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Chaotic attractors that exist only in fractional-order case
1Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia; College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia.
Journal of Advanced Research
|February 27, 2023
Summary
This study reveals that chaotic dynamics can exist exclusively in fractional-order systems, not their integer-order counterparts, using specific parameters in Matouk's hyperchaotic system. This finding challenges the assumption of chaos transmission between different order systems.
Area of Science:
- Non-linear dynamics
- Chaos theory
- Fractional calculus
Background:
- Fractional and integer-order dynamical systems are crucial for understanding non-linear phenomena.
- Phase transitions between fractional and integer-order systems are a key research area.
Purpose of the Study:
- To report the existence of chaotic attractors exclusively in the fractional-order case of a new hyperchaotic system.
- To investigate phase transitions and synchronization in fractional-order systems.
Main Methods:
- Stability analysis of steady-state solutions.
- Analysis of hidden and self-excited chaotic attractors.
- Computation of basin of attraction, bifurcation diagrams, and Lyapunov exponent spectrum.
- Projective synchronization using non-linear controllers.
Main Results:
- Chaotic attractors were found to exist only in the fractional-order Matouk's hyperchaotic system for specific parameter values.
- The corresponding integer-order system exhibited quasi-periodic dynamics under identical conditions.
- Successful projective synchronization of hidden chaotic attractors was achieved.
Conclusions:
- This work provides the first example demonstrating that chaotic states are not always transmitted between fractional- and integer-order systems.
- The findings highlight the unique behavior of fractional-order systems.
- Chaos synchronization in fractional-order systems offers new avenues for technological applications.
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