Beautiful chaotic patterns generated using simple untrained recurrent neural networks under harmonic excitation

Abstract

This paper focuses on the intrinsic chaotic behavior of recurrent neural networks under harmonic excitation. The chaotic behaviors of some untrained two-neuron RNN with strange attractors are studied in detail. Taking the input as an additional dimension, the intrinsic manifold shaped by an RNN is visualized using 3-D phase portraits. A series of isomorphic topological structures are generated to provide different aspects of the intrinsic manifold. Local fractal dimension is introduced to visualize the repetitive folding of attractors, which exhibits a spectrum of fractal dimensions concentrating around 1, 1.5 and 2. By using maximal Lyapunov exponent (MLE), our results show that when tuning the amplitude or frequency of the harmonic input, periodic and chaotic motions in the phase space emerge alternatingly. For a certain RNN, bifurcation with the topological transformation of the manifold can be observed when MLEs approach zero. Algorithm 1 is designed to search for various topological structures with potential chaotic patterns from massive randomly generated RNNs. Beautiful chaotic patterns are generated and presented in the form of a gallery appended in the end. This paper provides a novel prospect to generate topological structures with beautiful chaotic patterns using RNNs.