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In the present work we investigate the stability of the k-essence models allowing upto quadratic terms of the kinetic energy. The system of field equations is written as an autonomous system in terms of dimensionless variables and the stability criteria of the equilibria have been extensively investigated. The results strongly indicate that cosmologically consistent models dynamically evolve towards the quintessence model, a stable solution with a canonical form of the dark energy. PACS numbers: 98.80.-k; 95.36.+x

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