# Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1.**

## 3. System Description and Its Dynamical Behaviors

#### 3.1. Hidden Attractors in the Proposed System

#### 3.2. Coexistence of Different Hidden Attractors

## 4. Hybrid Synchronization Scheme

_{i}(i = 1, 2, 3) are controllers to be determined.

**Lemma 1 ([35]).**

**Lemma 2 ([36]).**

**Theorem 1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Hidden chaotic attractor of system (4) when $\alpha =0.28$. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 2.**Hidden periodic attractor of system (4) when $\alpha =0.5$. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 3.**Hidden quasi-periodic attractor of system (4) when $\alpha =0.95$. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 6.**Hidden chaotic attractor of system (4) when $\beta $ = 0.30. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 7.**Hidden periodic attractor of system (4) when $\beta $ = 0.4. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 8.**Hidden quasi-periodic attractor of system (4) when $\beta $ = 0.58. (

**a**) Phase trajectory diagram in $yz$ plane; (

**b**) Poincare map in $xz$ plane.

**Figure 11.**Coexistence of hidden chaotic attractors with symmetrical structure for different initial values. (

**a1**,

**a2**) are phase trajectory and Poincare maps with initial value (0, 2, 0, 0), respectively. (

**b1**,

**b2**) are phase trajectory and Poincare maps with initial value (1, −2, 0, 1), respectively.

**Figure 12.**Coexistence of hidden quasi-periodic and chaotic attractor for different initial values. (

**a1**,

**a2**) are phase trajectory and Poincare maps with initial value (0, 2, 0, 0), respectively. (

**b1**,

**b2**) are phase trajectory and Poincare maps with initial value (1, −2, 0, 0), respectively.

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**MDPI and ACS Style**

Jiang, H.; Zhuang, L.; Chen, C.; Wang, Z.
Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems. *Axioms* **2022**, *11*, 645.
https://doi.org/10.3390/axioms11110645

**AMA Style**

Jiang H, Zhuang L, Chen C, Wang Z.
Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems. *Axioms*. 2022; 11(11):645.
https://doi.org/10.3390/axioms11110645

**Chicago/Turabian Style**

Jiang, Haipeng, Lizhou Zhuang, Cheng Chen, and Zuolei Wang.
2022. "Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems" *Axioms* 11, no. 11: 645.
https://doi.org/10.3390/axioms11110645