### Abstract

Universal Learning Networks (ULNs) are proposed and their application to chaos control is discussed. ULNs provide a generalized framework to model and control complex systems. They consist of a number of inter-connected nodes where the nodes may have any continuously differentiable nonlinear functions in them and each pair of nodes can be connected by multiple branches with arbitrary time delays. Therefore, physical systems, which can be described by differential or difference equations and also their controllers, can be modeled in a unified way, and so ULNs may form a super set of neural networks and fuzzy neural networks. In order to optimize the ULNs, a generalized learning algorithm is derived, in which both the first order derivatives (gradients) and the higher order derivatives are incorporated. The derivatives are calculated by using forward or backward propagation schemes. These algorithms for calculating the derivatives are extended versions of Back Propagation Through Time (BPTT) and Real Time Recurrent Learning (RTRL) of Williams in the sense that generalized node functions, generalized network connections with multi-branch of arbitrary time delays, generalized criterion functions and higher order derivatives can be deal with. As an application of ULNs, a chaos control method using maximum Lyapunov exponent of ULNs is proposed. Maximum Lyapunov exponent of ULNs can be formulated by using higher order derivatives of ULNs, and the parameters of ULNs can be adjusted so that the maximum Lyapunov exponent approaches the target value. From the simulation results, it has been shown that a fully connected ULN with three nodes is able to display chaotic behaviors.

Original language | English |
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Pages (from-to) | 239-253 |

Number of pages | 15 |

Journal | Neural Networks |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2000 Mar |

### Keywords

- Chaos
- Higher order derivatives calculation
- Lyapunov exponent
- Neural networks
- Universal learning networks

### ASJC Scopus subject areas

- Cognitive Neuroscience
- Artificial Intelligence

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## Cite this

*Neural Networks*,

*13*(2), 239-253. https://doi.org/10.1016/S0893-6080(99)00100-8