Dynamics and Control of Nonlinear Variable Order Oscillators

Dı́az, Gerardo

doi:10.5772/6947

Cited by 6 publications

(3 citation statements)

References 22 publications

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“…Diaz and Coimbra [37] proposed a control technique and applied the VO approach as the control action to stabilize a chaotic dynamical system. At the same time, the VO fractional operators can be also applied to describe VO fractional noise and estimate the VO fractional derivative of an unknown signal in noisy environment based on the wavelet analysis [26].…”

Section: Vo Fractional Control Model Fractional Derivative Model Hasmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

284

138

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Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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Section: Vo Fractional Control Model Fractional Derivative Model Hasmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

284

138

View full text Add to dashboard Cite

show abstract

“…That is because of the adaptation of the initial conditions (HongGuang et al., 2009; Wang et al., 2014). We chose the Coimbra fractional derivative, because of its wide applications to different systems (Coimbra, 2003; Kobayashi et al., 2008; HongGuang et al., 2009; Diaz and Coimbra, 2010; Shen et al., 2012), and its derivative of a constant is zero (Zhang et al., 2013).…”

Section: Preliminary Definitions In Fractional Calculusmentioning

confidence: 99%

A freely damped oscillating fractional dynamic system modeled by fractional Euler–Lagrange equations

Agila

Băleanu

Eid

et al. 2017

Journal of Vibration and Control

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The behaviors of some vibrating dynamic systems cannot be modeled precisely by means of integer representation models. Fractional representation looks like it is more accurate to model such systems. In this study, the fractional Euler–Lagrange equations model is introduced to model a fractional damped oscillating system. In this model, the fractional inertia force and the fractional damping force are proportional to the fractional derivative of the displacement. The fractional derivative orders in both forces are considered to be variable fractional orders. A numerical approximation technique is utilized to obtain the system responses. The discretization of the Coimbra fractional derivative and the finite difference technique are used to accomplish this approximation. The response of the system is verified by a comparison to a classical integer representation and is obtained based on different values of system parameters.

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“…As a consequence, VO discrete fractional calculus (VODF) was available as an intriguing option for supplying an efficient computational scheme for precisely characterizing multifaceted natural mechanisms and procedures 22 . Following that, VO-FDEs have captured growing interest owing to their appropriateness in modeling a wide range of occurrences, such as anomalous diffusion 23 , 24 , viscosity mechanisms 25 , 26 , automation 27 , chemical engineering 28 , and numerous additional areas of science and technology 29 , 30 .…”

Section: Introductionmentioning

confidence: 99%

Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

Chu,

Rashid,

Alzahrani

et al. 2023

Sci Rep

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Complex networks have been programmed to mimic the input and output functions in multiple biophysical algorithms of cortical neurons at spiking resolution. Prior research has demonstrated that the ineffectual features of membranes can be taken into account by discrete fractional commensurate, non-commensurate and variable-order patterns, which may generate multiple kinds of memory-dependent behaviour. Firing structures involving regular resonator chattering, fast, chaotic spiking and chaotic bursts play important roles in cortical nerve cell insights and execution. Yet, it is unclear how extensively the behaviour of discrete fractional-order excited mechanisms can modify firing cell attributes. It is illustrated that the discrete fractional behaviour of the Izhikevich neuron framework can generate an assortment of resonances for cortical activity via the aforesaid scheme. We analyze the bifurcation using fragmenting periodic solutions to demonstrate the evolution of periods in the framework’s behaviour. We investigate various bursting trends both conceptually and computationally with the fractional difference equation. Additionally, the consequences of an excitable and inhibited Izhikevich neuron network (INN) utilizing a regulated factor set exhibit distinctive dynamic actions depending on fractional exponents regulating over extended exchanges. Ultimately, dynamic controllers for stabilizing and synchronizing the suggested framework are shown. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

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Dynamics and Control of Nonlinear Variable Order Oscillators

Cited by 6 publications

References 22 publications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

A freely damped oscillating fractional dynamic system modeled by fractional Euler–Lagrange equations

Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

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