Transient Chaos in Fractional Bloch Equations

Abstract

The Bloch equation provides the fundamental description of nuclear magnetic resonance (NMR) and relaxation (T-1 and T-2). This equation is the basis for both NMR spectroscopy and magnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalization of the integer-order equation that interrelates the precession of the x, y and z components of magnetization with time- and space-dependent relaxation. In this paper we examine transient chaos in a non-linear version of the Bloch equation that includes both fractional derivatives and a model of radiation damping. Recent studies of spin turbulence in the integer-order Bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory, which is concisely embedded in the order of the fractional derivative. Numerical analysis of this system shows different patterns in the stability behavior for alpha near 1.00. In general, when alpha is near 1.00, the system is chaotic, while for 0.98 >= alpha >= 0.94, the system shows transient chaos. As the value of alpha decreases further, the duration of the transient chaos diminishes and periodic sinusoidal oscillations emerge. These results are consistent with studies of the stability of both the integer and the fractional-order Bloch equation. They provide a more complete model of the dynamic behavior of the NMR system when non-linear feedback of magnetization via radiation damping is present. (C) 2012 Elsevier Ltd. All rights reserved.