On escape criterion of an orbit with s-convexity and illustrations of the behavior shifts in Mandelbrot and Julia set fractals
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View abstract on PubMed
Summary
This summary is machine-generated.This study introduces a new fractal orbit with s-convexity to analyze behavior shifts. It establishes an escape criterion for transcendental cosine functions and reveals connections between Mandelbrot and Julia sets.
Area Of Science
- Fractal Geometry
- Complex Dynamics
- Mathematical Analysis
Background
- Fractal geometry is essential for modeling complex natural structures.
- Understanding the behavior of transcendental functions is crucial in dynamical systems.
Purpose Of The Study
- To introduce a novel orbit with s-convexity for fractal behavior analysis.
- To establish an escape criterion for a specific class of transcendental cosine functions.
- To illustrate the influence of parameters on fractal formation and explore the relationship between Mandelbrot and Julia sets.
Main Methods
- Development of a theorem for the escape criterion of transcendental cosine functions Tα,β(u) = cos(um)+αu + β.
- Utilizing MATHEMATICA software for numerical examples and graphical illustrations.
- Algorithmic and colormap approaches to visualize fractal properties.
Main Results
- A novel orbit with s-convexity was presented, aiding in the illustration of fractal behavior shifts.
- An escape criterion theorem was proven for the defined transcendental cosine functions.
- Parameter impacts on fractal formatting were demonstrated, revealing that widening the Mandelbrot set at petal edges generates Julia sets.
Conclusions
- The study provides a new framework for analyzing fractal dynamics using s-convexity.
- The escape criterion offers insights into the stability and behavior of transcendental cosine functions.
- A significant observation is the inherent presence of Julia set data within Mandelbrot set points, deepening our understanding of their relationship.
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