Although Lyapunov exponents have been widely used to characterize the dynamics of nonlinear systems, few methods are available so far to obtain a priori bounds on their magnitudes. Recently, sufficient conditions to rule out the existence of attractors with positive Lyapunov exponents have been derived via a Lyapunov approach based on the second additive compound matrices of the system Jacobian. This paper first provides some insights into this approach by showing how the several available techniques for computing Lyapunov functions can be fruitfully applied to Lorenz and Thomas systems to derive explicit conditions on their system parameters, which ensure that there are no attractors with positive Lyapunov exponents. Then, the approach is extended to the case of nonlinear systems with a first integral of motion and its application to the memristor Chua’s circuit is discussed.