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We reviewed a nonlinear dynamical model in 2n-variables which has conservative nonlinear interactions defined in terms of Noether’s theorem in dynamics. The 2-variable (n = 1) conservative nonlinear model with external perturbations produced a possible explanation for problems such as the 10-year cycles of Canadian Lynx and snowshoe hair, interactions of microbes, stability and conservation law of nonlinear interacting systems. In this paper, the atto-fox (10-18-fox) problem on the LV nonlinear equation, properties of 4-variable conservative nonlinear interactions different from nonconservative nonlinear interactions are examined and emphasized. Properties of the 4-variable (n = 2) conservative interaction model and a method to construct numerical solutions are discussed by employing the 2-variable solution. The periodic times of component variables and the net periodic time defined by superposition of component variables are discussed in order to study stability of the net 4-variable system. With symmetries and conservation laws, nonlinear analyses would be useful to study microscopic and macroscopic complex systems.

Noether’s Conservation Laws and Stability in Nonlinear Conservative Interactions