# Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation

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## Abstract

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## 1. Introduction

## 2. Bifurcation Analysis

#### 2.1. Pitchfork Bifurcation

#### 2.2. Hopf Bifurcation

#### 2.3. Limit Cycle Bifurcation

## 3. Analysis of Cluster Oscillation

#### 3.1. The Point–Point Cluster Oscillation and Pitchfork Bifurcation

#### 3.2. The Point–Cycle Cluster Oscillation and Pitchfork/Hopf Bifurcation

#### 3.3. The Point–Cycle–Cycle Cluster Oscillation and Pitchfork/Hopf/Limit Cycle Bifurcation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Phase diagrams corresponding to the Hopf bifurcation. (

**a**) $q=1.13$. (

**b**) The limit cycle near the equilibrium $\left(\sqrt{F},0\right)$. (

**c**) The limit cycle near the equilibrium $\left(-\sqrt{F},0\right)$.

**Figure 5.**Phase diagrams corresponding to limit cycles bifurcation for $q=1.28$. (

**a**) The limit cycle near the equilibrium $\left(\sqrt{F},0\right)$. (

**b**) The limit cycle near the equilibrium $\left(-\sqrt{F},0\right)$. (

**c**) The large limit cycle for $F=0.5$.

**Figure 6.**The bifurcation with respect to the slowly varying parameter $F$ and the fractional order $q$ of the fast subsystem.

**Figure 8.**Cluster oscillation with $\beta =0.16$. (

**a**) Time history diagram. (

**b**) Transition phase diagram for the variable $x$ with respect to $F=0.16\mathrm{cos}\left(0.01t\right)$. (

**c**) Superposition of the transition phase diagram and bifurcation diagram on $\left(F,x\right)$ plane.

**Figure 9.**Cluster oscillation with $\beta =0.3$. (

**a**) Time history diagram. (

**b1**) Superposition of the transition phase diagram and bifurcation diagram on $\left(F,x\right)$ plane for the lower branch. (

**b2**) Superposition of the transition phase diagram and bifurcation diagram on $\left(F,x\right)$ plane for the upper branch.

**Figure 10.**Cluster oscillation with $\beta =0.4$. (

**a**) Time history diagram. (

**b**) Superposition of the transition phase diagram and bifurcation diagram on $\left(F,x\right)$ plane.

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**MDPI and ACS Style**

Li, X.; Wang, Y.; Shen, Y.
Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation. *Fractal Fract.* **2022**, *6*, 295.
https://doi.org/10.3390/fractalfract6060295

**AMA Style**

Li X, Wang Y, Shen Y.
Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation. *Fractal and Fractional*. 2022; 6(6):295.
https://doi.org/10.3390/fractalfract6060295

**Chicago/Turabian Style**

Li, Xianghong, Yanli Wang, and Yongjun Shen.
2022. "Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation" *Fractal and Fractional* 6, no. 6: 295.
https://doi.org/10.3390/fractalfract6060295