# Fractional Generalization of Gradient Systems

```@article{Tarasov2005FractionalGO,
author={Vasily E. Tarasov},
journal={Letters in Mathematical Physics},
year={2005},
volume={73},
pages={49-58}
}```
• V. E. Tarasov
• Published 1 July 2005
• Mathematics
• Letters in Mathematical Physics
We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems.
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## References

SHOWING 1-10 OF 31 REFERENCES
Fractional generalization of Liouville equations.
Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition, which is considered as anormalization condition for systems in fractional phase space.
Fractional systems and fractional Bogoliubov hierarchy equations.
• V. E. Tarasov
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2005
This work considers the fractional generalizations of the phase volume, volume element, and Poisson brackets, and fractional analogs of the Hamilton equations, which lead to an analog of thephase space and systems on this fractional phase space.
Fractional Integrals and Derivatives: Theory and Applications
• Mathematics
• 1993
Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of
Deterministic nonperiodic flow
Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with
Fractional differential forms
• Mathematics
• 2001
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined.
Fractional Fokker-Planck equation for fractal media.
It is proved that the fractional integrals can be used to describe the media with noninteger mass dimensions and the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives are derived.