Asymptotic cycles in fractional maps of arbitrary positive orders

Edelman, Mark; Helman, Avigayil B.

doi:10.1007/s13540-021-00008-w

Cited by 10 publications

(7 citation statements)

References 42 publications

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“…The requirement that ∑ ∞ k=1 U α (k) = ±∞, which was important for the finding of the periodic points in [35,36], was not used in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…When 0 < α < 1, the generalized universal α-family of maps can be written in the form (see [35] and Sections 2 and 3 in [36]):…”

Section: Preliminariesmentioning

confidence: 99%

“…In this formula G 0 (x) = h α G K (x)/Γ (α), x 0 is the initial condition, h is the time step of the map, α is the order of the map, G K (x) is a nonlinear function depending on the parameter K, U α (n) = 0 for n ≤ 0, and U α (n) ∈ D 0 (N 1 ). The space D i (N 1 ) is defined as (see [35])…”

Section: Preliminariesmentioning

confidence: 99%

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Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Edelman

2022

Preprint

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Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders 0 < α < 1 that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point - asymptotically period two bifurcation point. PACS 05.45.Pq · 45.10.Hj Mathematics Subject Classification (2000) 47H99 60G99 · 34A99 · 39A70

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“…The requirement that ∑ ∞ k=1 U α (k) = ±∞, which was important for the finding of the periodic points in [35,36], was not used in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…When 0 < α < 1, the generalized universal α-family of maps can be written in the form (see [35] and Sections 2 and 3 in [36]):…”

Section: Preliminariesmentioning

confidence: 99%

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Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Edelman

2022

Preprint

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“…Their properties to a high degree are defined by the common properties of their kernels which belong to a class of discrete functions whose series are diverging; however, the series of their differences belong to the space of absolutely converging series l 1 . For this reason, the generalized fractional maps, which include fractional and fractional difference maps, were introduced in [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…Although the fractional maps do not have periodic points, they have asymptotically periodic points. Asymptotically periodic points and stability of the fixed points in generalized fractional maps were investigated in the recent papers [11,12,24,25]. In this paper, we use the results of [11] to draw the bifurcation diagrams for the fractional difference logistic map.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < alpha < 1

Edelman¹,

Helman²,

Smidtaite³

2023

Preprint

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Generalized fractional maps of the orders 0 < α < 1 are Volterra difference equations of convolution type with kernels, which differences are absolutely summable, but the series of kernels are diverging. Commonly used in applications fractional (with the power-law kernels) and fractional difference (with the falling factorial kernels) maps of the orders 0 < α < 1 belong to this class of maps. We derived the algebraic equations which allow the calculation of bifurcation points of generalized fractional maps. We calculated the bifurcation points and drew bifurcation diagrams for the fractional and fractional difference logistic maps. Although the transition to chaos on individual trajectories in fractional maps may be characterized by the cascade of bifurcations type behavior and zero Lyapunov exponents, the results of our numerical simulations allow us to make a conjecture that the cas-

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Stability of fixed points in generalized fractional maps of the orders $$0< \alpha <1$$

Edelman

2023

Nonlinear Dyn

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Asymptotic cycles in fractional maps of arbitrary positive orders

Cited by 10 publications

References 42 publications

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < alpha < 1

Stability of fixed points in generalized fractional maps of the orders $$0< \alpha <1$$

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Asymptotic cycles in fractional maps of arbitrary positive orders

Cited by 10 publications

References 42 publications

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 &lt; α &lt; 1

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 &lt; α &lt; 1

Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < alpha < 1

Stability of fixed points in generalized fractional maps of the orders $$0< \alpha <1$$

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Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1