Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller

Abstract

Nuclear magnetic resonance is commonly used in engineering, chemistry, and medicine to study complex materials by relating magnetization to applied radiofrequency, gradient, and static magnetic fields through the Bloch equation. Chaos in the Bloch equation is a key factor in many important applications. This study focuses on the study of the Bloch equation under the influence of the Caputo fractional derivative, both with and without delay, and explores the underlying chaos using a sliding mode controller. The controller’s effectiveness is observed under uncertainty and external disturbances for both commensurate and incommensurate systems, and theoretical aspects such as the existence and uniqueness of solutions and the stability of the controlled system are examined. Lyapunov exponents are calculated for various fractional derivatives to demonstrate the presence of chaos in the system, and numerical simulations are used to verify theoretical assertions.