Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law

Universal embedding of autonomous dynamical systems into a Lotka-Volterra-like format

“…If some interesting feature emerges at such an abstract level of representation, it might be interesting to inspect how that feature manifests in the physical space. For instance, under this perspective the analysis of the quadratic format enabled to let emerge a definition of the slow manifolds observed in mass-action chemical kinetics [23] and to devise computational routes for their identification [24,25], to discover the existence of mutually orthogonal attracting subspaces in what was termed the 'hyper-spherical representation' of the dynamics [24,26], and even the existence of an intrinsic timing for Markov jump processes represented in the probability space [27].…”

“…Just by analogy, but without any real physical correspondence, these equations can be collectively seen as a Lotka-Volterra-like structure devoid of linear terms. The structure equation (2) was obtained for mass-action chemical kinetics [4,5,7], for some classes of phase-space dynamics under the extension to complex-valued h Q [26] and, recently, even for classical master equation dynamics in which the embedding into equation (2) implies the increase from linear to quadratic ODEs [27]. In all these situations, the change from x to h was devised ad hoc on the basis of the specific type of original ODEs.…”

“…Then, as 1 The V QQ¢ and the  V QQ¢ proved useful to construct suitable scalar fields to guide the localization of possible slow manifolds in the physical space [24,25]. Furthermore, the empirical inspection of the high-order time derivatives of the V t QQ ( ) ¢ has led to conjecture the existence of an intrinsic 'timing function' as long as the trajectory x(t) is inside an 'attractiveness region' of the physical space [24,27]. Concerning the hyper-spherical representation, let n dim be the total number of h Q variables.…”