On the piecewise-spectral homotopy analysis method and its convergence: solution of hyperchaotic Lü system

Abstract — In this paper, a novel modification of the spectral-homotopy analysis method (SHAM) technique for solving highly nonlinear initial value problems that model systems with chaotic and hyper-chaotic behaviour is presented. The proposed method is based on implementing the SHAM on a sequence of multiple intervals thereby increasing it’s radius of convergence to yield highly accurate method which is referred to as the piece-wise spectral homotopy analysis method (PSHAM). We investigate the application of the PSHAM to the Lü system [20] which is well known to display periodic, chaotic and hyper-chaotic behaviour under carefully selected values of it’s governing parameters. Existence and uniqueness of solution of SHAM that give a guarantee of convergence of SHAM, has been discussed in details. Comparisons are made between PSHAM generated results and results from literature and Runge–Kutta generated results and good agreement is observed.