Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations
In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and stu...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-02-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/2/478 |
| _version_ | 1797618045079781376 |
|---|---|
| author | Anita Tomar Vipul Kumar Udhamvir Singh Rana Mohammad Sajid |
| author_facet | Anita Tomar Vipul Kumar Udhamvir Singh Rana Mohammad Sajid |
| author_sort | Anita Tomar |
| collection | DOAJ |
| description | In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and study the consequences of underlying parameters on the variation of dynamics, colour, time of generation, and shape of generated fractals. The black points in the obtained fractals are the “non-chaotic” points and the dynamical behaviour in the black area is easily predictable. The coloured points are the points that “escape”, that is, they tend to infinity under one of iterative methods based on a four-step fixed-point iteration scheme extended with s-convexity. The different colours tell us how quickly a point escapes. The order of escaping of coloured points is red, orange, yellow, green, blue, and violet, that is, the red point is the fastest to escape while the violet point is the slowest to escape. Mostly, these generated fractals have symmetry. The Julia set, where we find all of the chaotic behaviour for the dynamical system, marks the boundary between these two categories of behaviour points. The Mandelbrot set, which was originally observed in 1980 by Benoit Mandelbrot and is particularly important in dynamics, is the collection of all feasible Julia sets. It perfectly sums up the Julia sets. |
| first_indexed | 2024-03-11T08:04:38Z |
| format | Article |
| id | doaj.art-2c4eb994aaa7495ea4fefd1ca429cd92 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T08:04:38Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2c4eb994aaa7495ea4fefd1ca429cd922023-11-16T23:33:51ZengMDPI AGSymmetry2073-89942023-02-0115247810.3390/sym15020478Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point IterationsAnita Tomar0Vipul Kumar1Udhamvir Singh Rana2Mohammad Sajid3Pt. Lalit Mohan Sharma Campus, Sridev Suman Uttarakhand University, Rishikesh 249201, Uttarakhand, IndiaD.A.V. College, Dehradun 248001, Uttarakhand, IndiaD.A.V. College, Dehradun 248001, Uttarakhand, IndiaDepartment of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi ArabiaIn this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and study the consequences of underlying parameters on the variation of dynamics, colour, time of generation, and shape of generated fractals. The black points in the obtained fractals are the “non-chaotic” points and the dynamical behaviour in the black area is easily predictable. The coloured points are the points that “escape”, that is, they tend to infinity under one of iterative methods based on a four-step fixed-point iteration scheme extended with s-convexity. The different colours tell us how quickly a point escapes. The order of escaping of coloured points is red, orange, yellow, green, blue, and violet, that is, the red point is the fastest to escape while the violet point is the slowest to escape. Mostly, these generated fractals have symmetry. The Julia set, where we find all of the chaotic behaviour for the dynamical system, marks the boundary between these two categories of behaviour points. The Mandelbrot set, which was originally observed in 1980 by Benoit Mandelbrot and is particularly important in dynamics, is the collection of all feasible Julia sets. It perfectly sums up the Julia sets.https://www.mdpi.com/2073-8994/15/2/478chaotic behaviourconvexityescape criterionescape radiifour-step fixed-point iterationiterative methods |
| spellingShingle | Anita Tomar Vipul Kumar Udhamvir Singh Rana Mohammad Sajid Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations Symmetry chaotic behaviour convexity escape criterion escape radii four-step fixed-point iteration iterative methods |
| title | Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations |
| title_full | Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations |
| title_fullStr | Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations |
| title_full_unstemmed | Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations |
| title_short | Fractals as Julia and Mandelbrot Sets of Complex Cosine Functions via Fixed Point Iterations |
| title_sort | fractals as julia and mandelbrot sets of complex cosine functions via fixed point iterations |
| topic | chaotic behaviour convexity escape criterion escape radii four-step fixed-point iteration iterative methods |
| url | https://www.mdpi.com/2073-8994/15/2/478 |
| work_keys_str_mv | AT anitatomar fractalsasjuliaandmandelbrotsetsofcomplexcosinefunctionsviafixedpointiterations AT vipulkumar fractalsasjuliaandmandelbrotsetsofcomplexcosinefunctionsviafixedpointiterations AT udhamvirsinghrana fractalsasjuliaandmandelbrotsetsofcomplexcosinefunctionsviafixedpointiterations AT mohammadsajid fractalsasjuliaandmandelbrotsetsofcomplexcosinefunctionsviafixedpointiterations |