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Forecasting models for chaotic fractional-order oscillators using neural networks

Bingi Kishore, Prusty B Rajanarayan

International Journal of Applied Mathematics and Computer Science

2021

Vol. 31, no. 3

387--398

This paper proposes novel forecasting models for fractional-order chaotic oscillators, such as Duffing’s, Van der Pol’s, Tamaševičius’s and Chua’s, using feedforward neural networks. The models predict a change in the state values which bears a weighted relationship with the oscillator states. Such an arrangement is a suitable candidate model for out-of-sample forecasting of system states. The proposed neural network-assisted weighted model is applied to the above oscillators. The improved out-of-sample forecasting results of the proposed modeling strategy compared with the literature are comprehensively analyzed. The proposed models corresponding to the optimal weights result in the least mean square error (MSE) for all the system states. Further, the MSE for the proposed model is less in most of the oscillators compared with the one reported in the literature. The proposed prediction model’s out-of-sample forecasting plots show the best tracking ability to approximate future state values.

Stability analysis of interconnected discrete-time fractional-order LTI state-space systems

Grzymkowski Łukasz, Trofimowicz Damian, Stefański Tomasz P.

International Journal of Applied Mathematics and Computer Science

2020

Vol. 30, no. 4

649--658

In this paper, a stability analysis of interconnected discrete-time fractional-order (FO) linear time-invariant (LTI) state-space systems is presented. A new system is formed by interconnecting given FO systems using cascade, feedback, parallel interconnections. The stability requirement for such a system is that all zeros of a non-polynomial characteristic equation must be within the unit circle on the complex z-plane. The obtained theoretical results lead to a numerical test for stability evaluation of interconnected FO systems. It is based on modern root-finding techniques on the complex plane employing triangulation of the unit circle and Cauchy’s argument principle. The developed numerical test is simple, intuitive and can be applied to a variety of systems. Furthermore, because it evaluates the function related to the characteristic equation on the complex plane, it does not require computation of state-matrix eigenvalues. The obtained numerical results confirm the efficiency of the developed test for the stability analysis of interconnected discrete-time FO LTI state-space systems.