Chaos in discrete fractional difference equations

Deshpande, Amey; Daftardar-Gejji, Varsha

doi:10.1007/s12043-016-1231-9

Cited by 38 publications

(19 citation statements)

References 12 publications

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“…In order to overcome the difficulties that arise from dealing with continuous time fractional order system and efficiently capture the memory effects, discrete fractional calculus was introduced [46][47][48][49][50]. The studies of dynamic behaviors and applications of fractional delta difference models attracted increasing interest in the last decade, see [49][50][51][52][53][54] and references therein.…”

Section: Preliminariesmentioning

confidence: 99%

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

et al. 2021

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Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.

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Section: Preliminariesmentioning

confidence: 99%

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

et al. 2021

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“…Many schemes for control of chaos have been applicable to both differential equations as well as maps. The notion of fractional order differential equation has been extended to fractional order difference equation and few definitions have been proposed [20]. Dynamics of linear and nonlinear systems have been investigated for fractional-order difference equations [21] and even spatially extended dynamical systems have been defined as well as investigated [22].…”

Section: Introductionmentioning

confidence: 99%

Stability and Dynamics of Complex Order Fractional Difference Equations

Bhalekar,

Gade,

Joshi

2021

Preprint

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We extend the definition of n-dimensional difference equations to complex order α ∈ C. We investigate the stability of linear systems defined by an n-dimensional matrix A and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where A = λ ∈ C, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for λ ∈ R, the solutions can be complex and dynamics in one-dimension is richer than the case for α ∈ R. These results can be extended to n-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.

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“…Due to the ergodic and non-repetition behaviours of chaotic maps, the combination of integer-order chaotic maps and meta-heuristic algorithms has proved to deliver significant improvements to the performance of algorithms [45], [46]. On the other hand, incorporation of fractional calculus with chaotic maps have enriched the dynamical behaviour of maps by demonstrating different distributions in comparison with integer-order counterparts [47]- [52]. According to the investigation results reported in the literature, the main superiorities of fractional-order chaotic maps compared with the integer-order chaotic maps can be summarized as (i) wider chaotic regions can be achieved due to the addition of fractional-order [50], (ii) more random chaotic sequences, more stability, and higher level of security are guaranteed [50], [51], and (iii) better ergodicity and distribution characteristic are illustrated [52].…”

Section: Introductionmentioning

confidence: 99%

Enhanced Fractional Chaotic Whale Optimization Algorithm for Parameter Identification of Isolated Wind-Diesel Power Systems

2020

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Parameters identification of isolated wind-diesel power systems (WDPS) is a significant issue in stability analysis of the power system as well as guaranteeing the power generation through the control system. In this paper, enhanced whale optimization algorithms (EWOA) are proposed to deal with the parameter identification problem of a WDPS system. The proposed EWOA effectively tackles the premature convergence problem of WOA by splitting the population into two subpopulations and updating the position of each whale according to the position of the best agent in its current subpopulation, the position of the other subpopulation's best agent, and the position of the best neighboring agent. Furthermore, fractional chaotic maps are embedded in the search process of EWOA to increase its performance in terms of accuracy. For validation purposes, the proposed algorithms are applied to identify the unknown parameters of WDPS, where different statistical analyzes and comparisons are carried out with other recent state-of-the-art algorithms. Simulation results confirm that the algorithms have less deviation in parameter estimation, more convergence speed, and higher precision in comparison with other algorithms. INDEX TERMS Optimization, parameter identification, whale optimization algorithm, wind-diesel power system.

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Chaos in discrete fractional difference equations

Cited by 38 publications

References 12 publications

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Stability and Dynamics of Complex Order Fractional Difference Equations

Enhanced Fractional Chaotic Whale Optimization Algorithm for Parameter Identification of Isolated Wind-Diesel Power Systems

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