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The goal of this paper is to catalog the chaotic and Pattern Recognition (PR) properties of a network of Logistic Neurons (LNs). Over the last few years, the field of Chaotic Neural Networks (CNNs) has been extensively studied because of their potential applications in PR, Associative Memory (AM), optimization, multi-value content addressing and image processing. The research in chaos theory has thus expanded to report numerous neural models that, by virtue of their inter-connections, yield chaotic behavior. Recently, the Adachi Neural Network (AdNN) and its variants have been shown to yield an entire spectrum of properties including chaotic, quasi-chaotic, PR and AM as its/their parameters change. To simplify the AdNN model and to also investigate the design philosophy of the CNN model, in this paper, we consider the consequences of networking a set of LNs, each of which is founded on principles of the Logistic map. By appropriately defining the input/output characteristics of a fully connected network of LNs, and by defining their set of weights and output functions, we have succeeded in designing a Logistic Neural Network (LNN). Although the LNN is much simpler than other CNNs such as the AdNN, it possesses some of those properties mentioned above. The chaotic properties of a single-neuron have been formally proven using the theory of Lyapunov analysis and by examining its Jacobian matrix. As far as we know, the results presented here, that the LNN can also demonstrate both AM and PR properties, are unreported, and we submit that it can, hopefully, lead to a new method of PR and AM.