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Experimental and theoretical studies of the dynamic magnetization in sweptmagnetic fields of the orthorhombic SrY$_2$O$_4$ single-crystals doped with theDy$^{3+}$ Kramers ions (0.01 and 0.5 at.%) with natural abundances of even andodd Dy isotopes are presented. Impurity ions substitute for Y$^{3+}$ ions attwo nonequivalent crystallographic sites with the same local $C_s$ symmetry butstrongly different crystal fields. Well pronounced double-loop hysteresis isobserved at temperatures 2, 4, 5 and 6 K for sweeping rates of 5 and 1 mT/s.The microscopic model of spectral, magnetic and kinetic properties of Dy$^{3+}$ions is developed based on the results of EPR, site selective optical spectraand magnetic relaxation measurements. The derived approach to the dynamicmagnetization in the sweeping field based on the numerical solution ofgeneralized master equations with time-dependent transition probabilitiesinduced by the electron-phonon interaction, quantum tunneling andcross-relaxation allowed us to reproduce successfully the evolution of thehysteresis loop shape with temperature, sweeping rate and concentration ofparamagnetic ions.

Double-loop hysteresis of multisite dilute Sr(Y$_{1-x}$Dy$_x$)$_2$O$_4$  single crystal Kramers paramagnets: electron-phonon interaction, quantum  tunneling and cross-relaxation