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Multistablity analysis and formation of spiral wave in the fractional-order nonlinear systems is a recent hot topic. In this paper, dynamics, coexisting attractors, complexity, and synchronization of the fractional-order memristor-based hyperchaotic Lü system are investigated numerically by means of bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and sample entropy (SampEn) algorithm. The results show that the system has rich dynamics and high complexity. Meanwhile, coexisting attractors in the system are observed and hidden dynamics are illustrated by changing the initial conditions. Finally, the network based on the system is built, and the emergence of spiral waves is investigated and chimera states are observed.

A new two-terminal circuit element called memristor characterized by a relationship between the charge and the flux was postulated as the fourth basic circuit element by Chua [

Memristor is a nonlinear electronic element. When the memristor is introduced, the chaotic oscillation of the circuit could be more complex due to its nonlinearity. So the research of memristor-based chaotic systems and its dynamic characteristics has become a new hot spot. In recent years, relevant research results have been obtained [

Recently, Qiao et al. [

The rest of this paper is organized as follows. Section

The most commonly used memristor-based dynamical system [^{2} is a memductance function describing the flux-dependent rate of change of charge. Qiao et al. [

Here, by introducing the fractional calculus to this system, a novel fractional-order memristor-based hyperchaotic Lü system is constructed, and the mathematical model is given by

(see [^{+} and

In order to analyze the dynamic and complexity characteristics of this fractional-order memristor-based hyperchaotic Lü system, we fix the system parameters as

It can be obtained by (

Let

The predictor-corrector method is an effective method for solving fractional-order equations. According to Kai et al. [

Let _{j} = ^{+}), and

The utilized source Matlab code of the predictor-corrector method can be downloaded online (_{k} (

Dynamics of the system with the variation of

Case 1. Fix

Case 2. Fix ^{−4}. Set the initial condition as

Case 3: Vary parameter ^{−4}. Thus, the parameter space is divided as a 400 × 500 grid. As with above cases, the initial condition is given by

Dynamical behaviours with derivative order

Dynamical behaviours with memristor gain

Maximum Lyapunov exponent contour plot in the

Phase portraits of the system with the initial condition

Phase diagrams with derivative order

Sample entropy (SampEn) [

Step 1. Given a time series {

Step 2. Define the Euclidean distance between

Step 3. Define the criterion of similarity _{r}^{m} (^{m} (

Step 4. Similarly, change ^{(m+1)} (

Finally, theoretically, the SampEn is defined as

The value of SampEn is related to the values of

SampEn complexity of the fractional-order memristor-based hyperchaotic Lü system with parameter

SampEn complexity analysis results with derivative order

Let parameter ^{−4}, and the initial condition be

SampEn complexity contour plots in the

Coexisting attractors can be observed in this system. Fix

Coexisting attractors with

Bifurcation diagram in the range

In order to further study the effect of initial conditions on the dynamics of the system, the SampEn complexity chaos diagram for different parameters is illustrated in Figure

SampEn complexity-based chaos diagram. (a) _{0} − _{0} plane. (b) _{0} − _{0} plane. (c) _{0} − _{0} plane. (d) _{0} − _{0} plane.

Compared with the basin attraction plots, the SampEn-based chaos diagrams illustrate the dynamics and complexity of the system with different initial conditions. If the complexity is different, the state of the system is different. Thus, it can show the multistability of the system in the initial plane through a complexity of view. Since complexity measure just needs a segment of time series, it is very convenient to detect the coexisting attractors in the multistability systems. However, it can only detect the coexisting states with different complexity. Usually, it means that there exists coexisting chaotic attractors and periodic circles.

To further capture the complex dynamics of the system, we construct a ring network of nonlinear systems which are coupled locally to the nearby 3 nodes with a coupling strength of

Input:

Output:

if

elseif

elseif

elseif

elseif

else

end if

Model of the ring network structure of the network (by Pajek 64 5.08).

In order to derive the dynamics of the neuronal ring network, we also fix the system parameters as

Spatiotemporal patterns of the model with the variation of

Case 1. Fix

Case 2. Fix

Case 3. Fix

Spatiotemporal patterns of the ring network for different coupling strengths of Case 1. (a)

Spatiotemporal patterns of the ring network for different coupling strengths of Case 2. (a)

Spatiotemporal patterns of the neuronal ring network for different coupling strengths of Case 3. (a)

As shown in Figures

In this paper, a fractional-order memristor-based hyperchaotic system with no equilibrium points has been numerically analyzed by employing the predictor-corrector method. It shows that the system has rich dynamical behaviours, and different states including periodic circle, chaotic attractor, and hyperchaotic attractor are plotted in the system. The minimum derivative order of generating chaos is found, and it is

All data, models, and code generated or used during the study appear in the submitted article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the National Natural Science Foundation for Theoretical Physics of China (grant nos. 61901530 and 11747150), the Hunan Provincial Department of Education General Project Fund (no. B08004056), the China Postdoctoral Science Foundation (grant no. 2019M652791), and the Postdoctoral Creative Talent Support Programme (grant no. BX20180386).