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Averaged number of visits
N Haydn1, E Lunedei, S Vaienti
1Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113, USA. nhaydn@math.usc.edu
Chaos (Woodbury, N.Y.)
|October 2, 2007
Summary
We introduce the averaged number of visits, a new indicator for dynamical systems, to estimate visit frequency in small regions. This quantity reveals insights into ergodic and topological properties, including periodic points.
Area of Science:
- Dynamical Systems
- Ergodic Theory
- Chaos Theory
Background:
- Understanding the long-term behavior of dynamical systems is crucial.
- Estimating the frequency of a system's visits to specific regions is a key challenge.
- Existing methods may not fully capture the nuances of regional visitation patterns.
Purpose of the Study:
- Introduce a novel indicator, the averaged number of visits, for dynamical systems.
- Quantify the frequency of visits to small regions within a dynamical system.
- Analyze the relationship between this indicator and the system's fundamental properties.
Main Methods:
- Developed an analytical framework to compute the averaged number of visits.
- Employed numerical simulations to calculate the indicator for diverse systems.
- Investigated the dependence of the indicator on system parameters.
Main Results:
- The averaged number of visits was successfully computed for various dynamical systems.
- Demonstrated that the indicator is sensitive to the ergodic properties of the systems.
- Showed a correlation between the averaged number of visits and topological features like periodic points.
Conclusions:
- The averaged number of visits provides a valuable new metric for characterizing dynamical systems.
- This indicator offers a quantitative link between a system's dynamics and its topological structure.
- Future research can leverage this indicator to explore complex system behaviors.
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