Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

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“…Applying Hopf-Like Bifurcation theory [2][3][4] and using Proposition 1 in [7], one obtains the Hopf-Like bifurcation value…”

Section: Classical Fractional Derivativementioning

confidence: 99%

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

Bourafa¹,

Lozi

Abdelouahab

2021

J. Innov. Appl. Math. Comput. Sci

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“…where  is the internal state of the memristor. The application of a voltage-controlled model of memristor to construct a third order Wien bridge oscillator is studied in [47].…”

Section: Fracmemristor Wien Bridge Oscillator (Fwo)mentioning

confidence: 99%

“…Using the above definition of a fractional order memristor [44], a third order Wien bridge oscillator is derived based on the one proposed in [47] as shown in Fig. 1.…”

mentioning

confidence: 99%

Chaotic Dynamics of Modified Wien Bridge Oscillator with Fractional Order Memristor

Karthikeyan¹,

Duraisamy²,

Nazarimehr³

et al. 2019

RADIOENGINEERING

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In this paper a modified third order Wien bridge oscillator with fractional order memristor is proposed. Various dynamical properties of the proposed oscillator are investigated such as equilibrium points, Eigenvalues, Lyapunov exponents and bifurcation diagrams. The Lyapunov spectrum of the system for various values of fractional order is derived. Using forward and backward continuation methods of plotting bifurcation diagram, the multistability of the oscillator is investigated. The proposed oscillator is realized using Field Programmable Gate Arrays and the experiment is conducted using hardwaresoftware co-simulation.

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“…It causes difficulties applying numerical methods for solving corresponding systems of ordinary differential equations (ODE). The problem is not limited to ODE with entire derivatives, but also for dynamical systems governed by fractional derivatives [2,3,4]. Many researchers use different numerical schemes based on classical methods, e.g.…”

Section: Introductionmentioning

confidence: 99%

A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities

Lozi¹,

Pogonin²,

Pchelintsev³

2016

Chaos, Solitons & Fractals

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7 pages; 3 figures.International audienceIn this article the authors describe the method of construction of approximate chaotic solutions of dynamical model equations with quadratic nonlinearities in a general form using a new accurate numerical method. Numerous systems of chaotic dynamical systems of this type are studied in modern literature.The authors find the region of convergence of the method and offer an algorithm of construction and several criteria to check the accuracy of the approximate chaotic solutions

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Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

Cited by 9 publications

References 20 publications

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

Chaotic Dynamics of Modified Wien Bridge Oscillator with Fractional Order Memristor

A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities

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