Stability properties of two-term fractional differential equations

Čermák, Jan; Kisela, Tomáš

doi:10.1007/s11071-014-1426-x

Cited by 35 publications

(25 citation statements)

References 19 publications

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“…Generally, the asymptotic stability of the trivial solution of system is not of exponential type,() because of the presence of the memory effect. A special type of nonexponential asymptotic stability concept has been defined for fractional‐order differential equations, called Mittag‐Leffler stability.…”

Section: Preliminariesmentioning

confidence: 99%

Stability of two‐component incommensurate fractional‐order systems and applications to the investigation of a FitzHugh‐Nagumo neuronal model

Brandibur

Kaslik

2018

Math Methods in App Sciences

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For two‐dimensional autonomous linear incommensurate fractional‐order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh‐Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two‐dimensional fractional‐order FitzHugh‐Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer‐order FitzHugh‐Nagumo model.

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Section: Preliminariesmentioning

confidence: 99%

Stability of two‐component incommensurate fractional‐order systems and applications to the investigation of a FitzHugh‐Nagumo neuronal model

Brandibur

Kaslik

2018

Math Methods in App Sciences

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“…, equipped with initial conditions [6,[8][9][10]. In [9], the authors investigated the Endolymph equation:…”

Section: Introductionmentioning

confidence: 99%

Positive properties of the Green function for two-term fractional differential equations and its application

Wang¹,

Liu²

2017

J. Nonlinear Sci. Appl.

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“…Most of these applications can be modeled as fractional-order linear time invariant systems (FLTI) [12]. Recently, the stability analyses of many linear and nonlinear fractional-order examples were introduced in [13][14][15][16][17][18][19], and the stability discussion of some delayed fractional-order examples were studied in [19][20][21][22]. Recently, the stability analyses of many linear and nonlinear fractional-order examples were introduced in [13][14][15][16][17][18][19], and the stability discussion of some delayed fractional-order examples were studied in [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Fundamentals of fractional‐order LTI circuits and systems: number of poles, stability, time and frequency responses

Semary

Radwan

Hassan

2016

Circuit Theory & Apps

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This paper investigates some basic concepts of fractional-order linear time invariant systems related to their physical and non-physical transfer functions, poles, stability, time domain, frequency domain, and their relationships for different fractional-order differential equations. The analytical formula that calculates the number of poles in physical and non-physical s-plane for different orders is achieved and verified using many practical examples. The stability contour versus the number of poles in the physical s-plane for different fractional-order systems is discussed in addition to the effect of the non-physical poles on the steady state responses. Moreover, time domain responses based on Mittag-Leffler functions for both physical and non-physical transfer functions are discussed for different cases, which confirm the stability analysis. Many fractional-order linear time invariant systems based on fractional-order differential equations have been discussed numerically in both time and frequency domains to validate the previous fundamentals.

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Stability properties of two-term fractional differential equations

Cited by 35 publications

References 19 publications

Stability of two‐component incommensurate fractional‐order systems and applications to the investigation of a FitzHugh‐Nagumo neuronal model

Stability of two‐component incommensurate fractional‐order systems and applications to the investigation of a FitzHugh‐Nagumo neuronal model

Positive properties of the Green function for two-term fractional differential equations and its application

Fundamentals of fractional‐order LTI circuits and systems: number of poles, stability, time and frequency responses

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