Affine‐Contractor Approach to Handle Nonlinear Dynamical Problems in Uncertain Environment

Abstract

Nonlinear dynamical problems form backbone in several fields of science and engineering, viz., structural mechanics, fluid dynamics, control theory, robotics, seismology, circuit analysis, etc. This chapter examines nonlinear ordinary differential equations (NODEs). There exist very few literature studies regarding the solution of nonlinear dynamical problems in uncertain environment. The chapter proposes a semianalytic method, the homotopy perturbation method (HPM), based on affine-contractor for evaluating the solution bounds of nonlinear dynamical problems in uncertain environment and discusses classical interval arithmetic and its properties. It mentions the dependency problem in the case of standard interval arithmetic, and incorporates affine arithmetic, contractor, SIVIA, etc. The chapter introduces the proposed method, affine HPM and affine-contractor HPM, for calculating the enclosures of INODE. Some illustrative application problems, viz., Rayleigh equation (for unforced NODE), Van der Pol-Duffing equation (for forced NODE), and nonhomogeneous Lane-Emden equation, are discussed.