Fractal dimension analysis and control of Julia set generated by fractional Lotka–Volterra models

Wang, Yu-Pin; Liu, Shutang; Wang, Wen

doi:10.1016/j.cnsns.2019.01.009

Cited by 18 publications

(8 citation statements)

References 22 publications

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“…The chain coupled predatory behavior of system (5) For both two species and multi species ecosystems, the complex coupling behavior among species are both complicated. For the previous works [21,27,35,36], the range of control parameter k is obtained by analyzing the stability of fixed point. There are still some deficiencies about the boundedness analysis of the original Julia set, and the boundedness change with different parameters k.…”

Section: T-mentioning

confidence: 99%

“…In [27], Sun and Zhang gave the definition of Julia set for a kind of discrete Lotka-Volterra system, and realized its control via feedback control and synchronization methods. Some follow-up studies about SIRS model [21] and fractional case [35,36] were successively given. Nevertheless, there are still some deficiencies about the boundedness analysis of the Julia set this work trol of sy are summ research A prelim of the Jul coupled p The Section 2 sets of s study on in Section tion proce eters.…”

Section: Introductionmentioning

confidence: 99%

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Fractals analysis and control for a kind of three-species ecosystem with symmetrical coupled predatory behavior

Sun²,

Zhang³

2020

ITC

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The Lotka-Volterra model plays an important role in the research area of population biology. This work presents the analysis of dynamical behaviours of a kind of three species GLV systemfrom the viewpoint of fractals. The definition of Julia set which describes the initial distribution of the three species’ densities is introduced. Then a gradient control method which contains both giant parameter and state feedback is applied to realize the control of Julia set. Coupling terms are designed to realize the synchronization of two Julia sets. Numerical examples are included to verify the conclusions

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Section: T-mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractals analysis and control for a kind of three-species ecosystem with symmetrical coupled predatory behavior

Sun²,

Zhang³

2020

ITC

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“…In [23,24], researchers investigated the citation profiles of researchers in fractional calculus, and proposed that the application areas of fractional calculus contain the fractal concept. Based on the control theory and method, Wang [25][26][27] investigates the Julia sets of a fractional Lotka-Volterra model and realizes its state feedback control. In [28], the numerical simulation of a Boussinesq equation with different fractal dimension and fractional order is carried out.…”

Section: Introductionmentioning

confidence: 99%

Fractals Parrondo’s Paradox in Alternated Superior Complex System

Zhang

Wang

2021

Fractal Fract

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This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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“…However, there exist in nature attractors with self‐similarities; this class cannot be captured neither with classical differentiation nor with classical fractional and new trends in fractional differentiation and integration. Many technics have been suggested to solve and model such problems; for instance, the concept of fractal including Julia set, the Mandelbrot set, and many other sets has been used to capture self‐similarities, but those self‐similarities do not come from chaotic attractors, neither they come from mathematical models 7–14 . Additionally, these self‐similarities obtained from these complex sets cannot be used to predict the evolution in time, as they are not time dependent; however, they are very useful for other purpose.…”

Section: Introductionmentioning

confidence: 99%

“…Many technics have been suggested to solve and model such problems; for instance, the concept of fractal including Julia set, the Mandelbrot set, and many other sets has been used to capture self-similarities, but those self-similarities do not come from chaotic attractors, neither they come from mathematical models. [7][8][9][10][11][12][13][14] Additionally, these self-similarities obtained from these complex sets cannot be used to predict the evolution in time, as they are not time dependent; however, they are very useful for other purpose. Up to 2016, the question was still unsolved as these types of differential and integral operators were not available in literature.…”

Section: Introductionmentioning

confidence: 99%

New chaotic attractors: Application of fractal‐fractional differentiation and integration

Gómez‐Aguilar

Atangana

2020

Math Methods in App Sciences

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Very recently, the concept of fractal differentiation and fractional differentiation has been combined to produce new differentiation operators. The new operators were constructed using three different kernels, namely, power law, exponential decay, and the generalized Mittag‐Leffler function. The new operators have two parameters: the first is considered as fractional order and the second as fractal dimension. In this work, we applied these new operators to model some chaotic attractors, and the models were solved numerically using a new and very efficient numerical scheme. We presented numerical simulations for some specific fractional order and fractal dimension. The classical fractional differential models could be recovered when the fractal dimension is equal to 1; in these cases, the obtained attractors with power law presented no similarities. Nevertheless, those obtained via Caputo‐Fabrizio and the Atangana‐Baleanu derivative show some crossover effects, which is due to non‐index law property. However, those obtained from fractal‐fractional, in particular, those with the Mittag‐Leffler kernel, show very strange and new attractors with self‐similarities; these results are obtained for the first time. We conclude that this new concept is the future to modelling complexities with self‐similarities.

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Fractal dimension analysis and control of Julia set generated by fractional Lotka–Volterra models

Cited by 18 publications

References 22 publications

Fractals analysis and control for a kind of three-species ecosystem with symmetrical coupled predatory behavior

Fractals analysis and control for a kind of three-species ecosystem with symmetrical coupled predatory behavior

Fractals Parrondo’s Paradox in Alternated Superior Complex System

New chaotic attractors: Application of fractal‐fractional differentiation and integration

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