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).