Special Characteristics and Synchronizations of Multi Hybrid-Order Chaotic Systems
Abstract
:1. Introduction
2. Mathematical Background
3. Characteristics Analysis of Hybrid Order Chaotic Systems
3.1. Hybrid Order Lorenz Systems
3.1.1. Dissipativeness and the Existence of Equilibriums
3.1.2. Equilibriums and Stability
3.1.3. Bifurcations of Hybrid Order Lorenz Systems
3.1.4. Relations of Different Chaotic Parameters in Hybrid Lorenz Systems
3.2. Other Classic Hybrid Order Chaotic ystems
3.2.1. Hybrid Order Chen Systems
3.2.2. Hybrid Order Lü Systems
3.2.3. Hybrid Order Complex Lorenz Systems
3.3. A Relation between Chaotic Parameters for Different Hybrid Order Chaotic Systems
- (1)
- always has inverse ratio relation with the lowest when .
- (2)
- Compared with complete fractional chaotic systems , we can always find lower in hybrid order systems.
4. Combination Synchronization of Hybrid Order Chaotic Systems
4.1. Combination Synchronization
4.2. Simulations
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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System | |||
---|---|---|---|
Integer order | 3 | 1 | 3 |
Equation (16) | 0 | 0.97 | 2.91 |
Equation (8) | 1 | 0.91 | 2.82 |
Equation (9) | 1 | 0.93 | 2.86 |
Equation (10) | 1 | 0.95 | 2.9 |
Equation (11) | 2 | 0.7 | 2.7 |
Equation (12) | 2 | 0.89 | 2.89 |
Equation (13) | 2 | 0.89 | 2.89 |
Type | Model | |||
---|---|---|---|---|
Complete | ||||
fractional | 0 | 0.78 | 2.34 | |
Chen | ||||
Hybrid | ||||
order | 1 | 0.68 | 2.36 | |
Chen | ||||
1 | 0.79 | 2.58 | ||
1 | 0.995 | 2.99 | ||
2 | 0.68 | 2.68 | ||
2 | 0.96 | 2.96 | ||
2 | 0.275 | 2.275 | ||
Type | Model | |||
---|---|---|---|---|
Complete | ||||
fractional | 0 | 0.78 | 2.34 | |
Lü | ||||
Hybrid | ||||
order | 1 | 0.653 | 2.306 | |
Lü | ||||
1 | 0.77 | 2.54 | ||
1 | 0.994 | 2.988 | ||
2 | 0.66 | 2.66 | ||
2 | 0.995 | 2.995 | ||
2 | 0.277 | 2.277 | ||
Type | Model | |||
---|---|---|---|---|
Fractional | ||||
complex | 0 | 0.959 | 2.877 | |
Lorenz | ||||
Hybrid | ||||
order | 1 | 0.908 | 2.816 | |
Lorenz | ||||
1 | 0.94 | 2.88 | ||
1 | 0.955 | 2.91 | ||
2 | 0.531 | 2.531 | ||
2 | 0.823 | 2.823 | ||
2 | 0.925 | 2.925 | ||
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Liu, J.; Wang, Z.; Zhang, F.; Yin, Y.; Ma, F. Special Characteristics and Synchronizations of Multi Hybrid-Order Chaotic Systems. Entropy 2020, 22, 664. https://doi.org/10.3390/e22060664
Liu J, Wang Z, Zhang F, Yin Y, Ma F. Special Characteristics and Synchronizations of Multi Hybrid-Order Chaotic Systems. Entropy. 2020; 22(6):664. https://doi.org/10.3390/e22060664
Chicago/Turabian StyleLiu, Jiaxun, Zuoxun Wang, Fangfang Zhang, Yankai Yin, and Fengying Ma. 2020. "Special Characteristics and Synchronizations of Multi Hybrid-Order Chaotic Systems" Entropy 22, no. 6: 664. https://doi.org/10.3390/e22060664