This question is similar to the following, but I have expanded the question moderately:
Nonlinearities arising from linear equations
The Bloch equations are described by the following vector equation (ignoring relaxation):
$$
\frac{d}{dt}\mathbf{M}(t) = \mathbf{M}(t) \times \gamma \mathbf{B}(t)
$$
It is frequently stated that the Bloch equations are non-linear.
For example,
In Principles of Magnetic Resonance Imaging – A Signal Processing Perspective by Liang and Lauterbur (pg. 89), it is stated without elaboration that :
The linear system assumption is not valid for a nuclear spin system during excitation.
Additionally, in Principles of Magnetic Resonance by Nishimura (pg. 124), it states :
"… the nonlinear behavior of the spin system becomes appreciable."
Lastly, in Magnetic Resonance Imaging – Physical Principles and Sequence Design by Brown et al. (pg. 661), "Bloch equation nonlinearities" are listed as a reason for possible measurement error.
The equation listed above can be reformulated in the following manner:
$$
\frac{d}{dt}\begin{bmatrix} M_x(t)\\M_y(t)\\M_z(t) \end{bmatrix}= \begin{bmatrix} 0 & \gamma B_z(t) & -\gamma B_y(t) \\ -\gamma B_z(t) & 0 & \gamma B_x(t) \\ \gamma B_y(t) & -\gamma B_x(t) & 0 \end{bmatrix} \begin{bmatrix} M_x(t)\\M_y(t)\\M_z(t) \end{bmatrix}
$$
This seems like a linear differential equation to me. What do people mean when they refer to the Bloch equations as non-linear?
Best Answer
The non linearities arise when you consider the feedback loop. The magnetic moment can generate a field of its own. $\mathbf{B}$ will no longer be the externally applied field, but will rather depend on $\mathbf{M}$ hence the nonlinearity. These nonlinearities give rise to new behaviors such as synchronization and chaotic motion.
Check out this article: Abergel D. "Chaotic solutions of the feedback driven Bloch equations." Phys Lett A 2002;302:17–22.
Hope this helps and tell me if you find some mistakes.