<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e142" altimg="si7.svg"><mml:mi>k</mml:mi></mml:math>-contraction: Theory and applications

A dynamical system is called contractive if any two solutions approach one another at an exponential rate. More precisely, the dynamics contracts lines This property implies highly ordered asymptotic behavior including entrainment to time-varying periodic vector fields and, in particular, global stability for time-invariant fields. Contraction theory has found numerous applications systems and control because there exist easy verify sufficient conditions, based on matrix measures, guaranteeing contraction. We provide a geometric generalization of contraction k-contraction. k-contractive k-parallelotopes For k=1 this reduces standard describe conditions k-contraction measure kth additive compound Jacobian field. also seminal work Muldowney Li, that can be interpreted framework 2-contraction, theory.