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If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
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Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area...
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Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...
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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half...
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Dynamical properties of the composed Logistic-Gauss map.

Luam Silva de Paiva1, Julia G S Rocha1, Joelson D V Hermes2,3

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This study introduces the Logistic-Gauss map, revealing complex sets of periodicity (CSP) within its parameter space. Findings highlight the organization of these structures through extreme orbits, enhancing our understanding of chaotic dynamics.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Dissipative mappings exhibit complex behaviors, including periodic and chaotic regions.
  • Understanding the parameter space is crucial for characterizing these dynamics.
  • Extreme orbits offer insights into the structural organization of one-dimensional maps.

Purpose of the Study:

  • To analyze the unique composition of the Logistic-Gauss map.
  • To explore its parameter space for complex periodic structures and chaotic regions.
  • To investigate the role of extreme orbits in understanding system behavior.

Main Methods:

  • Composition of the Logistic-Gauss map.
  • Exploration of the parameter space by manipulating control parameters.
  • Identification and analysis of extreme orbits and Complex Sets of Periodicity (CSP).

Main Results:

  • Discovery of Complex Sets of Periodicity (CSP) within the parameter space.
  • Identification of superstable curves characterizing CSP structures.
  • Observation of cascades of CSP structures with added periods, organized by extreme curves.

Conclusions:

  • The Logistic-Gauss map exhibits intricate dynamics with organized periodic structures.
  • Extreme orbits are key to understanding the structural organization and system behavior.
  • This research expands the understanding of chaos and periodicity in dissipative mappings.