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In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.
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In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction. In modern practice they are often replaced by bred vectors for this purpose.
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In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of papers, and nowadays various different approaches to the definition of Lyapunov dimension are used. Remark that the attractors with noninteger Hausdorff dimension are called strange attractors. Since the direct numerical computation of the Hausdorff dimension of attractors is often a problem of high numerical complexity, estimations via the Lyapunov dimension became widely spread. The Lyapunov dimension was named after the Russian mathematician Aleksandr Lyapunov because of the close connection with the Lyapunov exponents.
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Antonio Giorgilli is an Italian mathematical physicist, known for his work on the perturbative theory of Hamiltonian systems with applications to studies of orbital stability for major and minor planets.
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References
- ↑ Bezruchko, Boris P.; Smirnov, Dmitry A. (5 September 2010). Extracting Knowledge from Time Series: An Introduction to Nonlinear Empirical Modeling. Springer. pp. 56–57. ISBN 9783642126000.
- 1 2 Pierre Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, 2005. p. 7
- ↑ Friedland, G.; Metere, A. (2018). "Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity". arXiv: 1706.08638 .
{{cite journal}}: Cite journal requires|journal=(help) - ↑ Tancredi, G.; Sánchez, A.; Roig, F. (2001). "A Comparison Between Methods to Compute Lyapunov Exponents". The Astronomical Journal. 121 (2): 1171–1179. Bibcode:2001AJ....121.1171T. doi: 10.1086/318732 .
- ↑ Gerlach, E. (2009). "On the Numerical Computability of Asteroidal Lyapunov Times". arXiv: 0901.4871 .
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