On finite-time hyperbolicity (Q652013)

This interesting paper on the recent field of finite-time dynamics presents an invariant manifold theorem for finite-time hyperbolic solutions of nonautonomous ODEs. Here, finite-time hyperbolicity means an exponential dichotomy for the variational equation along this particular solution on a compact time interval. Since eigenvalues and -vectors are often unsuitable to describe hyperbolicity, a dynamic splitting of the extended state space is used to circumvent this problem. On this basis, it is shown that any solution staying clear of the so-called elliptic and degenerate parts of this splitting, is finite-time hyperbolic. An illustrating example compares the results to earlier approaches.