# Dynamics of Delay Logistic Difference Equation in the Complex Plane

Sk. Sarif Hassan
International Centre for Theoretical Sciences
Tata Institute of Fundamental Research
Bangalore $560012$, India
Email<EMAIL_ADDRESS>

###### Abstract

The dynamics of the delay logistic equation with complex parameters and
arbitrary complex initial conditions is investigated. The analysis of the
local stability of this difference equation has been carried out. We further
exhibit several interesting characteristics of the solutions of this equation,
using computations, which does not arise when we consider the same equation
with positive real parameters and initial conditions. Some of the interesting
observations led us to pose some open problems and conjectures regarding
chaotic and higher order periodic solutions and global asymptotic convergence
of the delay logistic equation. It is our hope that these observations of this
complex difference equation would certainly be an interesting addition to the
present art of research in rational difference equations in understanding the
behaviour in the complex domain.

Keywords: Pielou’s Difference equation, Delay Logistic equation, Chaotic,
Local asymptotic stability and Periodicity.
Mathematics Subject Classification: 39A10, 39A11

##  1 Introduction and Preliminaries

Consider the delay logistic difference equation

$z_{n+1}=\frac{\alpha z_{n}}{1+\beta z_{n-1}},n=0,1,\ldots$ (1)

where the parameter $\alpha$ is complex number and the initial conditions
$z_{-1}$ and $z_{0}$ are arbitrary complex numbers.

The same difference equation including some variations of the this are studied
when the parameter $\alpha$ and initial conditions are non-negative real
numbers, Eq.(1) was investigated in [2] and [3]. In this present article it is
an attempt to understand the same in the complex plane.

The set of initial conditions ${z_{-1},~{}z_{0}}\in\mathbb{C}$ for which the
solution of Eq.(1) is well defined for all $n\geq 0$ is called the good set of
initial conditions or the _domain of definition_. It is the compliment of the
forbidden set of Eq.(1) for which the solution is not well defined for some
$n\geq 0$. It is really hard to find out either the good set or forbidden set
for the second and higher order rational difference equations due to the lack
of an explicit form of the solutions. For the rest of the sequel it is assumed
that the initial conditions belong to the good set [14].

Our goal is to investigate the character of the solutions of Eq.(1) when the
parameters are complex and the initial conditions are arbitrary complex
numbers in the domain of definition.
Here, we review some results which will be useful in our investigation of the
behavior of solutions of the difference equation (1).

Let $f:\mathbb{D}^{2}\rightarrow\mathbb{D}$ where
$\mathbb{D}\subseteq\mathbf{C}$ be a continuously differentiable function.
Then for any pair of initial conditions $z_{0},z_{-1}\in\mathbb{D}$, the
difference equation

$z_{n+1}=f(z_{n},z_{n-1})\hskip 18.06749pt,\hskip 18.06749pt$ (2)

with initial conditions $z_{-1},z_{0}\in\mathbb{D}.$

Then for any _initial value_ , the difference equation (1) will have a unique
solution $\\{z_{n}\\}_{n}$.

A point $\overline{z}$ $\in$ $\mathbb{D}$ is called equilibrium point of
Eq.(2) if

$f(\overline{z},\overline{z})=\overline{z}.$

The _linearized equation_ of Eq.(2) about the equilibrium $\bar{z}$ is the
linear difference equation

$\displaystyle{z_{n+1}=a_{0}z_{n}+a_{1}z_{n-1}\hskip 18.06749pt,\hskip
18.06749ptn=0,1,\ldots}$ (3)

where for $i=0$ and $1$.

$a_{i}=\frac{\partial f}{\partial u_{i}}(\overline{z},\overline{z}).$

The _characteristic equation_ of Eq.(5) is the equation

$\lambda^{2}-a_{0}\lambda-a_{1}=0.$ (4)

The following are the briefings of the linearized stability criterions which
are useful in determining the local stability character of the equilibrium
$\overline{z}$ of Eq.(2), [1].

Let $\overline{z}$ be an equilibrium of the difference equation
$z_{n+1}=f(z_{n},z_{n-1})$.

  * •

The equilibrium $\bar{z}$ of Eq. (2) is called locally stable if for every
$\epsilon>0$, there exists a $\delta>0$ such that for every $z_{0}$ and
$z_{-1}$ $\in$ $\mathbb{C}$ with $|z_{0}-\bar{z}|+|z_{-1}-\bar{z}|<\delta$ we
have $|z_{n}-\bar{z}|<\epsilon$ for all $n>-1$.

  * •

The equilibrium $\bar{z}$ of Eq. (2) is called locally stable if it is locally
stable and if there exist a $\gamma>0$ such that for every $z_{0}$ and
$z_{-1}$ $\in$ $\mathbb{C}$ with $|z_{0}-\bar{z}|+|z_{-1}-\bar{z}|<\gamma$ we
have $\lim_{n\to\infty}z_{n}=\bar{z}$.

  * •

The equilibrium $\bar{z}$ of Eq. (2) is called global attractor if for every
$z_{0}$ and $z_{-1}$ $\in$ $\mathbb{C}$, we have
$\lim_{n\to\infty}z_{n}=\bar{z}$.

  * •

The equilibrium of equation Eq. (2) is called globally asymptotically
stable/fit is stable and is a global attractor.

  * •

The equilibrium $\bar{z}$ of Eq. (2) is called unstable if it is not stable.

  * •

The equilibrium $\bar{z}$ of Eq. (2) is called source or repeller if there
exists $r>0$ such that for every $z_{0}$ and $z_{-1}$ $\in$ $\mathbb{C}$ with
$|z_{0}-\bar{z}|+|z_{-1}-\bar{z}|<r$ we have $|z_{n}-\bar{z}|\geq r$. Clearly
a source is an unstable equilibrium.

##  2 Local Stability of the Equilibriums and Boundedness

In this section we establish the local stability character of the equilibria
of Eq.(1) when the parameters $\alpha$ and $\beta$ are considered to be
complex numbers with the initial conditions $z_{0}$ and $z_{-1}$ are arbitrary
complex numbers.

The equilibrium points of Eq.(1) are the solutions of the equation

$\bar{z}=\frac{\alpha\bar{z}}{1+\beta\bar{z}}$

Eq.(1) has two equilibria points $\bar{z}_{1,2}=0,\frac{\alpha-1}{\beta}$
respectively. The linearized equation of the rational difference equation(1)
with respect to the equilibrium point $\bar{z}_{1}=0$ is

$\displaystyle{z_{n+1}=\alpha z_{n},n=0,1,\ldots}$ (5)

with associated characteristic equation

$\lambda^{2}-\alpha\lambda=0.$ (6)

The following result gives the local asymptotic stability of the equilibrium
$\bar{z}_{1}$.

######  Theorem 2.1.

The equilibriums $\bar{z}_{1}=0$ of Eq.(1) is

locally asymptotically stable if and only if

$\left\lvert\alpha\right\rvert<1$

unstable if and only if

$\left\lvert\alpha\right\rvert<1$

and non-hyperbolic if and only if

$\left\lvert\alpha\right\rvert=1$

###### Proof.

The characteristic equation of the equilibrium as already mentioned above is
$\lambda^{2}-\alpha\lambda=0$. The zeros of this polynomials are $0$ and
$\alpha$. Therefore the result follows trivially by _Local Stability Theorem_.

∎

######  Theorem 2.2.

The equilibriums $\bar{z}_{2}$ $=\frac{\alpha-1}{\beta}$ of Eq.(1) is

locally asymptotically stable if and only if

$\frac{1}{3}\leq\left\lvert\alpha\right\rvert\leq\frac{4}{3}$

unstable if and only if

$0<\left\lvert\alpha\right\rvert<\frac{1}{3}$

###### Proof.

The characteristic equation of the equilibrium $\bar{z}_{2}$ is
$\lambda^{2}-\lambda+\frac{\alpha-1}{\alpha}=0$. The zeros of this polynomials
are $\frac{\alpha-\sqrt{(4-3\alpha)\alpha}}{2\alpha}$ and
$\frac{\alpha+\sqrt{(4-3\alpha)\alpha}}{2\alpha}$. The equilibrium
$\bar{z}_{2}$ is locally asymptotically stable if and only if the modulus of
the zeros of the characteristic equation are less than $1$. Simple algebraic
calculation of these two inequalities reduced to the condition
$\frac{1}{3}\leq\left\lvert\alpha\right\rvert\leq\frac{4}{3}$. In the similar
fashion the equilibrium is unstable if and only if the modulus of the zeros
are greater than $1$. Consequently, the condition reduces to
$0<\left\lvert\alpha\right\rvert<\frac{1}{3}$. Hence the theorem is proved.

∎

Now we would like to try to find open ball $B(0,\epsilon)\in\mathbb{C}$ such
that if $z_{n}\in B(0,\epsilon)$ and $z_{n-1}\in B(0,\epsilon)$ then
$z_{n+1}\in B(0,\epsilon)$ for all $n\geq 0$. In other words, if the initial
values $z_{0}$ and $z_{-1}$ belong to $B(0,\epsilon)$ then the solution
generated by the difference equation (1) would essentially be within the open
ball $B(0,\epsilon)$.

######  Theorem 2.3.

For every $\epsilon>0$ and for any complex parameter
$\left\lvert\alpha\right\rvert<1$ and $\left\lvert\beta\right\rvert<1$, if
$z_{n}$ and $z_{n-1}$ $\in B(0,\epsilon)$ then $z_{n+1}\in B(0,\epsilon)$
provided,

$0<\epsilon\leq\frac{1-\left\lvert\alpha\right\rvert}{\left\lvert\beta\right\rvert}$

###### Proof.

Let $\\{z_{n}\\}$ be a solution of the equation Eq.(1. Let $\epsilon>0$ be any
arbitrary real number. Consider $z_{n},z_{n-1}\in B(0,\epsilon)$. We need to
find out an $\epsilon$ such that $z_{n+1}\in B(0,\epsilon)$ for all $n$. It is
follows from the Eq.(1) that for any $\epsilon>0$, using Triangle inequality
for $\left\lvert\alpha\right\rvert<1$ and $\left\lvert\beta\right\rvert<1$

$\left\lvert z_{n+1}\right\rvert=\left\lvert\frac{\alpha z_{n}}{1+\beta
z_{n-1}}\right\rvert\leq\frac{\left\lvert\alpha\right\rvert\epsilon}{1-\left\lvert\beta\right\rvert\epsilon}$

In order to ensure that $\left\lvert z_{n+1}\right\rvert<\epsilon$, it is
needed to be

$\frac{\left\lvert\alpha\right\rvert\epsilon}{1-\left\lvert\beta\right\rvert\epsilon}<1\Leftrightarrow\left\lvert\alpha\right\rvert-1<-\left\lvert\beta\right\rvert\epsilon\Rightarrow\epsilon<\frac{1-\left\lvert\alpha\right\rvert}{\left\lvert\beta\right\rvert}$

Therefore the required is followed.

∎

##  3 Periodic Solutions

The global periodicity and the existence of solutions that converge to
periodic solutions of the difference equation (1 are adumbrated in this
section.

A solution $\\{z_{n}\\}_{n}$ of a difference equation is said to be _globally
periodic_ of period $t$ if $z_{n+t}=z_{n}$ for any given initial conditions.
solution $\\{z_{n}\\}_{n}$ is said to be _periodic with prime period_ $p$ if p
is the smallest positive integer having this property.

###  3.1 Existence and Local Stability of Prime Period Two Solutions

Here we find out prime period two solutions followed by the local stability
analysis of them.

####  3.1.1 Existence of Prime Period Two Solutions

Let $\ldots,\phi,~{}\psi,~{}\phi,~{}\psi,\ldots$, $\phi\neq\psi$ be a prime
period two solution of the difference equation (1). Then
$\phi=\frac{\alpha\psi}{1+\beta\phi}$ and
$\psi=\frac{\alpha\phi}{1+\beta\psi}$. This two equations lead to the set of
solutions (prime period two) except the equilibriums $0$ and
$\frac{\alpha-1}{\beta}$ as
$\left\\{\phi\to\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}},\psi\to\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}}\right\\}$
and
$\left\\{\phi\to\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}},\psi\to\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}}\right\\}$.

####  3.1.2 Local Stability of Prime Period Two Solutions

Let $\ldots,\phi,~{}\psi,~{}\phi,~{}\psi,\ldots$, $\phi\neq\psi$ be a prime
period two solution of the Pielou’s equation (1). We set

$u_{n}=z_{n-1}$ $v_{n}=z_{n}$

Then the equivalent form of the delay Logistic difference equation (1) is

$u_{n+1}=v_{n}$ $v_{n+1}=\frac{\alpha v_{n}}{1+\beta u_{n}}$

Let T be the map on $(0,\infty)\times(0,\infty)$ to itself defined by

$T\left(\begin{array}[]{c}u\\\ v\\\
\end{array}\right)==\left(\begin{array}[]{c}v\\\ \frac{\alpha v}{1+\beta u}\\\
\end{array}\right)$

Then $\left(\begin{array}[]{c}\phi\\\ \psi\\\ \end{array}\right)$ is a fixed
point of $T^{2}$, the second iterate of $T$.

$T^{2}\left(\begin{array}[]{c}u\\\ v\\\
\end{array}\right)==\left(\begin{array}[]{c}\frac{\alpha v}{1+\beta u}\\\ \\\
\frac{\alpha\frac{\alpha v}{1+\beta u}}{1+\beta v}\\\ \end{array}\right)$

$T^{2}\left(\begin{array}[]{c}u\\\ v\\\
\end{array}\right)==\left(\begin{array}[]{c}g(u,v)\\\ h(u,v)\\\
\end{array}\right)$

where $g(u,v)=\frac{\alpha v}{1+\beta u}$ and $h(u,v)=\frac{\alpha\frac{\alpha
v}{1+\beta u}}{1+\beta v}$. Clearly the two cycle is locally asymptotically
stable when the eigenvalues of the Jacobian matrix $J_{T^{2}}$, evaluated at
$\left(\begin{array}[]{c}\phi\\\ \psi\\\ \end{array}\right)$ lie inside the
unit disk. We have,

$J_{T^{2}}\left(\begin{array}[]{c}\phi\\\ \psi\\\
\end{array}\right)=\left(\begin{array}[]{cc}\frac{\delta g}{\delta
u}(\phi,\psi)&\frac{\delta g}{\delta v}(\phi,\psi)\\\ \\\ \frac{\delta
h}{\delta u}(\phi,\psi)&\frac{\delta h}{\delta v}(\phi,\psi)\\\
\end{array}\right)$

where $\frac{\delta g}{\delta
u}(\phi,\psi)=-\frac{\alpha\beta\psi}{(1+\beta\phi)^{2}}$ and $\frac{\delta
g}{\delta v}(\phi,\psi)=\frac{\alpha}{1+\beta\phi}$

$\frac{\delta h}{\delta
u}(\phi,\psi)=-\frac{\alpha^{2}\beta\psi}{(1+\beta\phi)^{2}(1+\beta\psi)}$ and
$\frac{\delta h}{\delta
v}(\phi,\psi)=-\frac{\alpha^{2}\beta\psi}{(1+\beta\phi)(1+\beta\psi)^{2}}+\frac{\alpha^{2}}{(1+\beta\phi)(1+\beta\psi)}$

Now, set

$\chi=\frac{\delta g}{\delta u}(\phi,\psi)+\frac{\delta h}{\delta
v}(\phi,\psi)=\frac{\alpha\left(-\beta\psi+\frac{\alpha(1+\beta\phi)}{(1+\beta\psi)^{2}}\right)}{(1+\beta\phi)^{2}}$
$\lambda=\frac{\delta g}{\delta u}(\phi,\psi)\frac{\delta h}{\delta
v}(\phi,\psi)-\frac{\delta g}{\delta v}(\phi,\psi)\frac{\delta h}{\delta
u}(\phi,\psi)=\frac{\alpha^{3}\beta^{2}\psi^{2}}{(1+\beta\phi)^{3}(1+\beta\psi)^{2}}$

In particular for the prime period $2$ solution,

$\left\\{\phi\to\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}},\psi\to\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}}\right\\}$,

$\chi=\frac{\alpha\left(-\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}+\frac{\alpha\left(1+\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2.-3.\alpha)\alpha)\beta^{2}}}{\beta}\right)}{\left(1+\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}\right)}{\left(1+\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}$

and

$\lambda=\frac{\alpha^{3}\left((-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}\right)^{2}}{\beta^{2}\left(1+\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{3}\left(1+\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}$

and for the prime period $2$ solution,

$\left\\{\phi\to\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}},\psi\to\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta^{2}}\right\\}$

$\chi=\frac{\alpha\left(-\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}+\frac{\alpha\left(1+\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2.-3.\alpha)\alpha)\beta^{2}}}{\beta}\right)}{\left(1+\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}\right)}{\left(1+\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}$

and

$\lambda=\frac{\alpha^{3}\left((-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}\right)^{2}}{\beta^{2}\left(1+\frac{(-0.5-0.5\alpha)\beta+0.5\sqrt{(1+(-2.-3.\alpha)\alpha)\beta^{2}}}{\beta}\right)^{3}\left(1+\frac{(-0.5-0.5\alpha)\beta-0.5\sqrt{(1+(-2-3\alpha)\alpha)\beta^{2}}}{\beta}\right)^{2}}$

Then it follows from the _Linearized Stability Theorem_ that both the
eigenvalues of the $J_{T^{2}}\left(\begin{array}[]{c}\phi\\\ \psi\\\
\end{array}\right)$ lie inside the unit disk if and only if
$|\chi|<1+|\lambda|<2$.

In particular, for $\alpha=i$ and $\beta=2+3i$ the prime period $2$ solutions
is $\left\\{\phi\to-0.294567+0.313317i,\psi\to-0.0900486-0.236394i\right\\}$.
The periodic trajectory is shown in _Fig.1_ of which the
$\left\lvert\chi\right\rvert=1.38112$ and
$\left\lvert\lambda\right\rvert=0.689678$. By the Linear Stability theorem
($|\chi|<1+|\lambda|<2$) the prime period $2$ solution is _locally
asymptotically stable_.

---

Figure 1: Periodic Cycle solution of period $3$ for $\alpha=i$ and
$\beta=2+3i$.

For $\alpha=1+i$ and $\beta=2+3i$ the prime period $2$ solutions is
$\phi\to-0.168166+0.534411i,$
$\psi\to-0.370295-0.226718i$. The periodic trajectory is shown in _Fig.2_ of
which the $\left\lvert\chi\right\rvert=1.5$ and
$\left\lvert\lambda\right\rvert=1.58114$. Hence the prime period $2$ solution
is _unstable_.

---

Figure 2: Periodic Cycle solution of period $3$ for $\alpha=i$ and
$\beta=2+3i$.

###  3.2 Higher Order Cycles in Solutions

Here we shall explore the higher order periodic cycle of the difference
equation (1) in a vivid manner.
For the parameters $\alpha=i$ and $\beta=2+3i$ of the difference equation, the
period $3$ cycle is the following:
$\left\\{z_{0}\to
0.316268+0.129975i,z_{1}\to-0.288941+0.157085i,z_{2}\to-0.181173-0.056291i\right\\}$.
The corresponding periodic trajectory is shown _Fig.3_

---

Figure 3: Periodic Cycle solution of period $3$ for $\alpha=i$ and
$\beta=2+3i$.

For the parameter $\alpha=\frac{1}{3}+i$ and $\beta=2+i$ a periodic cycle of
period $10$ has been encountered. The periodic trajectory is
$z_{0}\to-0.0197446-1.28723i,z_{1}\to
1.03398+0.925847i,z_{2}\to-0.406487+0.128166i,$
$z_{3}\to-0.125003-0.00142325i,z_{4}\to 0.63328-0.516259i,z_{5}\to
0.83925+0.756558i,z_{6}\to-0.223017+0.36021i,z_{7}\to-0.116754+0.0893373i,z_{8}\to-0.239031+0.16474i,z_{9}\to-0.382611-0.236912i$.
The plot of the trajectory is given in _Fig.4_.

---

Figure 4: Periodic Cycle solution of period $3$ for $\alpha=i$ and
$\beta=2+3i$.

Keeping fixed $\beta$ as $1$ computationally it is also seen that for
$\alpha=(15,26)$ and $(55,95)$, almost all solutions corresponding to the
initial values $z_{0},z_{-1}\in\mathbb{D}$ converges to the periodic point
$(-356.366,-194.0009)$ and (11.6656, -0.1928) of period $55$ and $199$
respectively. Certainly there are many more such examples of $\alpha$ for
which the same happens. The plot of the periodic trajectory are given the
_Fig.5_ and _Fig.6_ for the above two examples.

---

Figure 5: Periodic Cycle solution for $\alpha=(15,26)$ and $\beta=1$.

---

Figure 6: Periodic Cycle solution for $\alpha=(55,95)$ and $\beta=1$.

In both the _Fig.5_ and _Fig.6_ , the two figures are shown with different
number of iterations of the periodic trajectory.

Now we shall demonstrate few computational examples where periodic cycle
exists with _very high period_ roughly of order $1000$. We choose
$\alpha=(35,94)$ and $\beta=(88,55)$ and for arbitrary complex initial values
$z_{0}$ and $z_{-1}$, the trajectory of periodic cycles are plotted and
corresponding phase space also plotted in the _Fig.7_.

---

Figure 7: Periodic Cycle solution for $\alpha=(35,94)$ and $\beta=(88,55)$. In
Each figure 5 set of initial values are taken and corresponding trajectories
are plotted.

From the computational aspect, in each of the plots it is evident that for
$\alpha=(35,94)$ and $\beta=(88,55)$ the delay logistic difference equation
possess very high order periodic cycles eventually for almost all initial
values.

##  4 Chaotic Solutions

Finding chaotic solutions for the delay logistic equation (1) is interesting
indeed since in case of real parameter $\alpha$ and $\beta$ and the initial
values $z_{0}$ and $z_{-1}$ there does not exists any chaotic solutions.
The method of Lyapunov characteristic exponents serves as a useful tool to
quantify chaos. Specifically Lyapunov exponents measure the rates of
convergence or divergence of nearby trajectories. Negative Lyapunov exponents
indicate convergence, while positive Lyapunov exponents demonstrate divergence
and chaos. The magnitude of the Lyapunov exponent is an indicator of the time
scale on which chaotic behavior can be predicted or transients decay for the
positive and negative exponent cases respectively. In this present study, the
largest Lyapunov exponent is calculated for a given solution of finite length
numerically [11].

We are looking for complex parameter $\alpha$ and $\beta$ for which for every
initial values the solutions are chaotic. If we consider the parameter
$\beta=1$ then the difference equation is known as _Pielou’s equation_ and
which is well studied earlier in case of real line. Here are few examples
which we came across computationally.

Parameter $\alpha$, $\beta=1$ |  Internal of Lyapunav exponent
---|---
$\alpha=(8,43)$, $\beta=1$ |  $(1.205,2.623)$
$\alpha=(1,97)$, $\beta=1$ |  $(1.845,3.028)$
$\alpha=(6,53)$, $\beta=1$ |  $(0.785,1.718)$
$\alpha=(12,50)$, $\beta=1$ |  $(0.373,1.485)$
Table 1: Cycle solutions of the Pielou’s equation ($\beta=1$) for different
choice of $\alpha$ and initial values.

The largest Lyapunav exponents of the solutions for different initial values
are lying in the positive intervals as stated above in the table. This ensures
that the solutions are chaotic. It is observed that the chaos is bounded.

---

Figure 8: Chaotic Solutions for the Pielou’s equation ($\beta=1$) of four
different cases as stated in table 1.

##  5 Some Interesting Nontrivial Problems

The computational experiment endorses us to pose the following important open
problems in this context.

######  Open Problem 5.1.

Find out the subset of the $\mathbb{D}$ of all possible initial values $z_{0}$
and $z_{1}$ for which the solutions of the delay logistic equation possess
chaotic solutions for a given parameter $\alpha$ and $\beta$.

######  Open Problem 5.2.

Find out the complex parameters $\alpha$ and $\beta$ such that for any initial
values $z_{0}$ and $z_{-1}$ from the $\mathbb{D}$ the solutions of the delay
logistic equation are chaotic.

######  Open Problem 5.3.

Find out the parameters $\alpha$ and $\beta$ such that for any initial values
$z_{0}$ and $z_{-1}$ from the $\mathbb{D}$ the solution of the difference
equation are periodic (globally).

######  Open Problem 5.4.

Characterize the parameters $\alpha$ and $\beta$ such that for any initial
values $z_{0}$ and $z_{-1}$ from the $\mathbb{D}$ the solution of the
difference equation are periodic (globally). How large the period could be? Is
it possible to find an upper bound?

##  6 Future Endeavours

In continuation of the present work for a generalization of the delay logistic
equation, $\frac{\alpha z_{n-l}}{1+\beta z_{n-k}}$ with varies $\alpha$ and
$\beta$, where $l$ and $k$ are delay terms and it demands similar analysis
which we plan to pursue in near future.

## Acknowledgement

The author thanks _Dr. Esha Chatterjee_ and _Dr. Pallab Basu_ for discussions
and suggestions.

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