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In search of novel phononic crystals to effectively control the propagation of elastic waves, we propose a new single-material phononic crystal (PnC) with unit cells containing tapered resonators (TRs). The thickness of the circular taper radially decreases outward from the center. The device modulates dispersion of the wave by a local resonance mechanism and by slowly varying the group velocity of elastic waves. The TRs are layered on the top of a conventional PnC slab with a square arrangement of air holes. The band structure of the PnC is theoretically studied and a comparison is drawn between the avoided level crossings and the symmetry-protected ordinary degeneracies. In the absence of a bandgap, the zero group velocity at the band maximum restricts the waves from propagating. Moreover, the design shows anomalous dispersion phenomena such as self-collimation and bi-refringence, which are rare in conventional PnCs. We trace the origins of these phenomena by analyzing equifrequency contours associated with relevant frequencies. We show that the self-collimation effect persists even with a small variation in the angle of incidence and a perturbative hole at the center of each of the TRs. Within the classical limit, the scale invariance of the elastic wave equation makes the device useful in both the low frequency ultrasonic and the high frequency phononic regime. [ABSTRACT FROM AUTHOR]