Chaos in the fractional order nonlinear Bloch equation with delay

Abstract

The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ=0, we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α=0.8548 with subsequent period doubling that leads to chaos at α=0.9436. A periodic window is observed for the range 0.962<α<0.9858, with chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10ms delay is similar: two cycles appear at the value α=0.8532, and the transition from two to four cycles at α=0.9259. With further increases in the fractional order, period doubling continues until at α=0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α=0.8441, and α=0.8635, respectively. However, the system exhibits chaos at much lower values of α (α=0.8635). A periodic window is observed in the interval 0.897<α<0.9341, with chaos again appearing for larger values of α. In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a delay-dependent model that predicts instability in this non-linear fractional order system consistent with the experimental observations of spin turbulence.