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Studying changes in the dynamical patterns in two physical models involving new Caputo operator.

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.

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Summary

A novel Caputo fractional operator introduces new chaotic dynamics in physical models. Adjusting its parameter generates complex attractors, unlike classic fractional operators which yield non-chaotic states.

Keywords:

Chaos Generalization of Gamma function Liu model New Caputo fractional operator Van der Pol-Duffing model

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Area of Science:

Nonlinear Dynamics
Fractional Calculus
Physical Systems Modeling

Background:

Understanding phase transitions in physical models is crucial for describing system dynamics.
Fractional calculus offers advanced tools for modeling complex phenomena.

Purpose of the Study:

Investigate the impact of a new Caputo fractional operator on the dynamics of two physical models.
Explore the influence of a tunable fractional parameter on dynamical patterns.

Main Methods:

Introduced a new Caputo fractional operator and its associated Riemann-Liouville fractional integral.
Proved key properties and lemmas for the new fractional operator and integral.
Analyzed the fractional modified autonomous Van der Pol-Duffing and Liu systems.

Main Results:

The new fractional operator, with its additional parameter, induced diverse chaotic attractors.
The fractional Van der Pol-Duffing system displayed one-band, double-band chaos, and periodic cycles.
The fractional Liu system exhibited double scroll, self-excited, and hidden chaotic attractors.

Conclusions:

The new Caputo fractional operator effectively generates complex chaotic dynamics in physical models.
The operator's parameter is key to producing chaotic attractors, contrasting with classic operators.