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

Abstract

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 “hyper-spherical” since the new variables are a “radial” norm having physical units of inverse-of-time and a normalized “state vector” 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 “attracting subspaces” (one at a time). Correspondingly, the physical space can be split into “attractiveness regions.” 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

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

“…= where g(x) = − dU(x)/dx being U(x) a potential energy divided by the mass of the particle, and ξ a damping friction coefficient per unit of mass. Let us consider the case of the symmetric two-well potential already studied in [26]. The potential energy is U x x c 1 The computations were done for Δ = 5, c = 1 and ξ = 10, the same values employed in [26].…”

“…Let us consider the case of the symmetric two-well potential already studied in [26]. The potential energy is U x x c 1 The computations were done for Δ = 5, c = 1 and ξ = 10, the same values employed in [26]. Panel (a) of figure 2 shows some trajectories in the phase-space, while panels (b) and (c) show the percentage relative error that bears on the rate-field components approximated by equation (3) with the parameters obtained from the specific route adopted here.…”

“…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.…”