A Quantitative Method for Analyzing Symmetry Breaking of the Attractor in a Perturbed Reflection Invariant System

This paper discusses reflection invariant properties exist in some idealized mathematical systems without any external perturbations. This study is devoted to quantifying the symmetry breaking of the attractor for a three-dimensional perturbed reflection invariant system, which permits a further detailed investigation of the effect of the perturbation on the symmetry breaking. First, the attractor of a three-dimensional reflection invariant system with extremely tiny perturbation is reproduced employing the symplectic Runge–Kutta method. In the simulation of the attractor, the simulation results are validated by the tiny relative energy error in the conservative field. Subsequently, a general method for characterizing the degree of the symmetry breaking for the perturbed reflection invariant system is proposed, which is achieved by summing the quantized errors between the discrete points after the reflection transformation. This method may be beneficial for the design of complex encryption algorithms and pseudo-random number generators by controlling the perturbed strength to adjust the symmetry-breaking effect of the system. By means of the proposed quantitative method for the reflection invariant property, two important findings are reported. One is that, the strength and the action duration of the perturbation are the key factors affecting the reflection invariant property of the considered system. Another is that the symmetry breaking of the attractor for the perturbed reflection invariant system is irreversible.