Computing Poincaré map for One Parameter Pulsed Dipole System Using Hénon Trick
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Hénon trick, Hénon method, Poincaré map, Dipole system, Lorenz system
Abstract
For a trajectory generated by dynamical systems, Hénon has presented a method called the Hénon trick or Hénon method. In this method, a surface of a section (Poincaré surface) is defined, and the Poincaré map (i.e., the trajectory points distributed on it) is collected when the trajectory crosses the Poincaré surface. Whenever the Hénon trick is used to calculate the Poincaré map, the autonomous chaotic system's trajectory deviates from the original path, causing a deformation in its attractor. In this paper, the Hénon trick is discussed to calculate the Poincaré map for the attractor Lorenz system, after which a 1-parameter pulsed dipole (source-sink pairs) model is defined on an unbounded domain, and a Python data science code is built to plot the results. The paper provided a reformed Hénon trick to calculate the Poincaré map for a 1-parameter pulsed dipole model by defining a cross-section (Poincaré surface), then I calculate the Poincaré map of the intersection points between this cross-section and the streamlines generated by that pulsed dipole model. The Poincaré map is important to investigate the uniformity of the distribution of streamlines generated by the pulsed dipole system.
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References
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