Energy function for some maps and nonlinear oscillators

Continuous dynamical systems with different nonlinear terms can show rich dynamical characteristic, which can be presented and verified in nonlinear oscillators. Reliable algorithm for discretization is crucial to obtain suitable numerical solutions and then equivalent maps can be proposed to reproduce similar dynamics as the nonlinear oscillators. The application of Helmholtz theorem provides helpful way to define exact Hamilton energy function when the dynamical equations for a nonlinear oscillator are written in the vector form. However, reliable definition of energy function for maps is kept open from physical aspect. In this work, scale transformation for parameter and variables is applied to convert some maps into equivalent nonlinear oscillators and the definition of energy function is presented. Furthermore, a generic memristive term is introduced into the map for developing memristive map, and its equivalent memristive oscillator is obtained for clarifying the energy function and oscillatory characteristics. Then, the Hamilton energy function is dispersed to measure the shift of energy level in the map. Our scheme and results provide a criterion to check the reliability of maps derived from realistic system. When the nonlinear oscillator has the same nonlinear form for variables and local kinetics after linear transformation accompanying suitable parameters transformation, it indicates this map is discertized from dynamical systems for realistic systems rather than presenting it in mathematical form arbitrarily. It provides clues to convert periodic oscillators for nonlinear/linear circuits into low-dimensional map for inducing chaos, and digital circuits can be suggested for logic operation and signal processing.