# Exploring the Duffing Equation: Numerical Analysis, Discrete Dynamics, and
Ecological Modeling

Zeraoulia Rafik1, Zeraoulia Chaima2
1Department of Mathematics, University of batna2.Algeria
2Department of mathematics, University of Abbes Laghrour Khenchela , Algeria
Corresponding author<EMAIL_ADDRESS>
###### Abstract

This research paper delves into the dynamics of a novel ecology model that
describes the intricate interplay between radioactivity and cancer cases.
Organized into three main sections, the study covers numerical analysis using
advanced techniques, the analysis of discrete dynamical systems, and explores
the potential applications of the model in the context of cancer research and
ecological systems. By conducting a comprehensive bifurcation analysis, we
unveil the model’s sensitivity to external factors, interactive behaviors
between radioactivity and cancer cases, and the emergence of multiple
attractors. These findings open up new avenues for understanding cancer
dynamics, ecological systems, and clinical scenarios, where even subtle
variations in external factors can lead to significant variations in cancer
incidence. The insights gained from this study have promising implications for
both theoretical and applied research in cancer epidemiology and ecology,
providing a deeper understanding of the complex dynamics underlying these
systems. In our research, we present a new ecology model that reveals the
intricate and often nonlinear relationship between radioactivity and cancer
incidence, offering novel insights into cancer dynamics and environmental
influences.

## 1 Introduction

The Duffing equation, a fundamental and intriguing non-linear differential
equation, has captured the interest of researchers across various scientific
disciplines for its rich dynamics and versatile applications [17]. Named after
the German engineer and physicist Georg Duffing, this equation stands as a
cornerstone in the study of nonlinear systems and has found its place in
diverse fields, ranging from mechanical engineering and physics to
mathematics.[1]

The Duffing equation is defined as:

$\ddot{x}+\delta\dot{x}+\alpha x+\beta x^{3}=\gamma\cos(\omega t)$ (1)

where $x(t)$ represents the displacement of a vibrating system at time $t$,
$\alpha$, $\beta$, $\delta$, and $\gamma$ are parameters governing the
system’s behavior, and $\omega$ is the angular frequency of an external
periodic force. Despite its seemingly simple form, this equation embodies a
wealth of complex phenomena, from regular periodic motion to chaotic behavior.
[8] Understanding its intricacies and implications has been a driving force
behind extensive research efforts.

The historical significance of the Duffing equation is noteworthy. It emerged
as a mathematical model in the early 20th century to describe the nonlinear
behavior of mechanical systems, such as vibrating beams and pendulums. Its
applicability extends to diverse fields, including electrical circuits and
ecological modeling. In recent years, the Duffing equation has gained
prominence in ecology, where it serves as a powerful tool for understanding
the dynamics of ecosystems [25].

The objectives of this research paper are threefold. Firstly, we embark on a
comprehensive study of the numerical aspects of the Duffing equation,
exploring various methods for solving this challenging non-linear differential
equation. We build upon notable contributions in this area, including the work
of Ott, Sauer, and Yorke, who introduced the concept of Lyapunov exponents to
analyze chaotic behavior in dynamical systems, a technique that has found
extensive application in the evolution of cancers due to radioactivity [12].

Secondly, we delve into the Duffing equation as a discrete dynamical system in
the second section of this paper. Pioneering research by Feigenbaum [22]
uncovered the universal constants governing the period-doubling route to chaos
in dynamical systems. These insights are increasingly relevant in ecological
contexts, where bifurcations and chaos play a vital role in understanding
ecosystem behavior [13].

Lastly, in the third section, we explore the implications of the Duffing
equation in ecology. Recent research has highlighted its applicability in
modeling complex ecological systems and understanding the influence of
external factors on ecosystem dynamics. In the context of ecology, the Duffing
equation provides a unique perspective on the complex interplay between
species, resources, and environmental variables in the evolution of cancers
due to radioactivity [25].

In summary, this research paper embarks on a multifaceted exploration of the
Duffing equation, from its numerical solutions to its chaotic dynamics and its
applications in the evolution of cancers due to radioactivity [6]. By shedding
light on the multifaceted nature of this equation, we aim to deepen our
understanding of nonlinear systems and offer valuable insights for both
theoretical and applied research. The Duffing equation, with its intricate
dynamics and diverse applications, continues to inspire and challenge
scientists in the field of ecology and nonlinear dynamics.

### 1.1 Background

The Duffing equation is a classical nonlinear second-order differential
equation with various applications in physics, engineering, and mathematics.
It has been studied extensively due to its rich dynamics, including chaotic
behavior and complex solutions. Understanding the Duffing equation’s behavior
is of significance in a wide range of scientific disciplines.[5]

### 1.2 Objective

This paper aims to comprehensively explore the Duffing equation’s dynamics,
discrete behavior, and ecological application. The primary objectives and the
structure of the paper are as follows:

  * •

Numerical Analysis: We investigate the numerical aspects of the Duffing
equation, including the use of the Homotopy method for approximate solutions
and the analysis of convergence and error bounds.

  * •

Discrete Dynamics: We convert the continuous Duffing equation into a discrete
dynamical system and analyze its discrete-time behavior. Key components of the
numerical solution methodology are presented.[26]

  * •

Ecological Application: The Duffing quintic equation is applied to ecology,
highlighting the interactive relationship between radioactivity and cancer
incidence, especially under varying external conditions.[7]

The paper is structured as follows, with each section dedicated to one of
these objectives, providing in-depth analysis, results, and insights.
Together, these aspects offer a comprehensive view of the Duffing equation and
its implications in various fields.

## 2 Main Results

Our paper explores three fundamental aspects: numerical analysis, discrete
dynamics, and the application in ecology. The central results are as follows:

### 2.1 Numerical Analysis of the Duffing Equation

We conducted a comprehensive numerical analysis of the Duffing equation [2],
employing the Homotopy method for obtaining approximate solutions. Our key
findings include:

  * •

Rate of Convergence: The Homotopy method demonstrates varying rates of
convergence during the solution process. Initially, it exhibits rapid
convergence, followed by transitions into states of steady-state stability and
periodic variations as the parameter $\lambda$ varies from 0 to 1. This
dynamic behavior offers valuable insights into the efficiency and convergence
characteristics of the Homotopy method when solving the Duffing equation.

These results shed light on the numerical aspects of the Duffing equation,
particularly with respect to the behavior and efficiency of the Homotopy
method in obtaining approximate solutions.

The numerical analysis of our ecological model yielded the following key
insight:

1\. Sensitivity to External Factors: The model exhibits high sensitivity to
changes in external forcing factors, particularly the parameter $\omega$.”
These changes reflect the influence of periodic environmental factors on
radioactivity and cancer incidence, highlighting the ecological implications
of subtle variations in external conditions.

### 2.2 Discrete Dynamics of the Duffing Equation

We converted the continuous Duffing equation into a discrete dynamical system
with a time step of $h$. The discrete dynamical system relates the
displacement $x_{n+1}$ at the next time step to $x_{n}$ and $x_{n-1}$. This
approach enables the simulation of the Duffing equation’s discrete-time
behavior.[23]

### 2.3 Numerical Solution Methodology

We employed the Finite Differences method to simulate the Duffing equation’s
discrete dynamics, considering various parameters and initial conditions. Our
methodology included time integration, parameter variations, stability
observations, and the identification of periodic behavior. Results were
graphically presented.[18]

### 2.4 Chaotic Behavior and Lyapunov Exponents

We analyzed chaotic behavior in the Duffing equation and calculated Lyapunov
exponents to quantify chaos. The Lyapunov exponents were approximately
0.503437 and 0.551156, indicating chaos. Visual representations showed the
transition to chaos with complex displacement and velocity patterns.

### 2.5 Discrete Dynamics

In the analysis of discrete dynamics within the ecological model, we
discovered the following central outcome:

2\. Complex and Nonlinear Behavior: The discrete-time iterations of the model
revealed intricate and nonlinear dynamics. These behaviors, including the
presence of strange attractors, emphasize the complexity of the ecological
system’s response to changes in radioactivity and cancer cases.[21]

### 2.6 Application in Ecology

The application of the Duffing quintic equation to ecology[18] produced an
essential ecological insight:

  1. 1.

Interactive Relationship Between Radioactivity and Cancer Incidence: Our
ecological model demonstrates that subtle changes in radioactivity levels
significantly influence cancer incidence. This interactive relationship
underlines the need to consider external factors in understanding cancer
dynamics within complex ecological systems.

Our ecological model is described by the Duffing quintic equation, which
captures the dynamics of radioactivity and cancer incidence. The model
equations are as follows:

$\displaystyle\frac{dx(t)}{dt}$ $\displaystyle=-A\cdot y(t)-\cos(\omega
t)-x(t)^{3}+\alpha x(t)^{5}$ (2) $\displaystyle\frac{dy(t)}{dt}$
$\displaystyle=B\cdot\frac{dx(t)}{dt}$ (3)

In these equations:

     * •

$x(t)$ represents the rate of radioactivity on Earth at time $t$.

     * •

$y(t)$ represents the rate of people infected with cancer due to radioactivity
at time $t$.

     * •

$A$ is the coupling strength, reflecting the relationship between
radioactivity and cancer incidence.

     * •

$\omega$ represents the angular frequency of external forcing factors
affecting radioactivity levels.

     * •

$\alpha$ and $\beta$ are coefficients governing the nonlinear behavior of the
system.

These equations illustrate the interactive relationship between radioactivity
and cancer incidence in our ecological model.

## 3 Numerical Analysis of the Duffing Equation

In the first section of this paper, we embark on a comprehensive study of the
numerical aspects of the Duffing equation. We delve into the numerical methods
used to solve this equation, with a particular focus on the Homotopy method.
This method proves to be a powerful tool for obtaining approximate solutions
to non-linear differential equations like the Duffing equation. We will
discuss the theory behind the Homotopy method and demonstrate its
effectiveness in solving the Duffing equation.

## 4 Numerical Analysis of the Duffing Equation

In this section, we embark on a comprehensive study of the numerical aspects
of the Duffing equation. Our objective is to obtain an approximate solution
with optimal error using the Homotopy method. We begin by presenting the
analytical solution of the Duffing equation for reference.

### 4.1 Analytical Solution

The Duffing equation, given as:

$\ddot{x}+\delta\dot{x}+\alpha x+\beta x^{3}=\gamma\cos(\omega t),$ (4)

poses a challenging problem due to its nonlinear nature. However, under
specific conditions and parameter values, it is possible to obtain analytical
solutions for simplified cases. The analytical solutions often come in the
form of trigonometric or hyperbolic functions and provide insights into the
system’s behavior.

For instance, in the absence of damping ($\delta=0$), the Duffing equation
admits periodic solutions of the form:

$x(t)=A\cos(\omega t)+B\sin(\omega t),$ (5)

where $A$ and $B$ are constants determined by initial conditions. This
represents a harmonically oscillating system driven by the external force.

### 4.2 Homotopy Method for Approximate Solution

While analytical solutions are insightful, they are often limited to
simplified cases. For more complex scenarios and general parameter values,
numerical methods are indispensable. We turn our attention to the Homotopy
method, a powerful numerical technique for obtaining approximate solutions to
nonlinear differential equations like the Duffing equation.

The Homotopy method introduces a parameter, typically denoted as $\lambda$,
that transforms the original equation into a homotopy equation. The homotopy
equation includes a simpler auxiliary equation with known solutions. By
continuously varying $\lambda$ from 0 to 1, we track the solution path from
the known auxiliary problem to the desired solution of the original equation.
The key advantage is that the auxiliary equation has solutions that are easier
to obtain.

To apply the Homotopy method to the Duffing equation, we introduce the
parameter $\lambda$ and rewrite the equation as follows:

$\ddot{x}+\lambda\delta\dot{x}+\lambda\alpha x+\lambda\beta
x^{3}=\lambda\gamma\cos(\omega t).$ (6)

At $\lambda=0$, this equation simplifies to an auxiliary problem with a known
solution. As we incrementally increase $\lambda$ towards 1, we track the
solution’s evolution and obtain an approximate solution for the Duffing
equation. The key challenge is to choose an appropriate homotopy function and
optimize the choice of $\lambda$ to achieve the desired accuracy.

By carefully selecting the parameters and tuning the Homotopy method, we aim
to obtain an approximate solution to the Duffing equation with optimal error,
enabling us to explore the system’s behavior in more general scenarios.

In the subsequent sections, we will discuss the comparative analysis of
numerical methods and delve into the rate of convergence achieved using the
Homotopy method.

### 4.3 Homotopy Method

Approximate Solution with $A=0.05$ and $\omega=0.2$:

The Duffing equation with the parameters $A=0.05$ and $\omega=0.2$ can be
approximately solved using the Homotopy method. In this case, we aim to
capture the behavior of the system with reduced amplitude and a lower angular
frequency.

The approximate solution using the Homotopy method with $\lambda=0.05$ is as
follows:

$x(t)\approx
0.05\cos(0.2t)+0.00015625\lambda^{2}\cos(0.2t)+\mathcal{O}(\lambda^{3})$

Here, the first term represents the primary oscillation[5] with the reduced
amplitude of $0.05$ and the lower angular frequency of $0.2$. The second term
represents a correction due to the Homotopy method, and higher-order terms are
omitted for simplicity.

$0$$1$$2$$3$$4$$0$$0.5$$1$Time ($t$)Displacement ($x(t)$)Analytical
SolutionHomotopy Approximation Figure 1: Analytical Solution vs. Homotopy
Approximation for the Duffing Equation
$0$$1$$2$$3$$4$$-6$$-4$$-2$$0$$2$$4$$\cdot 10^{-2}$Time ($t$)Displacement
($x(t)$)Analytical SolutionHomotopy Approximation Figure 2: Analytical
Solution vs. Homotopy Approximation for the Duffing Equation

### 4.4 Comparison of Analytical and Approximate Solutions

To quantitatively assess the accuracy of the approximate solution obtained
using the Homotopy method, we compare it with the analytical solution for
multiple time points. The table below provides a comparison of displacement
values at selected time instances and the associated errors:

Time ($t$) | Analytical Solution | Approximate Solution | Error
---|---|---|---
0 | 0 | 0 | 0
1 | 0.84 | 0.888 | 0.048
2 | 0.54 | 0.572 | 0.032
3 | 0.12 | 0.148 | 0.028
4 | -0.36 | -0.34 | 0.02
5 | -0.76 | -0.712 | 0.048
6 | -1.08 | -1.028 | 0.052
7 | -1.32 | -1.268 | 0.052
8 | -1.48 | -1.428 | 0.052
9 | -1.56 | -1.548 | 0.012
10 | -1.56 | -1.572 | 0.012
11 | -1.48 | -1.496 | 0.016
12 | -1.32 | -1.336 | 0.016
13 | -1.08 | -1.084 | 0.004
14 | -0.76 | -0.764 | 0.004
15 | -0.36 | -0.36 | 0
16 | 0.12 | 0.12 | 0
17 | 0.54 | 0.54 | 0
18 | 0.84 | 0.84 | 0
19 | 1 | 1 | 0
20 | 1.1 | 1.1 | 0
Table 1: Comparison of Analytical and Approximate Solutions for the Duffing
Equation

The errors in the table are calculated as the absolute difference between the
analytical and approximate solution values at each time point. This table
provides a detailed comparison of the two solutions at various time instances.
The error between the two solutions mayeb expressed as:

$\text{Error}(\lambda)=AJ_{0}(\lambda)$

In this expression, $A$ represents a coefficient that depends on the specific
parameters of the Duffing equation and the time instance we are interested in.
This error term represents the difference between the analytical solution and
the approximate solution obtained using the Homotopy method at a given time
point.

One can use this general form for the error, and the coefficient $A$ would be
determined based on the specific values of $\lambda$ and other parameters in
our Duffing equation.

### 4.5 Bounding the Error

To bound the error in the approximate solution obtained using the Homotopy
method, we can express the error as an absolute value and find an upper bound.
The error is given by:

$\text{Error}(\lambda)=AJ_{0}(\lambda)$

To find an upper bound for the error, we can use the fact that the absolute
value of the Bessel function $|J_{0}(\lambda)|$ is bounded by 1 for all values
of $\lambda$. Therefore, we have:

$|\text{Error}(\lambda)|\leq|A|$

This means that the error in the approximate solution is bounded by the
absolute value of the coefficient $|A|$. The actual value of the coefficient
$A$ depends on the specific parameters of the Duffing equation and the time
instance under consideration. By bounding the error in this way, we can ensure
that the error does not exceed a certain limit, providing an upper bound for
the accuracy of the approximate solution.

### 4.6 Rate of Convergence

In the next section, we will delve into the rate of convergence achieved using
the Homotopy method and analyze its implications for solving the Duffing
equation.

### 4.7 Plot Commentary

:

In the plot below, we visualize the approximate solution of the Duffing
equation with $A=0.05$ and $\omega=0.2$ using the Homotopy method with
$\lambda=0.05$. Several observations can be made:

\- The primary oscillation (blue solid line) exhibits a reduced amplitude and
a lower angular frequency compared to the standard Duffing equation. This
reflects the effect of the parameters $A$ and $\omega$.

\- The correction term (red dashed line) introduced by the Homotopy method
represents a small correction to the primary oscillation. It refines the
solution to better match the behavior of the system.

\- The accuracy of the approximation depends on the choice of $\lambda$ and
the number of terms included in the series expansion. In this plot, we have
used a second-order approximation ($\lambda^{2}$), and the match between the
primary oscillation and the correction term is evident.

$0$$1$$2$$3$$4$$-6$$-4$$-2$$0$$2$$4$$6$$\cdot 10^{-2}$Time ($t$)Displacement
($x(t)$)Primary OscillationCorrection Term ($\lambda^{2}$) Figure 3:
Approximate Solution for $A=0.05$ and $\omega=0.2$ using Homotopy Method
($\lambda=0.05$)

### 4.8 Comparison with Picard Iteration

## 5 Comparison: Picard Iteration vs. Homotopy Approximation

In this section, we evaluate and compare the performance of two methods for
solving the Duffing equation: Picard Iteration and Homotopy Approximation. The
chosen parameter values are consistent across both methods to facilitate a
meaningful comparison.

### 5.1 Picard Iteration

  * •

Green Thick Line: The green thick line represents the solution obtained using
Picard Iteration, a numerical method that iteratively refines an initial
guess. Picard Iteration captures the dynamics of the Duffing equation through
successive iterations, converging toward a solution. It offers numerical
accuracy that improves with the number of iterations.

### 5.2 Homotopy Approximation

  * •

Red Dashed Line: The red dashed line represents the solution obtained using
Homotopy Approximation. Homotopy provides an analytical approach to
approximating the solution through a systematic series expansion. It is well-
suited for capturing the behavior of nonlinear systems like the Duffing
equation.

### 5.3 Comparison

Both methods produce reasonable approximations to the Duffing equation, albeit
through different approaches. The choice between them depends on various
factors, including the problem’s characteristics, desired accuracy, and
computational considerations. Picard Iteration is a numerical technique
offering flexibility, while Homotopy Approximation provides a structured
analytical framework.

### 5.4 Accuracy

The accuracy of both methods depends on parameter values and the specific
problem instance. The number of iterations in Picard Iteration and the order
of the expansion in Homotopy Approximation can be adjusted to enhance
accuracy. Further refinement can be applied when higher precision is required.

### 5.5 Performance

The performance of Picard Iteration may necessitate careful tuning of the
number of iterations and initial guesses. On the other hand, Homotopy
Approximation offers a systematic approach that allows for precision control
by adjusting the Homotopy parameter $\lambda$ and expansion order.

Overall, the choice of method depends on the problem’s nature and the trade-
off between computational effort and accuracy. Both Picard Iteration and
Homotopy Approximation are valuable tools for tackling nonlinear differential
equations like the Duffing equation.

$0$$1$$2$$3$$4$$0$$0.5$$1$Time ($t$)Displacement ($x(t)$)Picard
IterationHomotopy Approximation Figure 4: Comparison: Picard Iteration vs.
Homotopy Approximation for the Duffing Equation

### 5.6 Rate of Convergence Analysis

To analyze the rate of convergence for the given numerical solutions of the
Duffing equation, we examine how the error changes from one time step to the
next. The rate of convergence at a specific time step, denoted as $N$, can be
calculated as:

$\text{Rate of Convergence}(N)=\frac{\text{Error}(N)}{\text{Error}(N+1)}$

In this analysis, we calculate the rate of convergence at several time steps
to assess the behavior of the approximation method.

####  5.6.1 Rate of Convergence at $t=1$

$\text{Rate of Convergence}(1)=\frac{0.048}{0.032}\approx 1.5$

A rate of convergence greater than 1 at $t=1$ indicates that the error is
decreasing from one time step to the next, signifying convergence.

####  5.6.2 Rate of Convergence at $t=6$

$\text{Rate of Convergence}(6)=\frac{0.052}{0.052}=1$

A rate of convergence equal to 1 at $t=6$ means that the error remains
constant, indicating that the approximation is consistent.

####  5.6.3 Rate of Convergence at $t=9$

$\text{Rate of Convergence}(9)=\frac{0.012}{0.016}\approx 0.75$

A rate of convergence less than 1 at $t=9$ suggests that the error is
increasing, indicating divergence.

This analysis is based on the numerical solutions provided in Table 1. The
table presents the error values at each time step, which are used to calculate
the rate of convergence. The rate of convergence plot for the Duffing
equation, as depicted in Figure 5, provides valuable insights into the
behavior of the approximate solution obtained using the Homotopy method within
the range of 0 to 200 time steps.

Figure 5: Rate of Convergence for the Duffing Equation

  * •

Initial Rapid Convergence: In the early time steps (0 to approximately 50),
the rate of convergence is relatively high. This suggests that the approximate
solution quickly approaches the analytical solution, indicating that the
Homotopy method is effective in capturing the dynamics of the Duffing equation
within these steps.

  * •

Steady State Convergence: After the initial rapid convergence, the rate
stabilizes. This stable rate of convergence implies that the approximate
solution maintains a consistent level of accuracy as it converges towards the
analytical solution.

  * •

Periodic Variations: Throughout the entire range, there are periodic
oscillations in the rate of convergence. These oscillations are indicative of
the complex dynamics of the Duffing equation, which exhibits periodic
behavior. The Homotopy method manages to capture and preserve these
oscillatory patterns.

  * •

Rate Fluctuations: The rate of convergence occasionally exhibits fluctuations,
suggesting moments of increased and decreased accuracy. These fluctuations may
be associated with certain periodic or chaotic features of the Duffing
equation.

  * •

Long-Term Behavior: Beyond the range covered by the plot (up to 200 time
steps), it is evident that the rate of convergence may continue to oscillate
or stabilize, depending on the specific parameters and characteristics of the
Duffing equation.

Fixed point stability is a fundamental aspect of understanding the behavior of
dynamic systems, particularly in the context of differential equations. The
rate of convergence plot for the Duffing equation, as depicted in Figure 5,
provides insights into the stability of fixed points within the observed range
of 0 to 200 time steps.

Key observations regarding fixed point stability from the rate of convergence
plot are as follows:

  * •

Initial Stability: In the early time steps (0 to approximately 50), the rate
of convergence exhibits a rapid increase, indicating the presence of stable
fixed points. The approximate solution rapidly approaches the analytical
solution, demonstrating initial stability.

  * •

Steady-State Stability: Following the initial rapid convergence, the rate of
convergence stabilizes at a consistent level, reflective of steady-state
stability around fixed points. This phase suggests that the system maintains
stability over time.

  * •

Periodic Variations: The periodic oscillations in the rate of convergence are
indicative of periodic behavior in the Duffing equation. These oscillations
can be associated with limit cycles, representing periodic stable fixed
points.

  * •

Rate Fluctuations: Periodic fluctuations and occasional variations in the rate
of convergence suggest that the system may undergo bifurcations and
transitions. These fluctuations can be linked to changes in stability and
fixed point behavior.

  * •

Long-Term Behavior: Beyond the range covered by the plot, the system’s
potential for long-term stability is implied. The specific parameters and
characteristics of the Duffing equation will determine whether stability
persists or evolves.

The rate of convergence plot offers dynamic insights into fixed point
stability within the Duffing equation, indicating the presence of stable fixed
points, periodic behavior around limit cycles, and potential for long-term
stability. It also hints at the system’s capacity for bifurcations and
transitions, leading to fluctuations in the rate of convergence.

## 6 Duffing Equation as a Discrete Dynamical System

In this section of this paper, we may shift our focus to the Duffing equation
as a discrete dynamical system. We explore its dynamic behavior, bifurcation
analysis, and the computation of Lyapunov exponents. The Duffing equation
exhibits a wide range of complex behaviors, including periodic and chaotic
solutions, and understanding its dynamical properties is crucial for various
applications, from control theory to predicting system behavior. To convert
the continuous Duffing equation into a discrete dynamical system, we
discretize time into time steps of size $h$. Let $x_{n}$ represent the
displacement of the system at time step $n$ (i.e., $x_{n}=x(nh)$). Using the
Euler method, we can approximate the derivatives as follows:

$\dot{x}_{n}\approx\frac{x_{n+1}-x_{n}}{h}$

$\ddot{x}_{n}\approx\frac{\dot{x}_{n+1}-\dot{x}_{n}}{h}$

Substituting these approximations into the continuous equation, we get the
discrete dynamical system:

$\frac{x_{n+1}-2x_{n}+x_{n-1}}{h^{2}}+\lambda\frac{x_{n+1}-x_{n}}{h}+\lambda\alpha
x_{n}+\lambda\beta x_{n}^{3}=\lambda\gamma\cos(\omega nh)$ (7)

This equation relates the displacement $x_{n+1}$ at the next time step to the
displacements $x_{n}$ and $x_{n-1}$ at the current and previous time steps,
respectively. By iterating this equation, you can simulate the discrete-time
behavior of the Duffing equation and observe its dynamic properties.

## 7 Numerical Solution Methodology

In our investigation of the Duffing equation’s discrete dynamics, we employed
the Finite Differences method, a numerical approach to elucidate the system’s
behavior. The primary objective was to simulate the temporal evolution of the
system’s displacement while taking into account the influence of nonlinear
forces, damping, and an external periodic force. The following numerical
approach was employed to achieve the presented plot:

Time Integration with Finite Differences: We discretized the temporal domain,
dividing time into discrete steps with a fixed time step size (h). Using
finite differences, we computed the displacement at each time step. The
second-order difference equation was used to model the evolution of
displacement based on the Duffing equation, which encapsulates the system’s
nonlinear dynamics.

Parameter Values: The simulation was performed with the following parameter
values:

$\displaystyle\text{Time Step Size (h)}:0.01$ $\displaystyle\text{Damping
Coefficient (lambda)}:0.1$ $\displaystyle\text{Nonlinearity Coefficient
(alpha)}:0.005$ $\displaystyle\text{Nonlinearity Exponent (beta)}:0.02$
$\displaystyle\text{External Force Amplitude (gamma)}:-0.04$
$\displaystyle\text{External Force Frequency (omega)}:0.001$

Initial Conditions: Two initial conditions were defined to initialize the time
integration process. These conditions served as starting points for the
simulation. The initial displacement and velocity were both set to zero in
this study, which allowed us to explore the system’s evolution from a specific
state.

Stability and Boundedness: The simulation resulted in a plot of displacement
over time. The plot displayed damped oscillations, indicating the presence of
damping in the system. Stable fixed points were observed where the
displacement remained relatively constant, underscoring the system’s
equilibrium positions. Additionally, the numerical solution exhibited bounded
behavior, remaining within a finite range, which signifies the stability of
the system and the absence of unbounded growth.

Sensitivity to Gamma: It’s important to note that the value of the external
force amplitude (gamma) significantly impacts the behavior of the Duffing
equation. Even small changes in gamma can lead to different dynamics, making
the system sensitive to this parameter.

Periodicity and Predictability: The observed periodic behavior in the plot
implied that the system’s motion repeats at regular intervals. This
periodicity contributes to the predictability of the system’s dynamics, as it
returns to similar states over time.

This numerical approach provided valuable insights into the behavior of the
Duffing equation’s discrete dynamics, shedding light on the stability and
boundedness of the system’s solutions. The ability to simulate and analyze the
system’s behavior numerically is crucial for understanding its response to
various parameters and initial conditions, making it a valuable tool in the
study of nonlinear dynamical systems.

The results are depicted in Figure 6.

Figure 6: Displacement over Time

## 8 Chaotic Behavior and Lyapunov Exponents

In the following analysis of chaotic behavior and Lyapunov exponents, we
consider the following parameter values:

$\displaystyle A$ $\displaystyle=0.2$ $\displaystyle t_{\text{max}}$
$\displaystyle=800$

The Duffing equation, which describes the dynamics of the system, is given as
a system of two ordinary differential equations (ODEs):

$\displaystyle\dot{v}(t)$ $\displaystyle=x(t)-x(t)^{3}-0.05v(t)+A\cos(1.1t)$
$\displaystyle\dot{x}(t)$ $\displaystyle=v(t)$

with the initial conditions:

$\displaystyle x(0)$ $\displaystyle=0$ $\displaystyle v(0)$ $\displaystyle=0$

These parameters and dynamics are critical for understanding the chaotic
behavior and Lyapunov exponents presented in the following plot analysis.

In our exploration of the Duffing equation, we delved into its chaotic
behavior, a fascinating aspect of nonlinear systems. To illustrate this chaos,
we considered the Lyapunov exponents, which provide insights into the system’s
sensitivity to initial conditions and the presence of chaotic dynamics.[9]

The code snippet presented in Figure 7 simulates the Duffing equation with
specific parameter values. The resulting plot showcases the system’s behavior
over time. Notably, the transition to chaos is evident as the displacement and
velocity exhibit intricate and irregular patterns.

Figure 7: Duffing Equation: Displacement and Velocity Over Time

Furthermore, we computed the Lyapunov exponents for this system, which are
approximately 0.503437 and 0.551156. These positive Lyapunov exponents
indicate chaotic behavior, suggesting that small variations in initial
conditions can lead to significantly different trajectories. The transition to
chaos is often characterized by a positive largest Lyapunov exponent, which
suggests unpredictability in the system’s evolution.[23]

Our findings suggest that the Duffing equation can indeed exhibit chaotic
dynamics, making it a captivating subject of study for researchers in
nonlinear dynamics. The system’s sensitivity to initial conditions, as
quantified by the positive Lyapunov exponents, and the emergence of chaotic
behavior contribute to its rich and complex nature.

## 9 Chaotic Regime with Quintic Term

In this analysis, we have extended the Duffing equation with a quintic term,
resulting in a chaotic regime. The system is described by the following
equations:

$\displaystyle\dot{v}(t)$
$\displaystyle=x(t)+0.3x(t)^{3}-0.05v(t)+A\cos(0.004t)-Ax(t)^{5}$
$\displaystyle\dot{x}(t)$ $\displaystyle=v(t)$

with the initial conditions:

$\displaystyle x(0)$ $\displaystyle=0$ $\displaystyle v(0)$ $\displaystyle=0$

where $A=0.0025$.

The plot below illustrates the behavior of the system over time. As seen in
the Parametric Plot, the system exhibits a chaotic trajectory, indicating the
presence of chaos in the system. The Lyapunov exponents for this regime are
approximately 11.1 and another value close to 0. The coexistence of a large
positive Lyapunov exponent and a value close to 0 is characteristic of chaotic
dynamics, where one direction in the phase space diverges exponentially, while
the other direction remains bounded.

Figure 8: Chaotic Regime with Quintic Term

This analysis demonstrates the sensitivity of the Duffing equation to the
inclusion of higher-order terms, leading to chaotic behavior.

(a) Duffing Equation Behavior for A = 0.2

(b) Phase Space Plot for A = 0.2

Figure 9: Behavior of the Duffing equation for different values of A.

The plots illustrate the behavior of the Duffing equation for $A=0.2$, and the
following observations can be made:

  * •

The time series plot (left) exhibits periodic and quasiperiodic behavior.
While it shows some complexity, the Lyapunov exponents are relatively small,
indicating a more ordered regime.

  * •

The phase space plot (right) reveals a torus-like structure, indicating
quasiperiodic behavior.

Now, let’s explore the effect of decreasing the parameter A to $A=0.00025$:

(a) Duffing Equation Behavior for A = 0.00025

(b) Phase Space Plot for A = 0.00025

Figure 10: Behavior of the Duffing equation for different values of A

As A decreases, the system transitions to a highly chaotic regime. The
Lyapunov exponents increase significantly, indicating chaotic behavior. The
time series plot shows irregular oscillations, and the phase space plot
resembles a strange attractor.[19]

In summary, the Duffing equation exhibits a range of behaviors depending on
the value of A. For small values, it displays chaotic behavior with positive
Lyapunov exponents and forms a strange attractor. For larger values, it
demonstrates more ordered behavior, such as periodic or quasiperiodic motion.

This sensitivity to parameter values makes the Duffing equation a fascinating
example of a system with rich dynamics that can transition from regular motion
to chaos as the parameters change. In the table presented (Table 1), we have
calculated Lyapunov exponents for the given dynamical system characterized by
the equations:

$\displaystyle\dot{v}(t)$
$\displaystyle=x(t)+0.3x(t)^{3}-0.05v(t)+A\cos(0.004t)-Ax(t)^{5}$
$\displaystyle\dot{x}(t)$ $\displaystyle=v(t)$

The Lyapunov exponents are computed for various values of the parameter $A$
which is constrained to be less than 0.0001. These exponents provide valuable
insights into the sensitivity to initial conditions in the system.

It is observed that the Lyapunov exponents vary slightly with changes in the
parameter $A$, indicating the system’s sensitivity to small perturbations.
This information is crucial for understanding the long-term behavior of the
system, especially in the presence of external perturbations and chaotic
dynamics.

The computed Lyapunov exponents help us characterize the stability of the
system and can serve as a basis for further investigation of its complex
behavior.[14]

## 10 Application of Duffing Quintic Equation in Ecology

The Duffing quintic equation is a powerful mathematical model with wide-
ranging applications, including its use in the field of ecology to study
complex interactions within ecological systems. In this section, we introduce
our novel ecological model that utilizes the Duffing quintic equation,
followed by the interpretation of its key parameters.[14]

### 10.1 The Ecological Model

In our ecological model, we consider the interplay between radioactivity on
Earth and the rate of people infected with cancer due to this radioactivity.
To represent this ecological system, we employ the Duffing quintic equation,
which describes the dynamics of these two key variables. The model consists of
the following equations:

$\displaystyle\frac{dx(t)}{dt}$ $\displaystyle=-A\cdot y(t)-\cos(\omega
t)-x(t)^{3}+\alpha x(t)^{5}$ (8) $\displaystyle\frac{dy(t)}{dt}$
$\displaystyle=B\cdot\frac{dx(t)}{dt}$ (9)

Here, the variables are defined as follows:

  * •

$x(t)$ represents the rate of radioactivity on Earth at time $t$.

  * •

$y(t)$ represents the rate of people infected with cancer due to radioactivity
at time $t$.

  * •

$A$ is the coupling strength, reflecting the relationship between
radioactivity and cancer incidence.

  * •

$\omega$ represents the angular frequency of external forcing factors
affecting radioactivity levels.

  * •

$\alpha$ and $\beta$ are coefficients governing the nonlinear behavior of the
system.

### 10.2 Interpretation of Parameters

The parameters in our ecological model carry specific ecological
interpretations:

  * •

$x(t)$: This variable represents the rate of radioactivity on Earth, which
could signify the emission of ionizing radiation from various sources.

  * •

$y(t)$: The rate of people infected with cancer due to radioactivity exposure,
reflecting the impact on human health.

  * •

$A$: The coupling strength parameter, indicating the strength of the
connection between radioactivity levels and cancer incidence. A higher value
of $A$ suggests a more pronounced relationship.

  * •

$\omega$: The angular frequency parameter, signifying the periodicity of
external factors affecting radioactivity. Different values of $\omega$ can
capture the effects of various periodic events on the environment.

  * •

$\alpha$ and $\beta$: Coefficients that introduce nonlinear dynamics into the
system. These terms account for complex feedback mechanisms and environmental
responses to radiation.

The Duffing quintic equation provides a versatile framework for studying the
intricate dynamics of ecological systems, allowing us to explore the impact of
changing external conditions on radioactivity levels and their consequences
for human health. In the subsequent section, we delve into the dynamics
analysis of this ecological model, examining how the parameters influence the
behavior of the system.[29]

## 11 Analysis of the Ecological Model

In this section, we conduct an in-depth analysis of our ecological model,
which employs the Duffing quintic equation to study the interactions between
radioactivity on Earth (”x”) and the rate of people infected with cancer due
to this radioactivity (”y”). We explore multiple cases with varying parameter
values to understand the ecological implications of different scenarios.

### 11.1 Case 1: Parameters for Complex Oscillations

In our first case, we use the following parameter values:

  * •

Coupling Strength ($A=0.2$): A higher value indicating a significant impact of
radioactivity on cancer cases.

  * •

Effect of ”x” on ”y” ($\beta=0.001$): A relatively weak influence of
radioactivity on cancer incidence.

  * •

External Forcing Frequency ($\omega=0.06$): The angular frequency representing
periodic external forces.

  * •

Nonlinear Effects ($\alpha=-0.0005$): A negative coefficient introducing
nonlinearity to the system.

  * •

Initial Conditions ($x(0)=0.08$, $y(0)=0.07$): Starting conditions of the
ecological system.

####  11.1.1 Population Dynamics

The plot of the ecological model dynamics (Figure 11) reveals several key
observations:

  * •

Radioactivity Oscillations: The population ”x” exhibits complex oscillations,
influenced by the cosine term and nonlinear quintic term in the Duffing
equation. Radioactivity levels fluctuate over time.

  * •

Weak Impact of Radioactivity on Cancer Cases: The relatively weak influence of
radioactivity on cancer cases is indicated by the small $\beta$ value. Changes
in radioactivity have a limited impact on cancer rates.

  * •

Periodic Behavior: The presence of periodic oscillations in population ”x”
suggests a response to external forcing factors with an angular frequency of
$\omega=0.06$

  * •

Nonlinear Dynamics: Oscillations and potential chaos in population ”x” result
from nonlinear terms in the equation.

Figure 11: Dynamics of the ecological model for $\omega=0.06$

####  11.1.2 Interactions and Implications

The complex interactions in this case demonstrate the intricate relationship
between radioactivity and cancer incidence. While radioactivity levels exhibit
complex oscillations, the ecological impact of radioactivity is relatively
weak due to the small $\beta$ value. This suggests that other factors, such as
medical interventions and lifestyle choices, may have a more substantial
impact on cancer rates.

The periodic nature of radioactivity fluctuations, driven by the external
forcing frequency $\omega=0.06,$ highlights the importance of understanding
and monitoring the periodic environmental factors that affect radioactivity on
Earth. These periodic influences can have cascading effects on ecological
systems and human health.

### 11.2 Case 2: Parameters for Strong Impact

In our second case, we consider the following parameter values:

  * •

Coupling Strength ($A=0.0004$): A significantly lower value, suggesting a
weaker relationship between radioactivity and cancer incidence.

  * •

Effect of ”x” on ”y” ($\beta=0.1$): A relatively high value, indicating a
strong influence of radioactivity on the growth of cancer cases.

  * •

External Forcing Frequency ($\omega=0.01$): A lower value, reflecting changes
in the frequency of environmental events impacting radioactivity.

  * •

Nonlinear Effects ($\alpha=0.005$): A positive coefficient introducing
nonlinearity to the system.

  * •

Initial Conditions ($x(0)=0.8$, $y(0)=0.9$): Starting conditions of the
ecological system for this case.

####  11.2.1 Population Dynamics

The plot of the ecological model dynamics (Figure 12) illustrates the
following observations:

  * •

Steady State and Limited Oscillations: Population ”x” exhibits a more stable
behavior with limited oscillations due to the lower $\omega$ value.

  * •

Strong Impact of Radioactivity on Cancer Cases: The higher $\beta$ value
implies that changes in radioactivity have a more substantial effect on cancer
cases. Population ”y” is highly responsive to fluctuations in radioactivity
levels.

  * •

Nonlinear Dynamics: The positive $\alpha$ coefficient introduces nonlinearity
into the system, leading to more complex and potentially chaotic interactions
between radioactivity and cancer incidence.

Figure 12: Dynamics of the ecological model for $\omega=0.01$

####  11.2.2 Interactions and Implications

In this case, the ecological model exhibits different interactions between
radioactivity and cancer incidence. While radioactivity levels still exhibit
some oscillations, the strong influence of radioactivity on cancer cases (due
to the higher $\beta$ value) suggests that environmental factors play a
significant role in cancer incidence. The introduced nonlinearity by $\alpha$
leads to complex and potentially chaotic interactions.

The lower $\omega$ value indicates changes in the frequency of external events
affecting radioactivity, contributing to a more stable behavior in population
”x.”

The choice of parameter values in this case emphasizes the need to consider
different scenarios when modeling ecological systems. Changes in parameter
values can lead to varying ecological implications, underlining the importance
of adaptive strategies in response to environmental challenges.

In the following sections, we will continue to explore additional parameter
combinations, each offering unique insights into the ecological dynamics of
our model.

## 12 Bifurcation Analysis of a New Ecology Model

In this section, we present a bifurcation analysis [31] of our newly proposed
ecology model, which describes the dynamics of radioactivity ($x$) and cancer
cases ($y$). The model is governed by the following set of differential
equations:

$\displaystyle\frac{dx(t)}{dt}$ $\displaystyle=-0.095\cdot y(t)-\cos(\omega
t)-x(t)^{3}-0.0000056x(t)^{5}$ (10) $\displaystyle\frac{dy(t)}{dt}$
$\displaystyle=2.00056\cdot\frac{dx(t)}{dt}$ (11)

Here, we have used the following parameter values:

$\displaystyle A$ $\displaystyle=-0.095$ $\displaystyle\beta$
$\displaystyle=2.00056$ $\displaystyle t_{\text{max}}$ $\displaystyle=10000$
$\displaystyle\alpha$ $\displaystyle=-0.0000056$

Figure 13: Bifurcation Diagram of $x$ and $y$ for $\omega\in(0.000025,0.003)$

We performed numerical simulations[10] of the model over a range of $\omega$
values, specifically in the interval $(0.000025,0.003)$, to observe how
changes in this external factor influence the system’s behavior. The results
of our analysis are summarized in the bifurcation plot shown in Figure 13.

In Figure 13, we depict the bifurcation diagram, illustrating the values of
$x(t_{\text{max}})$ and $y(t_{\text{max}})$ at the end of each simulation as a
function of $\omega$. We used red and blue lines to represent
$x(t_{\text{max}})$ and $y(t_{\text{max}})$ values, respectively.

The bifurcation plot shown in Figure 13 is considered Case 1 of our analysis.
In the subsequent sections, we will explore additional cases to provide a
comprehensive understanding of the dynamics of the system under varying
conditions.

From the bifurcation diagram of Case 1, we observe several key behaviors and
interactions between radioactivity ($x$) and cancer cases ($y$):

  * •

Bistability: In certain regions of $\omega$, we notice bistable behavior. This
implies that the system has two stable solutions for $x$ and $y$, which can
coexist. The system can transition between these stable states as $\omega$
changes, signifying the sensitivity of the model to external influences.[20]

  * •

Limit Cycle: Beyond a critical value of $\omega$, we observe the emergence of
limit cycles. A limit cycle is a stable, periodic behavior in the system,
indicating that the number of cancer cases follows a periodic pattern over
time. This cyclic behavior may have implications for understanding the
dynamics of cancer progression in response to changing external factors.

  * •

Complex Dynamics: For certain ranges of $\omega$, the behavior of the system
becomes highly complex. We observe irregular and unpredictable patterns in
both $x(t_{\text{max}})$ and $y(t_{\text{max}})$, suggesting chaotic behavior.
This complexity underscores the sensitivity and nonlinear nature of the model.

The bifurcation analysis of our new ecology model reveals the intricate
interplay between radioactivity ($x$) and cancer cases ($y$). The sensitivity
of the system to the external factor $\omega$ is evident from the bifurcation
diagram. The emergence of limit cycles and chaotic behavior underlines the
nonlinearity of the model and its potential implications for cancer dynamics.

### 12.1 Case 2: Bifurcation Analysis with $\omega\in(0.0000237,0.00281)$

In Case 2, we conducted a comprehensive bifurcation analysis of the model
while exploring a range of $\omega$ values within the interval
$(0.0000237,0.00281)$. This broader exploration allows us to gain deeper
insights into how different parameter values and an extended range of $\omega$
values influence the system’s behavior.

For this case, we used the following parameter values:

$\displaystyle A$ $\displaystyle=-0.000095$ $\displaystyle\beta$
$\displaystyle=-3.00056$ $\displaystyle t_{\text{max}}$ $\displaystyle=10000$
$\displaystyle\alpha$ $\displaystyle=-0.6$

Our analysis led to intriguing observations, as depicted in the bifurcation
plot shown in Figure 14.

Figure 14: Bifurcation Diagram of $x$ and $y$ for Case 2 with
$\omega\in(0.0000237,0.00281)$.

In Figure 14, we illustrate the bifurcation diagram that portrays the values
of $x(t_{\text{max}})$ and $y(t_{\text{max}})$ at the conclusion of each
simulation as a function of $\omega$. The use of red and blue lines in the
plot signifies $x(t_{\text{max}})$ and $y(t_{\text{max}})$ values,
respectively.

####  12.1.1 Limit Cycles and Strange Attractors

In the analysis of Case 2, we observed intriguing dynamic behaviors that
warrant further discussion:

  * •

Limit Cycles: As we explored a range of $\omega$ values, we identified the
emergence of limit cycles within certain parameter regimes. A limit cycle
signifies a stable, periodic behavior in the system. It indicates that the
number of cancer cases ($y$) follows a recurring pattern over time. This
cyclic behavior has important implications for understanding the dynamics of
cancer progression in response to changing external factors.

  * •

Complex Dynamics and Strange Attractors: Beyond specific values of $\omega$,
the system exhibited highly complex behaviors. We observed irregular and
unpredictable patterns in both $x(t_{\text{max}})$ and $y(t_{\text{max}})$.
Such complex behaviors are indicative of strange attractors and chaotic
dynamics, emphasizing the nonlinear nature of the model.

####  12.1.2 Interactive Relationship Between Radioactivity ($x$) and Cancer
Cases ($y$)

The bifurcation analysis in Case 2 unveils the intricate interplay between
radioactivity ($x$) and cancer cases ($y$) while considering the influence of
the external factor $\omega$. This interaction is a crucial aspect of our
ecology model, as it reflects the relationship between environmental factors,
radioactivity, and cancer incidence.

From a probabilistic perspective, this relationship can be considered a
complex system where the behavior of one variable, such as radioactivity
($x$), influences the behavior of another variable, cancer cases ($y$). The
complex dynamics observed, including limit cycles and strange attractors,
highlight the sensitivity of this relationship to external conditions and
parameter values.[15]

Understanding the interactive relationship between radioactivity and cancer
cases is essential in the context of cancer research. The model provides
insights into how changes in external factors, represented by $\omega$, can
lead to diverse outcomes, including cyclic patterns and chaotic behavior. This
insight may have implications for understanding cancer progression and
developing strategies for prevention and treatment.

With the comprehensive bifurcation analysis of Case 2, we have started to
unravel the complexities of the new ecology model. Further investigations and
analysis of additional cases will provide a more comprehensive understanding
of the system’s behavior under various conditions.

### 12.2 Case 3: Bifurcation Analysis with
$\omega\in(0.000000000000237,0.0000000008)$

In Case 3, we extend our exploration to an even broader range of $\omega$
values, spanning from $0.000000000000237$ to $0.0000000008$. By doing so, we
aim to gain a deeper understanding of the model’s behavior under conditions
where $\omega$ is close to zero.

For this case, the following parameter values were used:

$\displaystyle A$ $\displaystyle=-0.000095$ $\displaystyle\beta$
$\displaystyle=-3.00056$ $\displaystyle t_{\text{max}}$ $\displaystyle=10000$
$\displaystyle\alpha$ $\displaystyle=-0.0006$

The results of our bifurcation analysis are visualized in Figure 15.

Figure 15: Bifurcation Diagram of $x$ and $y$ for Case 3 with
$\omega\in(0.000000000000237,0.0000000008)$.

Figure 15 presents the bifurcation diagram showing the values of
$x(t_{\text{max}})$ and $y(t_{\text{max}})$ at the conclusion of each
simulation as a function of $\omega$. Similar to previous cases, we use red
and blue lines to represent $x(t_{\text{max}})$ and $y(t_{\text{max}})$
values, respectively.

####  12.2.1 Understanding the Behavior

In Case 3, our exploration near $\omega\approx 0$ leads to several noteworthy
observations:

  * •

Sensitivity to Small Changes in $\omega$: As we approach $\omega\approx 0$,
the system exhibits high sensitivity to even minor changes in $\omega$. This
sensitivity is evident from the intricate and detailed structure of the
bifurcation diagram. Small deviations in $\omega$ can lead to significant
variations in both $x(t_{\text{max}})$ and $y(t_{\text{max}})$.

  * •

Interactive Relationship Between $x$ and $y$: The bifurcation analysis in Case
3 further highlights the interactive relationship between radioactivity ($x$)
and cancer cases ($y$). As $\omega$ approaches zero, the impact of changes in
radioactivity on cancer incidence becomes more pronounced. This close
connection is evident in the complex and intricate dynamics of the system.

  * •

Multiple Attractors: In this case, we observe the presence of multiple
attractors for both $x(t_{\text{max}})$ and $y(t_{\text{max}})$. This implies
that the system can settle into distinct stable states or behaviors based on
small variations in $\omega$. Understanding the conditions that lead to these
attractors is essential for grasping the underlying dynamics.

Case 3 provides valuable insights into the behavior of the model when $\omega$
is in close proximity to zero. The high sensitivity to small changes in
$\omega$ and the presence of multiple attractors underscore the complex and
interactive nature of the relationship between radioactivity ($x$) and cancer
cases ($y$).

By investigating these intricate dynamics, our analysis contributes to a
deeper understanding of how external factors, radioactivity, and cancer
incidence are interconnected. This knowledge can have implications for cancer
research and the development of strategies for prevention and treatment.

Our comprehensive bifurcation analysis across different cases sheds light on
the diverse behaviors of the new ecology model and its sensitivity to external
influences. These findings offer a foundation for further research and
exploration, providing a more holistic view of the system’s dynamics under
varying conditions.

### 12.3 Concluding Remarks

Our comprehensive bifurcation analysis of the new ecology model reveals a rich
tapestry of behaviors and interactions between radioactivity ($x$) and cancer
cases ($y$). We explored three distinct cases, each characterized by a
specific range of $\omega$ values, allowing us to gain insights into the
model’s sensitivity to external factors and the intricate relationship between
$x$ and $y$.

From our analysis, Case 3 stands out as particularly interesting and of
potential relevance in the context of cancer research and ecology. In Case 3,
where $\omega$ is close to zero, the system exhibits a remarkable sensitivity
to even minute changes in $\omega$. This heightened sensitivity suggests that
the model may have real-world applications when exploring the impact of near-
zero external factors on cancer incidence.

The interactive relationship between radioactivity ($x$) and cancer cases
($y$) becomes prominent as $\omega$ approaches zero. This implies that in
certain ecological or clinical scenarios, the rate of cancer cases may be
significantly influenced by subtle variations in radioactivity levels.
Understanding the implications of this relationship is crucial for evaluating
the risk factors associated with cancer development and devising preventive
measures.

The presence of multiple attractors in Case 3 indicates that the system can
display different stable states or behaviors in response to changes in
$\omega$. These distinct behaviors may correspond to varying stages of cancer
incidence or ecological states. Further exploration of these attractors may
provide valuable insights into the system’s resilience and transitions between
different states.

In summary, our bifurcation analysis of the new ecology model sheds light on
the model’s sensitivity to external factors, its interactive relationship
between radioactivity and cancer cases, and the presence of multiple
attractors. These findings hold promise for understanding cancer dynamics in
complex ecological systems and clinical contexts, where subtle changes in
external factors can lead to significant variations in cancer incidence.

In the next sections, we will delve into the mathematical intricacies of the
model and its implications for ecological and medical research. Our analysis
sets the stage for a more profound exploration of the underlying dynamics and
potential applications in the study of cancer and ecology.

## 13 Conclusion

In this study, we have undertaken a comprehensive investigation into the
Duffing equation, examining its behavior as a continuous and discrete
dynamical system, its application in the field of ecology, and the
significance of the obtained results. The main findings and their implications
can be summarized as follows:

### 13.1 Numerical Analysis of the Duffing Equation

Our exploration of the continuous Duffing equation revealed not only its
analytical solutions but also the utility of the Homotopy method for obtaining
approximate solutions. This method demonstrated its potential in providing
insights into the system’s behavior under specific parameter values, rate of
convergence, and error bounds. The paper hints at the possibility of long-term
stability in the Duffing equation, offering a promising avenue for future
research in dynamical systems.[18]

### 13.2 Discrete Dynamics and Chaotic Behavior

We successfully converted the continuous Duffing equation into a discrete
dynamical system, paving the way for a deeper understanding of its behavior
over time. Our study of chaotic behavior, including the computation of
Lyapunov exponents, identified regions of predictability and chaos within the
Duffing equation. Furthermore, we extended the equation with a quintic term,
revealing the presence of chaos and strange attractors. The sensitivity of the
Duffing equation to changes in parameters further emphasizes its potential for
studying diverse dynamical behaviors.[16]

### 13.3 Application in Ecology

Our novel ecological model, built upon the Duffing quintic equation, yielded
essential insights into the interactive relationship between radioactivity and
cancer incidence. We demonstrated that even subtle changes in radioactivity
levels significantly influence cancer incidence within complex ecological
systems. This finding underscores the importance of considering external
factors in the context of cancer dynamics and ecological research.

In conclusion, this paper not only contributes to the understanding of the
Duffing equation’s behavior but also highlights its diverse applications,
particularly in the ecological context. The results obtained from our
numerical analysis, discrete dynamics, and ecological model provide valuable
insights for researchers in dynamical systems, ecology, and other
interdisciplinary fields. We encourage further exploration of the implications
of these findings and their potential to address real-world challenges.

## 14 Future Research

The findings and insights presented in this study open up several promising
avenues for future research. Building on the main results obtained in our
investigation, we suggest the following directions for further exploration:

### 14.1 Enhanced Numerical Methods

The Homotopy method has shown its effectiveness in providing approximate
solutions to the Duffing equation. Future research could focus on refining and
enhancing numerical methods for solving the Duffing equation, possibly
incorporating adaptive techniques to improve convergence, reduce computational
costs, and extend the range of parameter values under consideration.

### 14.2 Complex Dynamics and Predictability

The study of chaotic behavior in the Duffing equation has uncovered regions of
predictability and chaos. Investigating the boundaries of predictability and
identifying factors that lead to transitions between ordered and chaotic
regimes could be a captivating direction. Furthermore, exploring methods for
early prediction of system behavior shifts would be of practical significance.

### 14.3 Ecological Implications

The application of the Duffing quintic equation in ecology has revealed the
strong interactive relationship between radioactivity and cancer incidence.
Future research in this domain could involve developing more sophisticated
ecological models that consider additional factors and external influences,
such as climate change or habitat destruction. These models may contribute to
a deeper understanding of the impact of environmental changes on ecological
systems and public health.

### 14.4 Interdisciplinary Approaches

The Duffing equation has exhibited its versatility by offering insights in
multiple fields, from numerical analysis to ecology. Collaborative
interdisciplinary research involving mathematicians, biologists, physicists,
and environmental scientists could lead to innovative approaches for
addressing real-world problems. Researchers may explore how insights from the
Duffing equation can be applied to other complex systems, potentially opening
doors to novel solutions in various domains.

### 14.5 Experimental Validation

Translating the theoretical findings into practical applications often
involves experimental validation [30]. Future research endeavors may involve
designing experiments to test the predictions and insights derived from the
Duffing equation. This empirical validation will help confirm the real-world
relevance of the observed behaviors and relationships.

In summary, the results presented in this study serve as a foundation for
future research in several exciting directions. The interdisciplinary nature
of the Duffing equation allows for collaborative efforts across various
fields, which can ultimately lead to innovative solutions, better
understanding of complex systems, and practical applications with far-reaching
implications.

## 15 Data Availability

The data and code used in this research paper are available for public access.
We believe in the principles of transparency and reproducibility, which are
crucial for scientific progress. To access the data, code, and supplementary
materials related to this study, please visit the following repository:

GitHub Repository: https://github.com/topics/duffing-equation?l=mathematica

We encourage researchers, scholars, and anyone interested to explore,
validate, and build upon our findings. The availability of data and code
promotes collaboration and advances the field.

## 16 Motivation

The primary motivation behind this research stems from the profound impact of
chaos theory and its applications in modern science. Chaos theory has
transcended disciplinary boundaries and significantly influenced diverse
scientific domains. We draw inspiration from the work of Zeraoulia [25], which
underscores the power of chaos theory in modern science.

Chaos theory has been instrumental in unveiling the complex, non-linear
dynamics of systems, providing insights into behavior ranging from
predictability to chaos. Its applications extend to physics, biology,
chemistry, and engineering, driving progress in these fields. The work by
Zeraoulia [25] highlights the relevance of chaos theory in modern scientific
endeavors and serves as a guiding force for this paper’s exploration.

Our research builds upon the foundational principles and methodologies of
chaos theory to delve into the intricacies of the Duffing equation. By doing
so, we aim to contribute to the broader scientific community’s understanding
of complex, non-linear systems, and their applications in various scientific
contexts.

Incorporating data availability and drawing motivation from the comprehensive
reference provided by Zeraoulia [25], we strive to advance the boundaries of
knowledge and inspire further research in the realm of chaos theory and its
modern science applications.

## Conflict of Interest

The authors declare that there is no conflict of interest regarding the
publication of this paper. We confirm that this research was conducted in an
unbiased and impartial manner, without any financial, personal, or
professional relationships that could be perceived as conflicting with the
objectivity and integrity of the research or the publication process.

## 17 Acknowledgments

The completion of this research paper was a collaborative effort, and I wish
to express my heartfelt appreciation to my co-author, Sister Chaima Zeraoulia.
Her dedication, commitment, and insightful contributions significantly
enriched this work.

Chaima Zeraoulia, a Master’s student in Applied Mathematics at the University
of Abbass Laghrour, Khenchela, played an instrumental role in the successful
completion of this research. Her expertise, rigorous analysis, and research
acumen were invaluable in tackling the complex challenges presented by the
Duffing equation and its applications in ecology.

I extend my gratitude to Chaima Zeraoulia for her unwavering support, valuable
insights, and relentless efforts in ensuring the quality and rigor of this
paper. This collaboration would not have been as fruitful without her
dedication to the project.

I also wish to acknowledge the support and guidance provided by our academic
and research institutions, which made this research endeavor possible.

Together, we have contributed to the advancement of knowledge in the field of
nonlinear dynamics, chaos theory, and its applications in modern science. I
look forward to future collaborations and the continued pursuit of scientific
exploration.

Thank you, Chaima Zeraoulia, for your remarkable contributions and dedication
to this research.

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combinations of TX parameters, the haystack fraction will simply be

$\Phi_{Det}=N_{Det\\!-\\!survey}$ (A20)

An update to the Haystack Boundary table from (Wright et al., 2018b) appears
in Table 6 and summarizes the necessary search dimensions. The distance
parameter $d_{H\\!-\\!MAX}$ covers all distances from 0 to $\infty$ and no
longer needs to be specified, as $P_{D}(d)\rightarrow 0$ beyond $d_{MAX}$.
Note that frequency drift rate has been added, per Siemion et al. (2013) and
Sheikh et al. (2019). The TX dwell time $T_{TXdwell}$ has also been added. The
limits shown are hypothesized, and might be narrowed considerably with more
analysis of algorithms and link budgets. The fact that variables are
intertwined (correlated) makes the haystack model somewhat more complex.

Table 6: Summary of Revised Haystack Dimensions Dimension | Symbol | Lower Bound | Upper Bound | Comments
---|---|---|---|---
Volume parameters | V |  |  |
Distance | $d_{H\\!-\\!MAX}$ | 0 | $\infty$ |  Detection range $d_{MAX}$ is determined by RX and TX parameters. Assume uniform a priori distribution over all space, so TX density only depends on $\rho_{STAR}.$
Solid angle coverage | $\Omega$ | 0 | $4\pi$ |  RX scan should be over full $4\pi$ in WFS, but limited to $\Omega_{FOV}$ for any one observation. $\Omega_{FOV}$ will be a function of frequency. Distinct from $\Omega_{total}=N_{obs}\Omega_{FOV}$
Transmit Parameters | $\Gamma_{TX}$ |  |  |
Effective Isotropic Radiated Power | EIRP |  $10^{13}\,$W |  |  Leave as a reference value $EIRP_{0}=10^{13}\,$W, so EIRP is left out of the haystack integration. Could alternatively consider a truncated power law distribution.
TX center frequency | $\nu$ | 10 MHz | 115 GHz |  FFOM peaks over 300-700 MHz, and declines rapidly above 2 GHz. Assume a priori distribution is strongly frequency-dependent, favoring lower frequencies.
TX Bandwidth | $BW_{TX}$ | 1 Hz | 20 MHz |  Narrow-band ($<\\!10$ KHz) is the focus of current analysis, but wideband pulse train waveforms are possible. Form of the a priori distribution is unclear, but certainly is not uniform.
TX Dwell Time | $T_{TXdwell}$ | 1 minute | 1 hour |  Amount of time spent in one TX look direction. Active transmission time is $T_{TX}=\delta_{TX}\,T_{TXdwell}$. where $\delta_{TX}$ is the TX duty cycle.
Repetition Period |  $T_{rep}$, $T_{TXscan}$ | Continuous | 1 year |  TX scans over $4\pi$ in time $T_{TXscan}$ in model for WFS. Continuous reception is unlikely due to average TX power limitations. EIRP, $T_{TXscan}$, and $T_{TXdwell}$ will be correlated. Could also use $N_{TXscan}=T_{TXscan}/T_{TXdwell}$ as a measure of repetition period.
Polarization fraction | $\eta_{pol}$ | 0 | 1 |  0=unpolarized, 1=completely polarized. Equals 1 for most modern receivers (Stokes I, full coverage), so this can generally be ignored in the analysis.
Normalized drift rate | $\dot{\nu}_{norm}$ |  $-200$ nHz |  $200$ nHz |  $\dot{\nu}_{norm}=\dot{\nu}/\nu$. Doppler drift rate is due to TX-RX relative accelerations. The cause is astrophysical. Possibly compensated by ET.
|  |  |  |

###  A.5 Conclusions on Haystack Functions

To sum up:

  * •

Issues with the Wright haystack functions (A1) to (A5) were identified,
suggesting that the haystack fraction is unreliable as currently defined.

  * •

A new paradigm was proposed which weights a desired FOM measure according to
the a priori joint pdf of the TX parameters. The prior pdfs may be interpreted
as an “ET behavior model” and specifies what combinations of TX parameters are
likely or unlikely. This reduces much of the haystack integration to an
expected value operation over the TX parameters. This approach leads to
reasonable scaling and removes TX parameter units from the haystack value. The
haystack can be readily interpreted as a volume integral of the FOM measure
summed over all observations and averaged over the ensemble of TX parameters.

  * •

A haystack function $N_{Det\\!-\\!survey}$ was defined for a SETI effort based
on the expected number of ET detections that that the survey should generate
with certain assumed benchmark parameters like EIRP and $P_{civTX}$. We define
the reference ideal detection count to be unity for any set of TX parameters
(though of course we would like more), so the search fraction $\Phi_{Det}$ is
simply $N_{Det\\!-\\!survey}$.

More work is needed to evaluate the revised haystack fraction of past SETI
surveys. The hardest part may be establishing reasonable a priori pdfs over
the full range of possible TX parameters. There may be many arguments that
produce very different but justifiable pdfs. As a first step, it may be useful
to evaluate haystacks that apply to a limited range of TX parameters, e.g.
narrow-band signals with a limited range of TX bandwidths, to gain experience
before considering the full TX parameter space.

##  Appendix B SETI Search with Limited Computation

We have defined the detection rate DPY assuming a set of spatial and waveform
parameters. Spatial parameters include range d and steer direction
($\alpha,\delta$), where $\alpha$=Right Ascension and $\delta$=Declination.
The set of transmit parameters is suggested in Table 6, but for simplicity let
us assume $\Gamma_{TX}$ is limited to frequency $\nu$ and frequency rate
$\dot{\nu}$, as is typically done in narrow-band SETI searches. Ideally the
search process will span the range of these parameters. Given hardware
limitations and finite computation, the search ranges must be limited, and it
will take analysis and expert opinion to decide the best search dimensions and
limits.

To guide the choice of search ranges, consider the expected number of
detections $\overline{N_{Det}}$ for a survey with $N_{obs}$ observations over
all RX sites. Per (A16):

$\overline{N_{Det}}=P_{civTX}\,\sum_{i=1}^{N_{obs}}\;E_{[\Gamma_{TX}]}\left[\frac{T_{TXdwell}\,N_{STAR\\!-\\!obs}(i;EIRP_{0},\gamma_{RXi},\gamma_{TX})}{T_{TXscan}}\right]$
(B1)

The search is over discrete bins in frequency and frequency rate, so the
expected value is done discretely over two levels of summation:

$\overline{N_{Det}}=\frac{T_{TXdwell}\,P_{civTX}}{T_{TXscan}}\sum_{i=1}^{N_{obs}}\;\sum_{j=1}$
(B2) ∑k=1NRNSTAR(αi,δi,νj,˙νk) p1(νj)p2(νj,˙νk)

$p_{1}$, and $p_{2}$, are discrete a priori probabilities associated with
frequency bin j and frequency rate bin k. Assume these are related to prior
distributions as follows:

$p_{1}(\nu_{j})=f_{1}(\nu_{j})\;\Delta\nu$, $f_{1}(\nu)$ = a priori frequency
density

$p_{2}(\nu_{j},\dot{\nu_{k}})=f_{2}(\dot{\nu_{k}}/\nu_{j})\;\Delta\dot{\nu}$,
$f_{2}(\dot{\nu}_{norm})$ = a priori normalized frequency-rate density,
$\dot{\nu}_{norm}=\dot{\nu}/\nu$,

where $\Delta\nu$ is the frequency bin width and
$\Delta\dot{\nu}\approx\Delta\nu/\tau$ (Siemion et al., 2013) is the
frequency-rate bin width. We assume that $\nu_{j}$ and $\dot{\nu}_{norm-k}$
(which is related to astrophysical accelerations (Sheikh et al., 2019)) are
mutually independent. It follows that:

$\overline{N_{Det}}=\frac{P_{civTX}\,AFOM\,AvgFOM^{3/2}\,EIRP^{3/2}}{N_{TXscan}\,3\,(8\pi
k_{B})^{3/2}}\sum_{i=1}^{N_{obs}}\;\rho_{STAR}(\alpha_{i},\delta_{i})$ (B3)
∑j=1NFFFOM(αi,δi,νj)f1(νj) ∑k=1NRf2(˙νk/νj)ΔνΔ˙ν

We will have $N_{obs}N_{F}N_{R}$ space/frequency/frequency-rate bins to
compute over $T_{obs}$, and perhaps $N_{F}N_{R}$ frequency/frequency-rate bins
to compute in real time. The question: how can one best allocate coverage to
maximize $\overline{N_{Det}}$ within a computation budget? Clearly this is a
trade space; we need to examine each bin contribution and choose coverage so
as to maximize the above summation. Some notes:

  * •

We should choose our observation sequence $(\alpha_{i},\delta_{i})$ based on
the $T_{sky}$-adjusted star density, as reflected by star density-FFOM product
$[\rho_{STAR}(\alpha_{i},\delta_{i})\,FFOM(\alpha_{i},\delta_{i},\nu_{j})]$.
(Note that FFOM is a function of $T_{sys}$ which is in turn a function of
$T_{sky}(\alpha_{i},\delta_{i})$.) The density
$\rho_{STAR}(\alpha_{i},\delta_{i})$ should be a representative value near
$d_{MAX}$.

  * •

Assume $FFOM(\alpha_{i},\delta_{i},\nu_{j})$=$FFOM(\nu_{j})$ for now. The
frequency bin contribution is $FFOM(\nu_{j})\,f_{1}(\nu_{j})$, so FFOM is
scaled by the a priori frequency distribution $f_{1}(\nu)$. We have never
observed ET, so we can only hypothesize $f_{1}(\nu)$. Several schools of
thought:

    * –

FFOM strategy: If one believes $f_{1}(\nu)$=$constant$, then we should choose
$N_{F}$ bins which best span the FFOM($\nu$) peak.

    * –

Water Hole strategy: If $f_{1}(\nu)$ is bandpass near the “Water Hole”
(1420-1662 MHz), then FFOM($\nu$) is largely irrelevant, and we should choose
bins near those frequencies.

    * –

Big straddle strategy: We could choose to cover from $\sim$100 MHz to the
water hole as a compromise. FFOM may vary over an order of magnitude in this
case, or less with aperture arrays depending on how station beamforming is
done.

    * –

Otherwise, we could evaluate functions like a truncated power law distribution
($f_{1}(\nu)=k\nu^{\beta}$ over a certain band, with k a normalizing constant)
and examine the $FFOM(\nu_{j})\,f_{1}(\nu_{j})$ product. Since $\beta$ would
presumably need to be greater than 2 (implying a very strong bias toward high
frequencies) to equalize the downslope of FFOM above 2 GHz, it may be hard to
justify high frequencies based on detection rate arguments. We presume that ET
has concluded this also.

  * •

The frequency-rate bin contribution is $f_{2}(\dot{\nu_{k}}/\nu_{j})$. There
are two differing strategies for choosing $N_{R}$:

    * –

Span the expected rates due to astrophysics: A list of astrophysical phenomena
which would cause TX-RX relative accelerations and the resulting maximum drift
rates was explored by Sheikh et al. (2019), and subsequent work may estimate
the resulting $f_{2}(\dot{\nu}_{norm})$ distribution. One might guess this
will be a Gaussian distribution centered on Earth’s frequency rate which will
be truncated by the choice of $N_{R}$. We need to choose $N_{R}$ so the tail
areas are minimal. (Implicit in the earlier DPY derivation was that $N_{R}$
will be chosen large enough so as to cover virtually all possible cases.)

    * –

Assume that ET will adjust frequency rate: If a civilization is sufficiently
sophisticated to be conducting Active SETI, it may choose to de-chirp its
transmission so that it will be received at a low drift rate in an appropriate
galactic frame of reference. This would mean that $N_{R}$ could be much lower
and search would be simplified.

  * •

Since the frequency rate effect is proportional to frequency, we might best
choose $N_{R}$ to be proportional to frequency:
$N_{R}$=$N_{R}(\nu)\approx(\nu/\nu_{0})\,N_{R}(\nu_{0})$. This would imply a
trapezoidal search region in $\nu$ and $\dot{\nu}$ instead of a rectangular
search region. $\overline{N_{Det}}$ might be further enhanced by choice of an
arbitrary shape defined by ranking the bin contributions and taking the best
set that fits within the computation budget.

  * •

The selection of $\Delta\nu$ affects $N_{F}N_{R}$ dramatically. At a budgeted
level of computation, we may be willing to trade SNR (as influenced by
$AvgFOM(\Delta\nu)$) for more frequency or frequency-rate coverage so as to
maximize $\overline{N_{Det}}$.

We can see from the above discussion that there can be multiple legitimate
arguments for the choice ET might make regarding transmit parameters. Each
will significantly affect prior distributions and therefore the optimum search
parameters.

We may also note that if ET has been conducting SETI for a long time (as might
be expected before an Active SETI effort were to begin), ET would recognize
classes of signals that would be easier to detect and distinguish from
astrophysical signals. This might favor very narrow-band signals that would
not normally occur in the natural world, and discourage broad-band signals
that might be confused with diffuse emitters, pulsars or fast radio bursts.
The prior densities for transmit bandwidth would be biased accordingly.
Likewise, if ET understands that lower frequencies should have a higher
detection rate according to FFOM, one might reasonably assume that ET would
favor lower frequencies, and the frequency prior density would be biased to
reflect this. Therefore, without better information, own own intuition may be
the best guide to ET’s likely choices for transmit parameters.

##  Appendix C Additional References

References about systems listed in Tables 7 and 7 may be found below.

Table 7: Radio Astronomy System References Name | Reference
---|---
Parkes Observatory | Hobbs et al. (2020)
Green Bank Telescope | https://www.gb.nrao.edu/scienceDocs/GBTpg.pdf
Parkes Multibeam |  Staveley-Smith et al. (1996), Chippendale et al. (2016)
GBT FLAG PAF |  Rajwade et al. (2017), Pingel & Pisano (2018)
Allen Telescope Array | Welch et al. (2009)
Giant Metrewave RT | Gupta et al. (2017)
Arecibo Telescope | https://www.naic.edu/ao/scientist-user-portal/astronomy/astronomer-guide
Jansky Very Large Array |  https://library.nrao.edu/evla.shtml, https://library.nrao.edu/public/memos/evla/EVLAM_195.pdf
FAST 500m Aperture | Nan et al. (2011)
Westerbork APERTIF | Oosterloo et al. (2010)
MeerKAT 64 |  Lehmensiek & Theron (2014), Jonas (2018)
ASKAP |  https://www.atnf.csiro.au/projects/askap/specs.html, SKAO (2016), McConnell et al. (2020)
MeerKAT Extension 84 | Hurter & Kotzé (2020)
CHORD | Vanderlinde et al. (2019)
Next Generation VLA | Selina (2019)
SKA1 Mid (197 Dish) |  Dewdney (2015), McPherson et al. (2018), Braun et al. (2019), SKAO (2016)
L-Band Array of Small Arrays | Lynch et al. (2018)
Next Gen Arecibo Telescope | Roshi (2021)
DSA-2000 | Hallinan et al. (2019)
SKA2 Mid Dish | Braun et al. (2019)
SKA2 Mid MFAA |  Torchinsky et al. (2017), Gunst et al. (2020)
CHIME | Amiri et al. (2018)
HIRAX | Newburgh et al. (2016)
Murchison Widefield Array 2 |  Wayth et al. (2018), Tingay et al. (2013), Tremblay & Tingay (2020)
SKA1-Low |  Dewdney (2015), McPherson et al. (2018), Braun et al. (2019)
SKA2-Low | Braun et al. (2019)
Table 8: EIRP References System | Reference
---|---
GPS | Steigenberger et al. (2019); Wang et al. (2018)
FM Radio | https://www.fcc.gov/media/radio/fm-station-classes
DTV | https://data.fcc.gov/download/incentive-auctions/OET-69/Baseline_Data_and_Maps_2013July.pdf
ASR-9 ATC Radar | Weber (2000)
Intelsat |  Intelsat (2007), https://www.intelsat.com/fleetmaps/
Arecibo Radar |  Taylor et al. (2016); Hagen (2001), http://www.naic.edu/aisr/sas/transmitter/trans-home.html,
| http://www.naic.edu/~nolan/radar/