Julia Set Continuity of Critical Portraits

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On the spatial Julia set generated by fractional Lotka-Volterra system with noise☆

Abstract

This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system's Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

Introduction

In mathematics, a fractal is an infinitely recursive and detailed set (cf. [1]). Julia set, proposed by Gaston Julia (cf. [2]) in 1918, is considered as one of the most attractive fractal sets. The Julia set is not only an attractor in the sense that the iteration sequence generated from a point in the set is still in it, but also represents a basin of attraction where the iteration of a dynamical system converges. Hence, the Julia set can be considered as a "map" for the boundedness or stability of nonlinear systems. In terms of the population dynamics, the most obvious advantage for knowing about Julia sets is to grasp the long-term behavior of population dynamics, and then control or avoid explosive solutions (cf. [3], [4], [5], [6]) that mean the "collapse" of the system. In biology and other interdisciplinary fields, Julia set also provides eximious explanations for many complicated phenomena (cf. [7], [8], [9]).

The study of noise-perturbed Julia sets has attracted interest in the early 2000s (cf. [10], [11]). Although more work has been developed on this topic, we mainly refer the readers to the following development. In 2010, Andreadis et al. built a Julia deviation distance and a Julia deviation plot to quantify and visualize the effect of noise on a Julia set (cf. [12]), respectively. Then, the authors studied the structural stability of Julia sets of a complex logistic system and found a power-law behavior of the Julia deviation distance. In 2012, Andreadis et al. calculated and obtained the critical noise strength that makes the Julia set generated by a family of complex quadratic systems completely lose its original (noise-free) structure (cf. [13]). In 2017, Wang et al. studied the Julia set of complex Lorenz system (cf. [14]) $\{\begin{array}{c}{z}_{n+1}=(1+a\tau )z_n-\tau z_n{w}_n,\hfill \\ w_{n+1}=(1-\tau )w_n+\tau z_n^2,\hfill \end{array}$ where $z_n,w_n,a,\tau \in \mathbb{C}$ and $n\in \mathbb{N}$ . The authors extended the study of noise-perturbed planar Julia sets to the spatial case, while proposed an index so called "symmetry criterion" to quantitatively analyze the influence of dynamic noise on the symmetry of system's Julia sets.

On the other hand, fractional operators are increasingly considered in modeling processes of biological systems due to their unique advantages in describing complex phenomena and behaviors related to memory or fractals (cf. [15]) that are essentially nonlinear, highly complex and hard to be explained by integer-order models. Ahmed et al. introduced a fractional model that emphasizes the nonlocal interaction besides the local interaction in an epidemic spreading (cf. [16]). These two actions are observed in foot-and-mouth disease, SARS and avian flu outbreaks in 2007. Rivero et al. found that the order of fractional derivation is an excellent controller of the velocity how a system's trajectories approach to the critical point, which represents the anomalous reality of the competition among some species (cf. [17]). Hamdan et al. discovered that the fractional differential equation is more appealing and realistic in modeling the dengue transmission as it possesses memory (cf. [18]).

Fractional models allow greater degrees of freedom in systems, which enable us to consider new intricate situations in population dynamics. In terms of the Lotka-Volterra model in ecology, it has been considered by Ahmed et al. (cf. [19]) since 2007. Afterwards, qualitative properties and numerical solutions to this kind of fractional models are widely studied. In 2015, Elsadany et al. studied fractional Lotka-Volterra system (cf. [15]) $\{\begin{array}{c}{}^C{D}_{0^{+}}^{\alpha }u\left(t\right)=u\left(t\right)(r-ku\left(t\right)-v\left(t\right)),\hfill \\ {}^C{D}_{0^{+}}^{\alpha }v\left(t\right)=v\left(t\right)(-1+\beta u\left(t\right)),\hfill \end{array}$ where ${}^C{D}_{0^{+}}^{\alpha }$ and ${}^C{D}_{0^{+}}^{\beta }$ are two Caputo fractional derivatives. Interestingly enough, the authors found that the corresponding discretized version of this model exhibits much richer dynamical behaviors. In 2018, Wang et al. investigated a discrete fractional Lotka-Volterra system (cf. [5], [6]) $\{\begin{array}{c}{\displaystyle u_{n+1}=r_1{h}_N^{\alpha }{u}_n-a_1{h}_N^{\alpha }{u}_n^2-b_1{h}_N^{\alpha }{u}_n{v}_n-\sum _{k=1}^{\mathcal{N}}{\omega }_k^{\left(\alpha \right)}{u}_{n-k+1},}\hfill \\ {\displaystyle v_{n+1}=-r_2{h}_N^{\beta }{v}_n+a_2{h}_N^{\beta }{u}_n{v}_n-b_2{h}_N^{\beta }{v}_n^2-\sum _{k=1}^{\mathcal{N}}{\omega }_k^{\left(\beta \right)}{v}_{n-k+1}}\hfill \end{array}$ from the fractal viewpoint for the first time, which extends the work of Sun et al. (cf. [4]) in 2017 by introducing of fractional elements into the classical fractal sets of discrete dynamical systems. After discussing the stability of fixed points of system (1.1), control and synchronization of system's Julia sets are realized by several control strategies.

Inspired by the work above, system (1.1) will be expanded to the complex field. Two kinds of systems with dynamic multiplicative noise are also generalized from system (1.1). Then, it is studied that features of the spatial fractal structures of these systems' Julia sets. The motivation of this work is providing a novel perspective and possible method to deeply grasp noise-perturbed dynamics of fractional systems by combining the ideas of fractional calculus and fractals. Because both fractional calculus and fractal theory have lots of practical applications, besides, there is an inseparable relationship between them (cf. [20]), it is worthwhile to study the fractional system from the viewpoint of fractal. But few such works are involved now.

The main novelties of this work can be summarized as follows:

(i)

Generalize system (1.1) into the complex domain. The spatial Julia set of the fractional complex system is obtained, and the dynamical characteristics of the system are studied from the fractal point of view. Fractional systems in the complex domain have richer dynamical behavior and more complex fractal structure, which brings new contents to the study of dynamical systems.

(ii)

Introduce noise into system (1.1). The influence of dynamic Gaussian (normal) noise on the fractal characteristics of fractional Lotka-Volterra system is analysed. Compared to the previous noise-free system in [5], [6], it has more practical significance, because noise disturbances are unavoidable for any process in the real environment. Various factors, such as the season, temperature, reproductive environment, will inevitably affect the state of the system, and these effects are uncertain and random.

(iii)

Propose the Julia deviation index. Compared with the Julia deviation distance in [12], [14], the new index has stronger immanence, that is, it fully reflects the role of the elements of the set itself and excludes the influence of the space where the set is located. So it can not only measure the effects of different noise on one system more distinctly, but also make it possible even to be utilized on different systems.

The remainder of the paper is organized as follows. In Section 2, some necessary knowledge is reviewed, and system (1.1) is derived and the system's Julia set is defined and visualized. In Section 3, two kinds of noise-perturbed fractional Lotka-Volterra systems are proposed. In Section 4, a Julia deviation index is proposed and its exponential behavior is observed for noise-perturbed Julia sets by fitting; a ratio index is also offered to assist in the analysis of the effect of noise. In Section 5, after theoretically proving the symmetry of the Julia set, the effect of noise on the symmetry is measured quantitatively. In Section 6, summary and prospect for our present work are made.

Section snippets

Preliminaries

At present, Grünwald-Letnikov, Riemann-Liouville and Caputo fractional differential operators are frequently used. In fact, these three definitions are equivalent under some conditions for a wide class of functions (cf. [21]). More relevant knowledge can be found in [21], [22], [23], [24] and references contained therein. In this and subsequent sections, the Grünwald-Letnikov definition is considered.

Definition 2.1

The Grünwald-Letnikov fractional derivative of order α > 0 of a continuous function y is given

The noise-perturbed fractional Lotka-Volterra system

From the perspective of population dynamics, there are two sources of noise: the intrinsic stochasticity associated with the dynamics of the individuals (intrinsic noise) and the random variability of the environment (environmental noise). The intrinsic noise can be interpreted as the fluctuation of model's parameters, while the environmental noise is a direct disturbance to system's states. Multiplicative noise source attracts more attention because of its better physical meaning in population

The structural changes of Julia sets of noise-perturbed fractional Lotka-Volterra system

This section contains our main results on quantification and visualization of noise on Julia sets. A new deviation index and an auxiliary ratio index are given to characterize and analyze these effects of noise.

The symmetry damages of Julia sets of noise-perturbed fractional Lotka-Volterra system

The investigation of the symmetry of Julia sets can be found in [14], [36], [37]. In this paper, Fig. 1 has already preliminarily shown that the Julia set of system (2.5) has a symmetrical structure, which can be treated as a visual evidence for the following theorem.

Theorem 5.1

$J_{\mathbf{F},{\tilde{v}}_0\equiv 0}$ is symmetric about the ${\overline{u}}_0{\overline{v}}_0$ -plane.

Proof

For two initial points ${\mu }_0=({\mu }_{01},{\mu }_{02},{\mu }_{03},0)\text{and}{\nu }_0=({\nu }_{01},{\nu }_{02},{\nu }_{03},0),$ set them be symmetric about the ${\overline{u}}_0{\overline{v}}_0$ -plane, i.e., ${\mu }_{01}={\nu }_{01},$ ${\mu }_{02}=-{\nu }_{02}$ and ${\mu }_{03}={\nu }_{03}$ . Then, denote the nth iterations of μ ₀ and

Conclusion

Combining qualitative study and quantitative analysis, we discuss the fractal characteristics, structural change and symmetry of Julia sets generated by a fractional Lotka-Volterra system with different noise realizations. After deriving the discrete system (2.5) from the continuous fractional system (2.2), the Julia set of system (2.5) is defined and visualized. By using the Julia deviation index, the effect of noise on its structural changes is analyzed. In addition, the symmetry of a 3-D

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

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Source: https://www.sciencedirect.com/science/article/abs/pii/S0960077919302966

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