limit cycles

This paper introduces a notion of non-oscillation, proposes a constructive method for its robust verification, and studies its application to biological interaction networks. The paper starts by revisiting Muldowney's result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. It then shows that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. The focus then turns ton nonlinear equations arising in biological interaction networks with general kinetics, the paper shows that the dynamics of the variational system can be embedded in a linear differential inclusion. This leads to algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, the paper applies the new techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.

This paper addresses the problem of giving conditions for transcriptional systems to be globally entrained to external periodic inputs. By using contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all solutions converge to fixed limit cycles. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems.

Contraction theory provides an elegant way of analyzing the behaviors of systems subject to external inputs. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that all trajectories converge to a unique solution. This property is especially interesting when forcing functions are periodic (globally attracting limit cycles result), as well as in the context of establishing synchronization results. The present paper provides a self-contained introduction to some basic results, with a focus on contractions with respect to non-Euclidean metrics.

Feedback loops play an important role in determining the dynamics of biological networks. In order to study the role of negative feedback loops, this paper introduces the notion of "distance to positive feedback (PF-distance)" which in essence captures the number of "independent" negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks it is shown that PF-distance has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same numbers of nodes and connectivity.