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When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals. [ABSTRACT FROM AUTHOR]

The majority of fractals' dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of "escape" is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine T c (z) = a s i n (z r) + b z + c and complex exponential T c (z) = a e z r + b z + c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters 'a' and 'b', and the parameters involved in the series expansion of the sine and exponential functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function h(z) = z2 + l. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented. [ABSTRACT FROM AUTHOR]

This work focuses on a kind of fractals Parrondo's paradoxial phenomenon "deiconnected+diconnected=connected" in an alternated superior complex system zn+1 = β b(zn² + ci) + (1-β) zn, i = 1, 2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. [ABSTRACT FROM AUTHOR]

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods. [ABSTRACT FROM AUTHOR]

In this paper, we define (p , q) -cosine and sine sigmoid polynomials. Based on this, the properties of each polynomial, and the structure and assumptions of its roots, can be identified. Properties can also be determined by the changes in p and q. [ABSTRACT FROM AUTHOR]