Mittag-Leffler stability and bifurcation of a nonlinear fractional model with relapse

Lahrouz, Aadil; Hajjami, Riane; Jarroudi, Mustapha El; Settati, Adel

doi:10.1016/j.cam.2020.113247

Cited by 12 publications

(8 citation statements)

References 33 publications

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“…Up to now, there have been quite a few characteristic investigations of control methods and dynamical behaviors for nonlinear systems [1]- [20], such as fixed-time control [1], event-triggered adaptive control [2], distributed control [3], piecewise control [4], fuzzy control [5], horizon control [6], U-control [7], passivity cascade technique-based control [8], stabilization control [9], iterative learning control [10], sliding set design [11], robustness control [12] and so forth. In addition, various dynamical behaviors of nonlinear systems have been explored [13]- [20], such as asymptotic stability [13], Mittag-Leffler stability [14], globally exponential stabilization [15], [16], synchronization [13], [17], dissipativity [18], robustness analysis [19], [20], etc. Nowadays, the characteristic application scenarios of nonlinear systems widely appears in reality, such as the computer-node information transmission, the circuit conduction, robot joint control, drive-by-wire control systems and so forth [21], [22], [23].…”

Section: Introductionmentioning

confidence: 99%

Robustness Analysis of Exponential Stability of Neutral-Type Nonlinear Systems With Multi-Interference

Xie

2021

IEEE Access

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With a view to the unfavorable impact of the inevitable exogenous interferences for the practical engineering and signal transmission, here we focus on the robustness of global exponential stability for nonlinear dynamical system subject to piecewise constant arguments, neutral terms and stochastic disturbances (SNPNDS). A new troublesome problem is that the neutral terms appeared in the derivative part affected on the other two interference factors is not a simple accumulation, so the Lipchitz condition is adopted to establish the ternary transcendental equations. However, different from previous transcendental equations with single or double variables, solving the transcendental equations with three variables becomes the bottleneck again. Hence, the special independent parameters & interdependent variables method targeted for SNPNDS is adopted here: firstly, all relative independent parameters are fixed. Next, the upper bounds of these three interdependent variables are orderly derived by their coupling relationship. Therein, the optimal constraint conditions for piecewise constant arguments and neutral terms are deduced. Through the strategies mentioned above, a class of algebraic problems of estimating three upper bounds by solving transcendental equations with three variables is settled. Besides, the main method ensures the relationship built among the interference factors is mutually restrictive and dynamic. Meanwhile, the optimized constraints make the linkage effect more comprehensive and valid. Furthermore, the established mechanism is practical enough to be generalized to more multivariable systems. Finally, the numerical simulation comparisons are given to illustrate the validity of the derived results.INDEX TERMS Robustness, nonlinear system, neutral term, piecewise constant argument, stochastic disturbance.

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Section: Introductionmentioning

confidence: 99%

Robustness Analysis of Exponential Stability of Neutral-Type Nonlinear Systems With Multi-Interference

Xie

2021

IEEE Access

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“…However, in most cases, it has been observed that the incidence rate starts to slow down after some time. In the last few decades, there have been studies considering nonlinear incidence rates given by a general function F (S, I) which satisfies certain mild conditions [18][19][20][21][22][23][24]. On the other hand, there are fractional-order epidemic models with a bilinear incidence rate and standard incidence rate, as well as specific nonlinear incidencerate functions [5,16,[25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…More recently, the authors in [26] considered a fractionalorder Susceptible-Exposed-Infected-Recovered (SEIR) model with a generalized incidence rate function of the type S f (I), where f satisfies some certain conditions. Furthermore, Lahrouz et al studied a fractional-order Susceptible-Infected-Recovered (SIR) model with a general incidence rate function and carried out Mittag-Leffler stability and bifurcation analysis for the model with and without delays [24]. It should be noted that the authors considered the model with a delay on the recovered/removed population (R(t)) and investigated the global dynamics.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Kashkynbayev

Rihan

2021

Mathematics

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In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

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“…For example, delays are often used to describe incubation time in biological models. Therefore, if the memory characteristics and delay effects of the model are taken into account at the same time, various time fractional order delay differential equations could be obtained, such as time fractional SIRI epidemic model with relapse and a general nonlinear incidence rate [LHEJS21].…”

Section: Introductionmentioning

confidence: 99%

Numerical stability of Grünwald-Letnikov method for time fractional delay differential equations

Wang²

2021

Preprint

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This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald-Letnikov (GL) approximation (also called fraction backward Euler scheme) for the Caputo fractional derivative, in particular, the numerical stability region and the Mittag-Leffler stability. Using the boundary locus technique, we first derive the exact expression of the numerically stability region in the parameter plane, and show that the fractional backward Euler scheme based on GL scheme is not τ (0)-stable, which is different from the backward Euler scheme for integer DDE models. Secondly, we also prove the numerical Mittag-Leffler stability for the numerical solutions provided that the parameters fall into the numerical stability region, by employing the singularity analysis of generating function. Our results show that the numerical solutions of F-DDEs are completely different from the classical integer order DDEs, both in terms of τ (0)-stabililty and the long-time decay rate.

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Mittag-Leffler stability and bifurcation of a nonlinear fractional model with relapse

Cited by 12 publications

References 33 publications

Robustness Analysis of Exponential Stability of Neutral-Type Nonlinear Systems With Multi-Interference

Robustness Analysis of Exponential Stability of Neutral-Type Nonlinear Systems With Multi-Interference

Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Numerical stability of Grünwald-Letnikov method for time fractional delay differential equations

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