  This paper derives new results for the analysis of nonlinear systems by
extending contraction theory in the framework of vector distances. A new tool,
vector contraction analysis utilizing a notion of the vector-valued norm which
evidently induces a vector distance between any pair of trajectories of the
system, offers an amenable framework as each component of vector-valued norm
function satisfies fewer strict conditions as that of standard contraction
analysis. Particularly, every element of vector-valued norm derivative need not
be strictly negative definite for convergence of any pair of trajectories of
the system. Moreover, vector-valued norm derivative satisfies a component-wise
inequality employing some comparison system. In fact, the convergence analysis
is performed by comparing the relative distances between any pair of the
trajectories of the original nonlinear system and the comparison system.
Comparison results are derived by utilizing the concepts of quasi-monotonicity
property of the function and vector differential inequalities. Moreover, the
results are also derived in the framework of the cone ordering instead of
utilizing component-wise inequalities between vectors. In addition, the
proposed framework is illustrated by examples.
