Attracting subspaces in a hyper-spherical representation of autonomous dynamical systems

In this work, we focus on the possibility to recast the ordinary differential equations (ODEs) governing the evolution of deterministic autonomous dynamical systems (conservative or damped and generally non-linear) into a parameter-free universal format. We term such a representation \u201chyper-spherical\u201d since the new variables are a \u201cradial\u201d norm having physical units of inverse-of-time and a normalized \u201cstate vector\u201d with (possibly complex-valued) dimensionless components. Here we prove that while the system evolves in its physical space, the mirrored evolution in the hyper-spherical space is such that the state vector moves monotonically towards fixed \u201cattracting subspaces\u201d (one at a time). Correspondingly, the physical space can be split into \u201cattractiveness regions.\u201d We present the general concepts and provide an example of how such a transformation of ODEs can be achieved for a class of mechanical-like systems where the physical variables are a set of configurational degrees of freedom and the associated velocities in a phase-space representation. A one-dimensional case model (motion in a bi-stable potential) is adopted to illustrate the procedure

Approaches to dimensionality reduction and model simplification of dynamics in the chemical context

Much of the effort in the modern chemical and physical sciences is devoted to the study of complex dynamical phenomena. Such a study is often hampered by the considerable complexity (i.e., the high dimensionality) exhibited by the systems of interest. In this research project, of theoretical and methodological character, we explore some facets of the topics of model reduction and simplification of complex dynamics, both deterministic and stochastic. In particular, in the first part of the work (chs. 2-5), we focus on deterministic systems. In chapter 2, starting from the findings of two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013) and P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234102 (2013)] we introduce the concept of "canonical format" of the evolution law for mass-action-based chemical kinetics, and show that the study of such a type of formats could lead to the discovery of new interesting features and to a rationalization of already well-known ones. Specifically, we unveil the existence of "attracting subspaces" in an abstract "hyper-spherical" representation of the dynamics of a reacting system. In chapter 3, based on the theory devised in ch. 2, we develop an algorithm (implemented in the companion software DRIMAK, acronym of Dimensional Reduction for Isothermal Mass-Action Kinetics) aimed at detecting the neighborhood of the Slow Manifold, which is a hypersurface, in the concentration space, in the proximity of which the slow evolution takes place. The detection of the Slow Manifold for a reacting system is a potential key-step to elaborate dimensionality reduction strategies. In chapter 4 we extend the theory to open reaction networks, i.e., reaction networks with one or more reactants continuously injected in the reaction environment. Finally, in chapter 5 we further generalize the theory to general phase-space dynamics, possibly damped. The second part of the work (chs. 6-8) is devoted to stochastic systems. In chapter 6 we move the first steps towards the model reduction of stochastic chemical kinetics. Specifically, we show the existence of geometric structures (in the space of the number of molecules of each species) analogous to the Slow Manifold in the macroscopic counterpart. Still in the context of stochastic chemical kinetics, in chapter 7 we make a critical study of two common continuous approximations of the chemical master equation and of the associated Gillespie's stochastic simulation algorithm; namely, we investigate on the physical reliability of the chemical Fokker-Planck and chemical Langevin equations. In particular, we prove that both the approximations suffer from nonphysical probability currents at equilibrium, even for fully reversible and detailed-balanced chemical reaction networks. Finally, in chapter 8 we focus on general overdamped fluctuating systems, which, apart from very simple and low-dimensional cases, are often mathematically intractable. In this context, given the well-known difficulties for the mathematical treatment of such systems, we aim only at achieving a partial, but easily computable, information. Namely, we devise a set of mathematical time-dependent bounds for key-quantities describing the systems of interest