When the frequency is swept through resonance, as the driving voltage is increased, the resonance curve bends over due to the nonlinear constants of quartz. The peak of each curve is the resonance frequency.  The AT-cut, illustrated above, bends towards higher frequencies, i.e., it behaves as a hard spring (a hard spring’s stiffness increases with increasing displacement).  Some other cuts behave as soft springs; the resonance curve bends towards lower frequencies.  The locus of the maxima of the resonance curves varies as the square of the current, f/f = aI2, where a is typically in the range of 0.02/A2 to 0.2/A2; a depends on resonator design - angles of cut, overtone, plate contour, etc.
	At high drive levels, the amplitude vs. frequency curves can be triple valued functions, but only the highest and lowest values are accessible experimentally (upon increasing and decreasing voltages, respectively).  The current vs. frequency exhibits discontinuities as the driving voltage (or frequency) is increased or decreased around the resonance peak.
	D. Hammond, C. Adams, and L. Cutler, "Precision Crystal Units," Proc. 17th Annual Symposium on Frequency Control, pp. 215-232, 1963, AD-423381.
	R. L. Filler, "The Amplitude-Frequency Effect in SC-Cut Resonators," Proc. 39th Annual Symposium on Frequency Control, pp. 311-316, 1985, IEEE Catalog No. 85CH2186-5.
	H. F. Tiersten and D. S. Stevens, “The Evaluationof the Coefficients of Nonlinear Resonance for SC-cut Quartz Resonators,” Proc. 39th Annual Symposium on Frequency Control, pp. 325-332, 1985, IEEE Catalog No. 85CH2186-5.
	J.J. Gagnepain and R. Besson, “Nonlinear Effects in Piezoelectric Quartz Crystals,” in Physical Acoustics, Vol. XI, pp. 245-288, W.P. Mason & R.N. Thurston, editors, Academic Press, 1975