Chaos in the incommensurate fractional order system and circuit simulations

Chaos in the incommensurate fractional order system and circuit simulations In this paper, the dynamics of an autonomous three dimensional system with three nonlinearity quadratic terms is investigated. With the help of stability analysis of equilibrium points, the Lyapunov exponent, and the phase portraits we study the dynamical behavior of the fractional order system. We find that system can display double scroll and double-four wing chaotic attractor. The commensurate order has been investigated and we find that the necessary condition for chaos to appear is 0.84 <α ≤ 1. For the incommensurate order, we show that the instability measure of the equilibrium points for all the saddle points of index 2 must be non-negative for a system with five equilibrium points to be chaotic. Numerical simulations and the analog simulations are carried out in Multisim. Keywords Chaos · Fractional order system · Commensurate order · Incommensurate order · Circuit realization 1 Introduction erature is exponentially growing. The Lorenz equation was derived from the Oberbeck–Boussinesq approximation, and The topic of chaos is gaining more attention, and its appli- describes the fluid circulation in a shallow layer of fluid [7]. cation field becomes more and more important. Nowadays, That system has only two nonlinear quadratic terms and can we found the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Dynamics and Control Springer Journals

Chaos in the incommensurate fractional order system and circuit simulations

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Engineering; Vibration, Dynamical Systems, Control; Control; Complexity
ISSN
2195-268X
eISSN
2195-2698
D.O.I.
10.1007/s40435-018-0442-y
Publisher site
See Article on Publisher Site

Abstract

In this paper, the dynamics of an autonomous three dimensional system with three nonlinearity quadratic terms is investigated. With the help of stability analysis of equilibrium points, the Lyapunov exponent, and the phase portraits we study the dynamical behavior of the fractional order system. We find that system can display double scroll and double-four wing chaotic attractor. The commensurate order has been investigated and we find that the necessary condition for chaos to appear is 0.84 <α ≤ 1. For the incommensurate order, we show that the instability measure of the equilibrium points for all the saddle points of index 2 must be non-negative for a system with five equilibrium points to be chaotic. Numerical simulations and the analog simulations are carried out in Multisim. Keywords Chaos · Fractional order system · Commensurate order · Incommensurate order · Circuit realization 1 Introduction erature is exponentially growing. The Lorenz equation was derived from the Oberbeck–Boussinesq approximation, and The topic of chaos is gaining more attention, and its appli- describes the fluid circulation in a shallow layer of fluid [7]. cation field becomes more and more important. Nowadays, That system has only two nonlinear quadratic terms and can we found the

Journal

International Journal of Dynamics and ControlSpringer Journals

Published: May 29, 2018

References

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