On QUAD, Lipschitz, and Contracting Vector Fields for Consensus and Synchronization of Networks

In this paper, a relationship is discussed between three common assumptions made in the literature to prove local or global asymptotic stability of the synchronization manifold in networks of coupled nonlinear dynamical systems. In such networks, each node, when uncoupled, is described by a nonlinear ordinary differential equation of the form <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">x=f(x,t)</tex> </formula> . In this paper, we establish links between the QUAD condition on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$f(x,t)$</tex></formula> , i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">(x-y)T[f(x,t)-f(y,t)] - (x-y)T Δ (x-y) ≤</tex></formula> <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">-ω(x-y)T(x-y)</tex> </formula> for some arbitrary <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">Δ</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">ω</tex> </formula> , and contraction theory. We then investigate the relationship between the assumption of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">f</tex> </formula> being Lipschitz and the QUAD condition. We show the usefulness of the links highlighted in this paper to obtain proofs of asymptotic synchronization in networks of identical nonlinear oscillators and illustrate the results via numerical simulations on some representative examples.