A numerical solution to the shown equations with given parameters leads to the shown paths, and it is known as the Coullet Attractor.
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This nonlinear system with specific initial conditions is solved
numerically and the resulting trajectory is shown through a 3 dimensional animation.
[Ref: www.3d-meier.de...]
Initial condition 1: (0.1, 0.41, 0.31)
Initial condition 2: (0.2, 0.51, 0.41)
Time step: 0.01
"In a chaotic system, the trajectory moves around on the attractor as time goes on, but two
nearby points separate exponentially so that eventually they are very far apart. Although their
future is determined uniquely and precisely by the governing equations, very small differences
in the starting point can make large differences in the future conditions. Although tomorrow’s
weather depends on the conditions today, and the weather the day after tomorrow depends on
the conditions tomorrow, small errors in measuring the current weather eventually grow until
all hope of predictability is lost - the ‘butterfly effect.’ "
[Ref: SPROTT / lorenz EC JOURNAL . Winter 2008 sprott.physics....]
Since Lorenz found the first chaotic attractor in a smooth three-dimensional autonomous system, later chaotic attractors were developed, for example the Rossler system, the Sprott system, the Chen system, the Lu system, the generalized Lorenz system family, and the hyperbolic type of the generalized Lorenz canonical form. Here one of such attractor is shown in this video.
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#Coullet|#ChaoticSystem #ButterflyEffect| thinkeccel
27 сен 2024
