and have finite Lyapunov exponents. For any $n\in \mathbb {N}$, these exponents form the following sets: $S_{1n}$ for solutions with initial values $(c_1,0)\ne 0$, where $|c_1|\in (n-1,n]$, and $S_{2n}$ for solutions with initial values $(c_1,c_2)$ where $|c_2|\in (n-1,n]$. In particular, for any bounded Suslin sets $S_{-}\subset (-\infty ,0)$ and $S_{+}\subset (0,+\infty )$ we have also established the existence of a nonlinear system whose Lyapunov exponents for all nontrivial solutions form these two sets (singletons in the Perron case).