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Researchers combined two chaotic attractors using a composite operator to create new dynamics. This study introduces a novel numerical method for solving the resulting fractional integro-differential equations, demonstrating high accuracy.

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Dynamical Systems

Background:

  • The composite operator, a concept from functional analysis, has real-world applications.
  • Modeling chaotic phenomena and natural processes often involves advanced mathematical techniques like fractional calculus.
  • Composing equations, analogous to composing functions, can potentially lead to novel dynamics and problem-solving approaches.

Purpose of the Study:

  • To explore the creation of new chaotic attractors by composing existing ones using the composite operator.
  • To investigate the resulting linear integro-differential equations, including classical and fractional forms in the Caputo-Fabrizio sense.
  • To develop and validate a new numerical scheme for solving these novel equations.

Main Methods:

  • Application of the composite operator to combine two distinct chaotic attractors.
  • Derivation of linear integro-differential equations (classical and fractional) in the Caputo-Fabrizio sense.
  • Development of a numerical scheme employing finite difference, Simpson, and Lagrange polynomial approximations.
  • Validation of the numerical scheme through comparison with exact solutions for specific examples.

Main Results:

  • Successful generation of new chaotic attractors through the composition of existing ones.
  • Formulation of novel fractional integro-differential equations.
  • Demonstration of a highly accurate numerical scheme with errors on the order of 10-4.
  • Validation of the proposed numerical method against exact solutions.

Conclusions:

  • The composite operator provides a novel method for generating new chaotic attractors.
  • The proposed numerical scheme accurately solves the newly derived fractional integro-differential equations.
  • This approach offers a promising tool for modeling complex phenomena and solving real-world problems with enhanced dynamics.