Transient chaos in fractional Bloch equations

The Bloch equation provides the fundamental description of nuclear magnetic resonance(NMR) and relaxation(T1 andT2). This equation is the basis for both NMR spectroscopy andmagnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalizationof the integer-order equation that interrelates the precession of the x, y and z componentsof magnetization with timeand space-dependent relaxation. In this paper we examinetransient chaos in a non-linear version of the Bloch equation that includes both fractionalderivatives and a model of radiation damping. Recent studies of spin turbulence in theinteger-order Bloch equation suggest that perturbations of the magnetization may involvea fading power law form of system memory, which is concisely embedded in the orderof the fractional derivative. Numerical analysis of this system shows different patternsin the stability behavior for α near 1.00. In general, when α is near 1.00, the system ischaotic, while for 0.98 ≥ α ≥ 0.94, the system shows transient chaos. As the value of αdecreases further, the duration of the transient chaos diminishes and periodic sinusoidaloscillations emerge. These results are consistent with studies of the stability of both theinteger and the fractional-order Bloch equation. They provide a more complete model ofthe dynamic behavior of the NMR system when non-linear feedback of magnetization viaradiation damping is present.© 2012 Elsevier Ltd. All rights reserved.