Example dynamical systems

# LyapunovExponents.Examples.ExampleBase.LEDemo — Type.

LEDemo(example::LEExample; <keyword arguments>)

Here is an example code for constructing an example dynamical system, calculate its LEs and plot them:

using LyapunovExponents
using Plots
demo = solve!(LyapunovExponents.lorenz_63())
plot(demo)

Create a LEDemo holding an example and an appropriate LEProblem created from the example.

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# LyapunovExponents.Examples.ExampleBase.LEExample — Type.

A type to hold an example dynamical system and its known Lyapunov exponents.

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Discrete systems

# LyapunovExponents.Examples.HenonMap.henon_map — Function.

Return a LEDemo for the Hénon map.

• M. Hénon, Commun. Math. Phys. Phys. 50, 69-77 (1976)
• http://sprott.physics.wisc.edu/chaos/comchaos.htm
• https://en.wikipedia.org/wiki/H%C3%A9non_map

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# LyapunovExponents.Examples.LoziMap.lozi_map — Function.

Return a LEDemo for the Lozi map.

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# LyapunovExponents.Examples.StandardMap.standard_map — Function.

Return a LEDemo for the Chirikov standard map.

• B. V. Chirikov, Physics Reports 52, 263-379 (1979)
• http://sprott.physics.wisc.edu/chaos/comchaos.htm
• https://en.wikipedia.org/wiki/Standard_map
• http://www.scholarpedia.org/article/Chirikov_standard_map

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# LyapunovExponents.Examples.BakersMap.bakers_map — Function.

Baker's map

• https://en.wikipedia.org/wiki/Baker%27s_map

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# LyapunovExponents.Examples.ArnoldCatMap.arnold_cat_map — Function.

Return a LEDemo for the Arnold's cat map

• https://en.wikipedia.org/wiki/Arnold%27s_cat_map

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Continuous systems

# LyapunovExponents.Examples.Lorenz63.lorenz_63 — Function.

Return a LEDemo for the Lorenz system.

• https://en.wikipedia.org/wiki/Lorenz_system
• http://sprott.physics.wisc.edu/chaos/comchaos.htm
• E. N. Lorenz, J. Atmos. Sci. 20, 130-141 (1963)

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# LyapunovExponents.Examples.LinzSprott99.linz_sprott_99 — Function.

Return a LEDemo for the simplest piecewise linear dissipative chaotic flow.

• http://sprott.physics.wisc.edu/chaos/comchaos.htm
• S. J. Linz and J. C. Sprott, Phys. Lett. A 259, 240-245 (1999)

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# LyapunovExponents.Examples.VanDerPol.van_der_pol — Function.

Return a LEDemo for the van der Pol oscillator with periodic forcing.

.known_exponents are extracted from Figure 6 of Geist, Parlitz & Lauterborn (1990).

• http://scholarpedia.org/article/Van_der_Pol_oscillator
• https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
• van der Pol and van der Mark. “Frequency Demultiplication.” Nature 120, no. 3019 (September 1927): 363. https://doi.org/10.1038/120363a0.
• Parlitz, Ulrich, and Werner Lauterborn. “Period-Doubling Cascades and Devil’s Staircases of the Driven van Der Pol Oscillator.” Physical Review A 36, no. 3 (August 1, 1987): 1428–34. https://doi.org/10.1103/PhysRevA.36.1428. (Figure 10a)
• Geist, K., Parlitz, U., & Lauterborn, W. (1990). Comparison of Different Methods for Computing Lyapunov Exponents. Progress of Theoretical Physics, 83, 875–893. https://doi.org/10.1143/PTP.83.875. (Figure 6)

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# LyapunovExponents.Examples.RNN.beer_95 — Function.

Return a LEDemo for a low-dimensional chaotic continuous-time recurrent neural networks by Beer (1995).

• Beer, R. D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3(4), 469–509. https://doi.org/10.1177/105971239500300405. (Figure 9D)

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