# Chaos Controllability in Non-Identical Complex Fractional Order Chaotic Systems via Active Complex Synchronization Technique

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## Abstract

**:**

## 1. Introduction

- The suggested HCPS methodology considers two dissimilar complex fractional order chaotic systems.
- It designs a robust HCPS strategy-based control input to achieve hybrid complex projective synchronization among considered fractional order complex systems and performs oscillation for synchronization errors with a fast rate of convergence.
- The description of HCPS scheme-based active control inputs is executed in a simplistic manner utilizing LSC and drive-response/master–salve configuration.
- Simulation outcomes depict the efficacy and superiority of the suggested HCPS strategy.

## 2. Preliminaries

**Property**

**1.**

**Property**

**2.**

## 3. Problem Formulation

**Definition**

**1.**

## 4. Synchronization Phenomena

## 5. Numerical Simulation and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Phase diagrams of fractional order complex Lorenz system (11) and (12) at $q=0.95$; (

**a**) between ${u}_{1}\left(t\right)-{u}_{2}\left(t\right)-{u}_{3}\left(t\right)$ space; (

**b**) between ${u}_{3}\left(t\right)-{u}_{5}\left(t\right)-{u}_{4}\left(t\right)$ space; (

**c**) between ${u}_{5}\left(t\right)-{u}_{4}\left(t\right)-{u}_{1}\left(t\right)$ space; (

**d**) between ${u}_{1}\left(t\right)-{u}_{4}\left(t\right)$ plane.

**Figure 2.**Phase diagrams of slave fractional order complex chaotic T-system for (13) and (14) at $q=0.95$; (

**a**) between ${v}_{1}\left(t\right)-{v}_{2}\left(t\right)-{v}_{3}\left(t\right)$ space; (

**b**) between ${v}_{3}\left(t\right)-{v}_{5}\left(t\right)-{v}_{4}\left(t\right)$ space; (

**c**) between ${v}_{3}\left(t\right)-{v}_{2}\left(t\right)-{v}_{5}\left(t\right)$ space; (

**d**) between ${v}_{1}\left(t\right)-{v}_{4}\left(t\right)$ plane.

**Figure 3.**State trajectories of master fractional order complex chaotic system and controlled slave fractional order complex chaotic system for (11)–(14) at $q=0.95$; (

**a**) ${v}_{1}\left(t\right)-{u}_{1}\left(t\right)-{u}_{2}\left(t\right)$ and ${v}_{2}\left(t\right)-{u}_{1}\left(t\right)-{u}_{2}\left(t\right)$; (

**b**) ${v}_{3}\left(t\right)-{u}_{3}\left(t\right)-{u}_{4}\left(t\right)$ and ${v}_{4}\left(t\right)-{u}_{3}\left(t\right)-{u}_{4}\left(t\right)$; (

**c**) ${v}_{5}\left(t\right)-{u}_{5}\left(t\right)$.

**Figure 4.**Synchronization error between master system and slave system at $q=0.95$ with controllers; (

**a**) $(t,{e}_{1})$; $(t,{e}_{2})$; $(t,{e}_{3})$ and $(t,{e}_{4})$; (

**b**) $(t,{e}_{5})$ and simultaneous plot $(t,{e}_{1},{e}_{2},{e}_{3},{e}_{4},{e}_{5})$.

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**MDPI and ACS Style**

Sajid, M.; Chaudhary, H.; Kaushik, S.
Chaos Controllability in Non-Identical Complex Fractional Order Chaotic Systems via Active Complex Synchronization Technique. *Axioms* **2023**, *12*, 530.
https://doi.org/10.3390/axioms12060530

**AMA Style**

Sajid M, Chaudhary H, Kaushik S.
Chaos Controllability in Non-Identical Complex Fractional Order Chaotic Systems via Active Complex Synchronization Technique. *Axioms*. 2023; 12(6):530.
https://doi.org/10.3390/axioms12060530

**Chicago/Turabian Style**

Sajid, Mohammad, Harindri Chaudhary, and Santosh Kaushik.
2023. "Chaos Controllability in Non-Identical Complex Fractional Order Chaotic Systems via Active Complex Synchronization Technique" *Axioms* 12, no. 6: 530.
https://doi.org/10.3390/axioms12060530