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Fractal structures of normal and anomalous diffusion in nonlinear nonhyperbolic dynamical systems
1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany.
Physical Review Letters
|November 22, 2002
Summary
The smooth nonlinear climbing sine map creates fractal patterns in diffusion. These patterns reveal self-similar structures linked to the system's nonlinear dynamics.
Area of Science:
- Nonlinear dynamics
- Statistical mechanics
- Chaos theory
Background:
- Deterministic diffusion is a key concept in understanding complex systems.
- Nonhyperbolic dynamical systems offer unique insights into diffusion processes.
- The smooth nonlinear climbing sine map is a known model for deterministic diffusion.
Purpose of the Study:
- To investigate the fractal nature of diffusion generated by the smooth nonlinear climbing sine map.
- To analyze the relationship between control parameters and the emergence of diffusive regions.
- To connect fractal structures to the underlying nonlinear microscopic dynamics.
Main Methods:
- Analysis of the smooth nonlinear climbing sine map.
- Identification and characterization of normal and anomalous diffusive regions.
- Application of the Green-Kubo formula.
- Development of fractal Takagi-like functions.
Main Results:
- The map generates fractal hierarchies of normal and anomalous diffusive regions.
- These self-similar sets are parameter-dependent and have positive measure.
- A fractal diffusion coefficient was observed in regions of normal diffusion.
- Fractal structures were linked to nonlinear dynamics via fractal Takagi-like functions.
Conclusions:
- The smooth nonlinear climbing sine map exhibits complex fractal behavior in its diffusion properties.
- The study establishes a link between macroscopic diffusion patterns and microscopic nonlinear dynamics.
- Fractal analysis provides a powerful tool for characterizing deterministic diffusion in dynamical systems.
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