Fractals are widely studied for their ability to model complex phenomena, and Julia sets are prominent fractal structures in complex dynamics. These sets are defined as the closure of repelling periodic points of rational maps. For a rational map R(z)=P(z)Q(z), where P and Q are polynomials with deg(P)>deg(Q), infinity acts as a fixed point, and the Julia set is the boundary of the basin of attraction of infinity. The system zn+1=12(zn+λzqn), studied here, introduces poles at the origin, transforming the dynamics significantly.

A rational map R(z)=P(z)Q(z)​,whereP(z)andQ(z) are polynomials, forms the basis for studying Julia sets. The degree of R(z) is defined as max(deg(P),deg(Q)). Julia sets are fractal structures and are defined as:

J(R)={z∈C∣the family {Rn(z)}n≥1 is not normal in the sense of Montel}.

Equivalently, J(R) is the closure of the set of repelling periodic points of R(z). For R(z), if |z|→∞, then |R(z)|→∞, making infinity a fixed point. The Julia set J(R) forms the boundary of the basin of attraction of this fixed point.

A Julia set J(R) is a fractal that serves as the boundary between:

1. Points in the complex plane that exhibit stable, predictable dynamics under a rational map R(z).

2. Points that diverge to infinity or exhibit chaotic behavior.

In a physical context:

• Julia sets often represent systems that alternate between stable and chaotic regimes.

• They serve as models for phase transitions, where the boundary represents critical points between phases.

• In fields like optics, Julia sets help describe patterns in nonlinear systems, such as wave front distortions or light propagation through media

1. Perturbed Rational Map:

   zn+1=zqn+c+λzpn...(1)

   where:

   • q controls the degree of the polynomial term.

   • λzpn​ introduces a singularity (pole) at the origin.

2. Impact of λ:

   • For λ=0, the map reduces to the classical form zn+1=zqn+c, a polynomial.

   • For λ≠0, the pole dramatically changes the system’s dynamics, increasing the map’s degree to p+q and altering the structure of Julia and Mandelbrot sets.

3. Special Cases:

   • When c=0, the study explores how adding a pole affects the Julia set, transforming it from the unit circle (for λ=0 ) to a more complex fractal structure.

The complex perturbed rational map studied is zn+1=12(zn+λzqn), where q≥1, λ∈C, and z∈C. Key features include:

• zn+1​ contains poles at the origin, altering the dynamics of classical Julia sets.

• When λ=0, the system reduces to a polynomial map of degree q+1.

This map generalizes rational maps R(z)=P(z)Q(z), where P(z) and Q(z) are polynomials, by introducing a pole at the origin. This perturbation significantly alters the dynamics and the structure of the Julia sets associated with the map.

In rational maps, fixed points satisfy R(z)=z. The stability of these fixed points depends on the derivative R′(z):

• Attracting: R′(z)<1,

• Repelling: R′(z)>1,

• Neutral: R′(z)=1.

In this work, infinity acts as a fixed point, and its behavior is analyzed through escape criteria and control mechanisms.

A control parameter k is introduced to modify the dynamics of the map. The controlled map is defined as:

g(z)=f(z)−k⋅(f(z)−z)

Interpretation:

1. Base Map Contribution: The term f(z) represents the original rational Julia map.

2. Feedback Control: The term k⋅(f(z)−z) acts as a damping factor. The control parameter k (with 0≤k≤1) adjusts the extent to which the new value depends on the previous iteration.

When k=0, g(z)=f(z), recovering the original Julia map. When k=1, the map becomes stationary.

The base system is:

zn+1=fλ,q(zn)=12(zn+λzqn)...(2)

which can also be written as:

fλ,q(z)=zq+1+λ2zq.

This is a rational map where:

• The numerator P(z)=zq+1+λ,

• The denominator Q(z)=2zq.

To control the dynamics, a nonlinear control term un is introduced:

un=−k[fλ,q(zn)−zn].

The controlled system becomes:

zn+1=fλ,q(zn)+un

or equivalently:

zn+1=fλ,q(zn)−k[fλ,q(zn)−zn]

Substitute fλ,q(zn)=12(zn+λzqn) into the controlled system:

zn+1=12⋅(zn+λzqn)−k⋅[12⋅(zn+λzqn)−zn]...(3)

This is a concise symbolic representation called controlled rational map, and we define this following way:

R(z)=(1+k)zq+1n+λ(1−k)2zqn

• Fixed Points: The fixed points are solutions of R(z)=z, which reduce to solving:

  (1+k)zq+1+(1−k)λ2zq=z

  Rearrange:

  (1+k)zq+1+(1−k)λ=2zq+1

  Factorize:

  (1+k−2)zq+1=−(1−k)λ

• Attracting Fixed Point: If ∣1+k∣>2, infinity becomes an attracting fixed point.

That’s how we can write:

|R(z)|>|z|for|z|>M

where M is a threshold beyond which iterates escape to infinity. Using bounds, the escape criterion simplifies to:

|R(z)|=|(1+k)zq+1+(1−k)λ2zq|>|z|

• The control parameter k influences the structure of the Julia set.

• As ∣1+k∣>2, infinity becomes an attracting fixed point, shrinking the Julia set.

• The escape criterion ensures iterates either escape to infinity or remain bounded within the Julia set boundary.

1. Iterative Function:

   zn+1=zqn+c+λzpn​where:

   • p,q∈Z+, are positive integers.

   • λ,c∈C are complex parameters.

   • zn∈C is the iterated sequence.

2. Boundedness Criterion:

   • A point z0​ in the complex plane belongs to the filled Julia set if its orbit {z0,z1,z2,…} remains bounded under iteration.

   • For practical purposes, we compute boundedness using an escape radius R:

     • If |zn|>R, the point escapes to infinity.

     • Otherwise, it is bounded.

3. Key Parameters:

   • q: Controls the growth of the polynomial term.

   • λ: Perturbation parameter that introduces singularities at the origin.

   • p: Governs the strength of the singularity.

   • c: Determines the central point of the hyperbolic component in the Mandelbrot set.

Steps:

1. Initialize Grid:

   • Define a grid of points in the complex plane z0​, e.g., x,y∈[−2,2].

2. Iterative Mapping:

   • For each z0​, compute its orbit:

     zn+1=zqn+c+λzpn

3. Escape-Time Calculation:

   • Track the number of iterations n required for |zn| to exceed a given escape radius R.

   • Assign a colour to z0​ based on n (e.g., gradient or palette).

4. Edge Cases:

   • Points where zn​ diverges rapidly (|zn|→∞) are coloured differently.

   • Points that remain bounded ( |zn|≤R after the maximum iteration) are typically coloured black (belong to the Julia set).

This phenomenon arises in the study of coupled systems, particularly in complex dynamics, where multiple Julia sets, associated with different parameters, exhibit coordinated behavior. Synchronization involves aligning the trajectories and fractal properties of two systems:

Driving System:
wn+1=12(wn+μwpn)with parametersp,μ....(4)

Response System:
zn+1=R(zn)−k[R(zn)−R(wn)]

Synchronization is achieved when the trajectory difference: |zn+1−wn+1|→0,ask→1.

Why Synchronization Works

Synchronization is feasible because:

• Both systems originate from the same functional framework.

• Differences are limited to parameter variations (λ , q).

• The coupling mechanism effectively bridges the gap created by these parameter differences, aligning the trajectories and fractal properties.

The equation provided is:

zn+1=12(zn+λzqn)−k[12(zn+λzqn)−12(wn+μwpn)]...(5)

where:

• zn: Represents the primary iterated complex variable.

• wn: Represents a secondary complex variable interacting with zn​.

• λ: A complex parameter controlling the rational map dynamics.

• μ: Another complex parameter that modifies the wn​ dependent term.

• p,q: Positive integers controlling the degree of the respective rational maps.

• k: A feedback control parameter.

a modified version of a Julia set generated by introducing a control mechanism into the dynamics of a rational map.

This mechanism alters the behavior of the system, enabling adjustments to its fractal properties based on a coupling or control parameter k. Let’s break down this system and analyze its mathematical structure:

1. Base Map Contribution:

   12(zn+λzqn)

   forms the unperturbed Julia set, as seen in System (2).

2. Secondary Influence:

   12(wn+μwpn)

   introduces a coupling between the zn​ and wn​ dynamics, with the parameter μ adjusting this interaction.

3. Control Mechanism: The term:

   −k[12(zn+λzqn)−12(wn+μwpn)]

   applies feedback that synchronizes the zn and wn based maps, creating a controlled fractal structure. The control parameter k determines the extent of this feedback.

This system describes a coupled rational map with feedback control, where:

1. Coupling: The term involving wn and μ/wpn introduces a synchronized perturbation to the primary Julia set, ensuring dynamic interactions between zn and wn.

2. Feedback: The parameter k acts as a regulator, controlling the influence of the coupling term and modifying the convergence or divergence of points in the fractal.