This study considers linearisation with regard to a particular trajectory rather than a fixed point, introducing a novel method for linearising nonlinear signals in dynamical systems. The method is implemented on approximating a non-linear system by a linear time-varying system in the vicinity of a reference trajectory. The theoretical basis is established through a systematic mathematical derivation culminating in a proof of the linearization theorem. This approach will improve accuracy, allow applicability to new applications and experimental findings, increase flexibility, and enhance the possibility of specific application predictions in statistics for dielectric breakdown and operations for RRAM. The results confirm the applicability of the proposed approach for approximate description of non-linear system behavior and the prospects of using the method for stability analysis, control design, and understanding the behavior of complex systems. The theoretical and mathematical results presented in this paper provide a solid foundation for the application of the proposed linearization method in the analysis and control of complex dynamical systems.