Non-normality and non-monotonic dynamics in complex reaction networks

Abstract

Complex chemical reaction networks, which underlie many industrial and biological processes, often exhibit non-monotonic changes in chemical species concentrations. Such non-monotonic dynamics are in principle possible even in a linear model if the matrix defining the model is non-normal, as characterized by a necessarily non-orthogonal set of eigenvectors. However, the extent to which non-normality is responsible for non-monotonic behavior remains an open question. Here, using a master equation to model the reaction dynamics, we derive a general condition for observing non-monotonic dynamics of individual species, establishing that non-normality promotes non-monotonicity but is not a requirement for it. In contrast, we show that non-normality is a requirement for non-monotonic dynamics to be observed in the Rényi entropy. Using hydrogen combustion as an example application, we demonstrate that non-monotonic dynamics under experimental conditions are supported by a linear chain of connected components, at variance with the dominance of a single giant component observed in typical random reaction networks. The exact linearity of the master equation enables development of a rigorous theory and simulations for dynamical networks of unprecedented sizes (approaching 10 5 dynamical variables, even for a network of only 20 reactions and involving less than 100 atoms). Our conclusions are expected to hold for other combustion processes, and the general theory we develop is applicable to all chemical reaction networks, including biological ones.

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality

“…Thus they have not been able to evidence the nonlinear behavior and in particular the restriction of the synchronization basin. Furthermore the affirmations made in [1] are in neat contrast with ones available in the literature, in particular with a very recent contribution of the first two authors of the Comment note [13] agreeing with our statements (pg. 1 Introduction section) "Even when a fixed point is linearly stable in a nonlinear system described by ordinary differential equations, if the corresponding Jacobian matrix is non-normal, a small but finite perturbation can transiently grow beyond the validity of the linear approximation and enter into the nonlinear regime, preventing the perturbation from decaying to zero."…”

“…In fact, in the non-normal dynamics regime, a finite perturbation regarding a stable state can undergo a transient instability [ 12 ], which, because of the nonlinearities, could never be reabsorbed [ 13 , 14 ]. The effect of non-normality in dynamical systems has been studied in several contexts, such as hydrodynamics [ 19 ], ecosystems stability [ 20 ], pattern formation [ 21 ], chemical reactions [ 22 ], etc. However, it is only recently that the ubiquity of non-normal networks and the related dynamics have been put to the fore [ 13 , 14 , 15 , 16 , 17 , 18 ].…”

Network structural origin of instabilities in large complex systems

“…2, the observable is the 2-norm of the system state vector. This choice is standard in the literature [see, e.g., ( 5 , 14 – 16 , 19 , 21 , 25 , 26 )], as it permits a convenient characterization through eigenvalues, but for certain network processes, different measures of deviations may be more natural and would lead to different definitions of reactivity [see, e.g., the 1-norm used in ( 42 – 44 ) and the absolute value of a one-dimensional projection used in ( 17 )].…”

Non-Normal Interactions Create Socio-Economic Bubbles

“…But of course, as for general dynamics following non-normal operators, not all perturbations go through a non-monotonous transient amplification. The realized trajectory of the transient very much depends on the projection of the perturbations onto the pseudo-eigenvectors [7,9,57]. Not all large Kreiss constant values should thus lead to a bubble, as the market dynamics is more complex and cannot just be reduced to one source of influence.…”