Lower characteristic exponents of nontrivial solutions to Pfaffian systems (Q2703959)

The linear system NEWLINE\[NEWLINE \partial{x}/\partial{t_i}=A_i(t)x,\quad x\in \mathbb R^{n},\;t=(t_1,t_2)\in \mathbb R^{2}_{\geq 1},\;i=1,2, \tag{1} NEWLINE\]NEWLINE with bounded continuously differentiable matrices \(A_i(t)\) satisfying the condition NEWLINE\[NEWLINE A_1(t)A_2(t)+\partial{A_1}/\partial{t_2}= A_2(t)A_1(t)+\partial{A_2}/\partial{t_1},\;t\in \mathbb R^{2}_{\geq 1}, NEWLINE\]NEWLINE for the complete integrability of (1) is considered. NEWLINENEWLINENEWLINEThe author introduces the notion of the lower characteristic exponent: a finite vector \(d\in \mathbb R^2\) is called a lower characteristic exponent of a nontrivial solution to (1) corresponding to a lower characteristic vector \(p\) if NEWLINE\[NEWLINE \underline{\ln}_{x}(p,d)=\mathop{\underline{\lim}}\limits_{t\to \infty}\frac{\ln{\|x(t)\|}-(p,t)-(d,\ln{t})}{\|\ln{t}\|}=0,\;\ln{t}=(\ln{t_1},\ln{t_2});NEWLINE\]NEWLINE NEWLINE\[NEWLINE \underline{\ln}_{x}(p,d+\varepsilon e_i)<0\quad \forall \varepsilon >0,\;i=1,2. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe set of all lower exponents corresponding to an inner point of the lower characteristic set is described. It is proved that for any countable set \(\{c_k\}\) of distinct numbers there exists a system (1) with infinitely differentiable bounded coefficients such that the lower characteristic vector-exponent set corresponding to the interior of the lower characteristic set of any nontrivial solution to this system consists of countably many lines \(d_1+d_2=c_k\) on the two-dimensional plane. A similar result is proved for any finite interval \(|a,b|.\)