This paper demonstrates effective capabilities of a relatively simple deep convolutional neural network in estimating the Lyapunov exponent and detecting chaotic signals. A major difference between this study and existing research is that our networks take raw data as input, automatically generate a selection of informative features, make a direct estimation of the Lyapunov exponent and form a decision whether a chaotic signal is present. The proposed method does not require attractor reconstruction. It also can be used for processing relatively short signals - in the experiment described here the signal length is 1024 sequence elements. The study has demonstrated that deep convolutional neural networks are effective in applications involving chaotic signals (down to narrowband or broadband stochastic processes), as well as distinct patterns, and can, therefore, be used for a number of signal processing tasks.