Higher-Order Nonlinear Contraction Analysis
  Nonlinear contraction theory is a comparatively recent dynamic control system design tool based on an exact differential analysis of convergence, in essence converting a nonlinear stability problem into a linear time-varying stability problem. Contraction analysis relies on finding a suitable metric to study a generally nonlinear and time-varying system. This paper shows that the computation of the metric may be largely simplified or indeed avoided altogether by extending the exact differential analysis to the higher-order dynamics of the nonlinear system. Simple applications in economics, classical mechanics, and process control are described. 
Winfried Lohmiller and Jean-Jacques E. Slotine