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The cosmological evolution in Nonlinear Born-Infeld(hereafter NLBI) scalarfield theory with negative potentials was investigated. The cosmologicalsolutions in some important evolutive epoches were obtained. The differentevolutional behaviors between NLBI and linear(canonical) scalar field theoryhave been presented. A notable characteristic is that NLBI scalar field behavesas ordinary matter nearly the singularity while the linear scalar field behavesas "stiff" matter. We find that in order to accommodate current observationalaccelerating expanding universe the value of potential parameters $|m|$ and$|V_0|$ must have an {\it upper bound}. We compare different cosmologicalevolutions for different potential parameters $m, V_0$.Comment: 18 pages, 18 figures, some references added, revised version for  Int.J.Mod.Phys.A, appeared in Int.J.Mod.Phys.

COSMOLOGY IN NONLINEAR BORN–INFELD SCALAR FIELD THEORY WITH NEGATIVE POTENTIALS