The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds

Abstract

The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton-Maclaurin expansion is proposed and applied to those difference equations including no a small parameter.

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

“…Therefore, for these kinds of equations, we only need to solve the usual ordinary differential equation (9). This kind of fractional equation can be used to describe anomalous diffusion based on a continuous time random walk model with incorporated memory effects [22].…”

Exact dynamical behavior for a dual Kaup–Boussinesq system by symmetry reduction and coupled trial equations method

“…These methods have been extensively developed and applied to a lot of nonlinear problems [26][27][28][29][30][31][32][33][34][35][36][37]. Liu's renormalization method and its applications can be found in [38][39][40][41][42][43][44][45]. Some new methods and results on fractional differential equations can be seen in [46][47][48][49][50][51][52][53][54][55][56][57] and the references therein.…”