This study describes the mathematical representation of nonlinear viscoelastic coupling involving rolling elements, which are crucial components in various models of deformable bodies systems. It examines the coupling of different deformable bodies interconected with layers of different viscoelastic nonlinear properties that can be integratet with inerters. By employing the same mathematical formalism and mapping, these diverse systems are analysed. The study reveals that similarities in dynamic behaviours can be observed and explained due to the mathematical analogy found in the time domain of solutions for these different natural phenomena. The nonlinear dynamics in these systems, influenced by various parameters, can result in extreme phenomena ranging from limit cycles and resonant jumps to chaotic attractors. The analysis focused on examining the mutual influence and transition between different nonlinear modes of system dynamics. Despite the complexity arising from multiple elements, parameters, and nonlinearity, identical synchronisation (IS) can be identified and utilised within these systems. Due to the significant number of contributing parameters, a multi-parameter analysis was utilised to investigate how system components synchronise, mutually and with external periodic forces. It has been concluded that despite nonlinearity and complexity, there is a coupling parameter relationship that can ensure identical synchronisation in the system.