On the simultaneous instability of exponential exponents of linear differential systems (Q2387984)

The authors study the upper exponential exponent \[ \nabla(A) = \lim_{\Theta \to 1+} \limsup_{k \to \infty} \Theta^{-k} \sum_{i=1}^k \ln \| X_A(\Theta^i, \Theta^{i-1})\| , \] where \(X_A(t,\tau)\) denotes the principal solution matrix of the linear system \( \dot x = A(t)x . \) The coefficient \(\nabla(A)\) is negative iff a small nonlinear perturbation of the linear system is exponentially stable. In the definition of the lower exponential exponent \(\Delta(A)\) the \(\limsup\) is replaced by \(\liminf\). The author studies both exponents under certain perturbations of the linear system.