Long-range Interactions, Stochasticity and Fractional Dynamics

Luo, Albert C. J.; Afraimovich, Valentin

doi:10.1007/978-3-642-12343-6

Cited by 28 publications

(1 citation statement)

References 98 publications

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“…The results and methods proposed in these works were developed in article on quantum theory [126][127][128][129]. We assume that these applications and others possible applications [90,[150][151][152][153][154][155][156][157][158][159][160][161] can be generalized for fractional exact finite-difference to describe processes and systems with non-locality in space and time.…”

Section: Discussionmentioning

confidence: 99%

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Tarasov

2024

Mathematics

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In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

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Section: Discussionmentioning

confidence: 99%

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Tarasov

2024

Mathematics

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Dynamical study of fractional order differential equations of predator‐pest models

Mandal

Sha

Chattopadhyay

2019

Math Methods in App Sciences

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To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.

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On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

Almeida

Malinowska

Odzijewicz

2019

Math Methods in App Sciences

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Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order 0 < < 1 of the -Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader-follower consensus for fractional multi-agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples.

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Long-range Interactions, Stochasticity and Fractional Dynamics

Cited by 28 publications

References 98 publications

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Dynamical study of fractional order differential equations of predator‐pest models

On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

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