Physical approach of higher order nonlinear maps and neurons

Nonlinear oscillators derived from nonlinear circuits have clear physical significance, and bifurcation analysis provides helpful guidance for further dynamical control and energy regulation. Most mathematical map models have found wide applications in digital signal processing and nonlinear dynamical analysis; however, the assumptions of high order nonlinear terms in the theoretical models lack clear physical interpretations. Euler forward algorithm provides accessible bridge between nonlinear oscillators and maps by applying linear transformation on the variables and intrinsic parameters, and the time step is incorporated as one intrinsic parameter so that the oscillator and its approximate equivalent map model can show similar dynamical characteristic. In particular, energy definition for the obtained map model becomes important when discrete memristor with high order term is introduced, e.g. a cubic term emerges in the map. In this work, a generalized continuous and discrete transformation framework suitable for high order nonlinear maps is proposed. The present approach extends the conventional continuousization theory from quadratic nonlinear systems to cubic and more general high order polynomial maps. By introducing appropriate scaling transformations and time step normalization, discrete high order maps can be approximately transformed into equivalent continuous nonlinear oscillators while preserving their principal dynamical properties, and the corresponding Hamilton energy functions are further constructed. Based on Helmholtz’s theorem, the Hamilton energy functions are derived through the decomposition of the vector field into rotational and gradient components, where the physical oscillator is represented in vector form. Furthermore, a unified transformation form for arbitrary high order polynomial maps is established. The results show that high order nonlinear maps naturally correspond to continuous systems with multi-well potential-energy structures, which may exhibit multistability, and complex oscillatory behaviors. The proposed framework provides an approximate continuous representation for high order discrete maps while preserving their principal dynamical characteristics. The proposed framework not only provides a unified theoretical approach for constructing Hamilton energy functions of high order nonlinear maps, but also offers new insights into the investigation of complex dynamics in memristive systems, nonlinear circuits with multistability, and high order nonlinear systems.