Constructing bifurcation diagram for a chaotic time-series data through a recurrent neural network model

Abstract

The Bifurcation Diagram (BD) of a given dynamical system gives the idea of the behaviour of one of the outputs of that system with different values of one of the control input parameters keeping all the other input parameters constant. It also gives the idea of iterative behaviour of the system for the particular input conditions. Plotting the BD through the mathematical models is popular in control / chaos theory domain. In this work, a methodology to construct the BD from the available time-series data using Recurrent Neural Networks o ( " ) and Chaos Theory has been developed. The ability of the developed methodology is first demonstrated on a time-series data from a mathematical dynamical system and then on a reallife complex system (submerged arc furnace). The model developed for the dynamical system has shown a high sensitivity to the training MSE level of RNN rather than to the network architecture and the recurrence level of the model, etc.

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

“…Most real-world processes as observed in [24] often have rich array of Bifurcation Diagrams and attractors. However, the Hénon map which has been selected as a model for processes has a relatively simple attractor.…”

“…1, when trained on the time-series data of a chaotic system can be used to predict either the next one step or several steps in a recursive mode [23,24]. It is also known that most RNN architectures have a capability to suppress noise.…”

Modelling and control of chaotic processes through their bifurcation diagrams generated with the help of recurrent neural network models. Part 2: An industrial study

“…The presence of high temperatures, huge electrical arcs with varying currents, multiple chemical reactions and the fact that a large mass always remains inside the furnace, compared to one which is periodically tapped, makes it a complex and difficult to model process. In [11] it has been shown that the working of SAF can be chaotic in nature, based on the calculated value of the one of the Lyapunov exponents turning out to be positive for one of the output variables evolving in time.…”