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Error distribution in randomly perturbed orbits.
Ph Marie1, G Turchetti, S Vaienti
1Centre de Physique Theorique, CNRS, UMR-6207, Universites d'Aix-Marseille I, II, Provence-Alpes-Cote-d'Azur 13284, France. pmarie@cpt.univ-mrs.fr
Chaos (Woodbury, N.Y.)
|January 12, 2010
Summary
This study models numerical noise in dynamical systems by analyzing the convergence of errors. It shows that errors in computed observable values stabilize over time, forming an asymptotic error distribution.
Area of Science:
- Dynamical Systems
- Numerical Analysis
- Chaos Theory
Background:
- Dynamical systems are often studied using discrete time steps.
- Numerical computations can introduce errors, particularly roundoff errors, affecting long-term predictions.
- Understanding the impact of these errors is crucial for accurate modeling.
Purpose of the Study:
- To model the effects of numerical noise, specifically roundoff error, on dynamical systems.
- To analyze the convergence of errors in computed observable values over time.
- To characterize the distribution of the resulting asymptotic error.
Main Methods:
- Defining an observable on the phase space of a dynamical system.
- Comparing exact map computations with computations using randomized neighborhood maps.
- Analyzing the distribution of the difference (error) between exact and randomized computations.
- Investigating the asymptotic behavior of this error as time steps increase.
- Employing rigorous mathematical analysis and numerical simulations for perturbed systems.
Main Results:
- The error variable, dependent on initial conditions and random realizations, converges in distribution to an asymptotic error.
- The study investigates the density of the asymptotic error distribution for systems with additive noise.
- Rigorous results are presented for some systems, while others undergo numerical investigation.
Conclusions:
- The asymptotic error distribution provides a model for the impact of numerical roundoff noise.
- This framework helps in understanding the long-term predictability of dynamical systems under computational constraints.
- The findings are applicable to various scientific fields relying on numerical simulations of dynamic processes.
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