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Indian stock market has a long-term significant association with the behaviour of American stock exchange. The present work is an effort in this direction and the purpose of the present work is to search for any kind of nonlinearity and chaos present in the two prime Indian stock market indices viz. SENSEX and NIFTY and the prime American stock market index DOW-JONES. We have analyzed delay vector variance to identify nonlinearity in these indices. As considerable amount of nonlinearity is observed in the said stock exchange indices, there is a possibility that the time series may have chaotic nature. For that purpose, we have employed 0-1 test, largest Lyapunov exponent and recurrence plot to examine chaotic behaviour of these time series. For the present study we have considered SENSEX close data during the period from 1st January, 1990 to 31st December, 2013, NIFTY close data during the period from 3rd July, 1990 to 31st December, 2013 and DOW-JONES close data during the period from 8th May, 1969 to 31st December, 2013. Study reveals that the all these three stock market indices are nonlinear, stable, deterministic and non-chaotic in nature.