I. Introduction
Characterizing and analyzing the long-term behavior of solutions of a dynamical system is a classical topic in the study of differential equations. The question is closely related with assessment of (global) stability and instability properties of solutions, and many complementary approaches exist to these days, including Lyapunov analysis techniques [1], contraction analysis [2], monotonicity-based considerations [3], density functions and dual Lyapunov analysis [4], to name a few. For a finite dimensional time-invariant nonlinear system defined by sufficiently smooth equations , contraction analysis guarantees convergence of solutions towards one another by imposing a Linear Matrix Inequality constraint fulfilled by the Jacobian of the vector field , defining the system’s equations. A related Jacobian-based approach was introduced by James Muldowney in the seminal paper [5], where conditions are provided based on the second additive compound of the Jacobian to rule out existence of periodic solutions in nonlinear systems of differential equations. Roughly speaking Muldowney’s conditions rule out the possibility of periodic solutions on the ground of contraction assumptions prescribed on arbitrary infinitesimal bidimensional surfaces. A self-contained introduction to the topic of additive and multiplicative compound matrices was recently proposed in [6]. Indeed such matrices have been widely adopted in the qualitative study of attractors, with special focus on their geometry and dimensionality, as documented in the monograph [7]. In the same spirit, compound matrices of fractional order were recently introduced in [8]. Closer to this letter’s contribution, Muldowney’s results were taken up and extended in [9] to rule out periodic and almost periodic solutions (more precisely existence of invariant toruses of arbitrary dimension) and applied to the qualitative study of biochemical networks. Conditions were specifically formulated using piecewise linear Lyapunov functions, defined with respect to an auxiliary variational equation involving the second-additive compound matrix of the Jacobian: \begin{align*} \dot {x}=&f(x) \\ \dot { \delta }^{[{2}]}=&\frac { \partial f}{ \partial x}^{(2) } (x) \delta ^{[{2}]},\tag{1}\end{align*} whose precise definition will be later detailed. This auxiliary equation also serves as the starting point of our paper. Our main contribution is to show how the same Lyapunov conditions proposed in [9], and in fact in a slightly relaxed form, are also suitable to rule out existence of attractors with positive Lyapunov exponents. In this respect, they complement the results of [9] by ruling out another significant class of non-convergent dynamical behaviors and have a complementary scope of application to numerical algorithms for the computation of maximal Lyapunov exponents, (see [10] and references therein).