Using Soft Computing and Chaos Theory in investigating the Deformed Stadium
محورهای موضوعی : نظریه آشوب
1 - Energy and Environment Research Center, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran
کلید واژه: Chaos, Billiard, Poincaré map, Cross section, Birkhoff mapping,
چکیده مقاله :
This paper analyzed the dynamic system of billiards from a classic perspective. For this purpose, mapping and cross-section methods were first employed to study the behavior of this system and the results indicated that it was a chaotic one. Then a deformed stadium was introduced and its long-term behavior was analyzed. Considering changes in the behavior of this system following the slightest deformation at the boundaries, Poincaré map was used to demonstrate the occurrence of regular and irregular motions, indicating the completely chaotic behavior of the system. The shape of the cross-section of the regular motion shows that the points of contact with the boundary are located on a line in the phase space. On the other hand, the cross-sectional surface of a chaotic motion, the surface is covered with collision points and the empty spaces are surrounded by invariant curves. These spaces are also filled in case of and they eventually disappear and the surface is covered with collision points, completely. This behavior is characteristic of chaotic systems.
This paper analyzed the dynamic system of billiards from a classic perspective. For this purpose, mapping and cross-section methods were first employed to study the behavior of this system and the results indicated that it was a chaotic one. Then a deformed stadium was introduced and its long-term behavior was analyzed. Considering changes in the behavior of this system following the slightest deformation at the boundaries, Poincaré map was used to demonstrate the occurrence of regular and irregular motions, indicating the completely chaotic behavior of the system. The shape of the cross-section of the regular motion shows that the points of contact with the boundary are located on a line in the phase space. On the other hand, the cross-sectional surface of a chaotic motion, the surface is covered with collision points and the empty spaces are surrounded by invariant curves. These spaces are also filled in case of and they eventually disappear and the surface is covered with collision points, completely. This behavior is characteristic of chaotic systems.
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