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 “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.