The ideal model for a conservative system is the pendulum devoid of friction, which in theory maintains an invariable movement and a constant energy.
This type of system may be represented by equations integrable without restrictions and its past and future behavior can be completely computed.
The theoretical simplification thus obtained is useful for the prediction of the behavior of those concrete systems whose energy is nearly constant and are controlled by only one dominant variable, as for example the planets in the solar system.
Perfect conservative systems (which do not exist in the real world), having a constant energy, would not be submitted to the 2d. Principle of thermodynamics, and be perfectly reversible and independent from time and, thus, have no attractor.
"Their evolution does not provoque any contraction of the areas in the phases space. "(P. BERGÉ et al., 1984, p.43).
L. BRILLOUIN did consider this question in depth, starting from POINCARÉ's ideas, in Chapter VI of his book "Vie, Matiere et Observation", under the heading: "POINCARÉ: l'Energétique et le déterminisme" (1959, p.195-217). See also his comments on the related LIOUVILLE theorem (Ibid, p.181-191).
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Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]
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