logistic dynamical systems with oscillating...

LOGISTIC DYNAMICAL SYSTEMS WITH

OSCILLATING PARAMETERS

by

KATHERINE T. BRYANT

SUBMITTED TO SCRIPPS COLLEGE IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF ARTS

PROFESSOR MARIO MARTELLI

PROFESSOR ANI CHADERJIAN

MARCH 12, 2004

Table of Contents

List of Tables iii

List of Figures iv

Abstract v

Acknowledgements vi

Introduction 1

1 Nonlinear Dynamical Systems 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Aperiodic and Chaotic Behavior . . . . . . . . . . . . . . . . . . . . . 91.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Oscillations Between Two a Values 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Aperiodic Orbits and Chaotic Behavior . . . . . . . . . . . . . . . . . 19

3 Three and Beyond 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Three and Four Equations . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Conclusions & Directions for Further Research . . . . . . . . . . . . . 25

A Resulting Periodic Behavior Data 27

Bibliography 29

iii

List of Tables

1.1 Periodic behavior of F (x) = ax(1− x) . . . . . . . . . . . . . . . . . 6

2.1 Periodic behavior of F2(x) . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Periodic behavior of F5(x) . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Periodic behavior using with an aperiodic a . . . . . . . . . . . . . . 20

3.1 Periodic behavior of F(x) when it oscillates between 3 values for a . . 22

3.2 Periodic behavior of F(x) when it oscillates between 4 values for a . . 24

A.1 Periodic behavior of F3(x) . . . . . . . . . . . . . . . . . . . . . . . . 27





iv

List of Figures

1.1 F (x) = 2x(1− x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Fixed Points of F (x) = 2x(1− x) . . . . . . . . . . . . . . . . . . . . 41.3 Points of period 2 of F (x) = 3.4x(1− x) . . . . . . . . . . . . . . . . 61.4 Bifurcation diagram of F (x) = ax(1− x) . . . . . . . . . . . . . . . . 71.5 Chaotic behavior of F (x) = 3.81x(1− x) . . . . . . . . . . . . . . . . 10

2.1 Stable Periodic behavior of (x0, x1) = (1− 1a2 ,1a2

) = (23, 1

3) . . . . . . . 14

2.2 Values that result in period 2 orbits of G(x) = a1a2x(1−x)(1−a2x(1−x)) 162.3 Possible Period 2 Orbits of G(x) =1 a2x(1− x)(1− a2x(1− x)) . . . . 16

v

Abstract

The goal of this thesis is to analyze what happens to dynamical systems when they

have a nonconstant oscillating parameter. Specifically, I use the logistic map

F (x) = ax(1 − x) as an example. As a starting point, I discuss the well known sit-

uation when a is constant. I go on to analyze the presence and stability of orbits of

period 2 when the parameter a oscillates between 2 values. Then, I briefly examine

what happens when there are 2 values for the parameter, but the oscillation does not

occur every iteration, such as when it changes on the 5th iteration. I proceed to men-

tion some numerical results of the behavior of F (x) when it oscillates between three

parameters. The study of two and three (or more) oscillating parameters provides

many directions for further study.

vi

Acknowledgements

I would like to thank my advisor, Professor Mario Martelli, Claremont McKenna

College, and my reader, Ani Chaderjian, Scripps College, as well as Professor Ami

Radunskaya, Pomona College, for their support as well as their time and assistance.

I am also grateful to Professor Jim Cushing, University of Arizona Tuson, for sending

me his work on oscillating parameters. Thank you Mom and Dad for all your love

and encouragement. Also, to all my friends, thank you so much for all the smiles and

laughter. I wouldn’t have gotten here without all of you.

Katherine T. BryantClaremont, CaliforniaMarch 12, 2004

vii

Introduction

The examination of relatively simple first order difference equations through the eyes

of chaos theory led investigators to the conclusion that these models are quite complex.

The basic logistic map, xn+1 = axn(1 − xn), which models population growth, is an

example of a discrete dynamical system that given certain values of a displays not

only fixed and periodic behavior, but aperiodic and chaotic properties as well. I begin

my research by considering the case where xn+1 = axn(1− xn) with a constant value

for the parameter a. Then I discuss what happens when a is no longer constant and

present my research on when a oscillates between two and three values. I offer both

new theoretical and numerical evidence for further study.

1

Chapter 1

Nonlinear Dynamical Systems

1.1 Introduction

One of the simplest nonlinear systems available for study is that of population growth

over time. It is possible to study how the previous generation’s population relates

to the next by examining the model Xt+1 = F (Xt) [5]. This first order nonlinear

equation models many systems other than population growth, including those in

economics, genetics and various social sciences. In many of these cases the X increases

when it is small and decreases when it is large [5]. In other words, F (0) = 0 and F (x)

increases monotonically for 0 < x < m, until it attains a maximum at x = m when

it decreases monotonically beyond x = m [5].

One example of F is the logistic difference equation:

Nt+1 = Nt(a− bNt) (1.1.1)

2

Often this equation is rewritten as:

Xn+1 = aXn(1−Xn) (1.1.2)

by substituting X = bNa

. Equation 1.1.2 is governed by the logistic function

F (x) = ax(1− x) (1.1.3)

This is a basic quadratic map and a classic example of how complex chaotic

behavior can result from very simple non-linear dynamical systems. It is necessary

to place constraints on both a and x values in equation 1.1.3 so that we only study

real solutions. In order to prevent the iterations from diverging to −∞, x is limited

so that x ∈ [0, 1]. (In a population model, when x > 1, the population becomes

extinct.) F (x) (1.1.3) attains a maximum at x = 12, which occurs when F (x) = a

4,

so the system only has trivial dynamical behavior if a > 4. Thus a is limited to

a ∈ [0, 4]. One example of equation 1.1.3 is presented below (Figure 1.1):

Figure 1.1: F (x) = 2x(1− x)

3

1.2 Fixed Points

One important area of study of dynamical systems is that of fixed points. Fixed

points, also known as stationary states, are points where the iterations of a dynamical

system (such as equation 1.1.2) become equal over time.

Definition 1.2.1. A fixed point is any point xs such that F (xs) = xs, given that

F (xn) = xn+1, a discrete dynamical system [4].

Therefore every fixed point of equation 1.1.3 satisfies F (a, x) = ax(1 − x) = x.

This implies that the fixed points for this system are xs1 = 0 and if x 6= 0 then

xs2 = 1 − 1a , when a > 1. Graphically this can be seen by examining when F (x)

(1.1.3) intersects the line y = x. This intersection is show below for F (x) = 2x(1−x)

in Figure 1.2. The fixed points are xs1 = 0 and xs2 =12.

Figure 1.2: Fixed Points of F (x) = 2x(1− x)

It is important to examine the stability of the fixed points. There are two ways

to define stability:

4

Definition 1.2.2. A fixed point xs is stable if for every r > 0 there exists δ > 0 such

that ‖x0 − xs‖ ≤ δ implies that ‖xn − xs‖ ≤ r for all n ≥ 1. If xs is not stable it is

unstable [4].

Definition 1.2.3. Let I be an open interval and xs be a fixed point of a continuous

function F : I → I. Assume there exists an r > 0 such that F is differentiable on

(xs − r, xs + r), except possibly at xs and |F ′(x)| ≤ 1. Then xs is stable [4].

A change of stability occurs at a = 1 because that is where xs1 = 0 and xs2 = 1− 1a

intersect. F (x) = ax(1− x) (1.1.3) so F ′(1− 1a) = 2− a and F ′(0) = a. Thus xs1 is

stable for all a ∈ (−1, 1) and xs2 is stable for all a ∈ (1, 3). When a > 3 the fixed

points are both unstable which leads to the examination of periodic orbits.

1.3 Periodic Orbits

A fixed point is considered to be a point of period 1 because every iteration returns

to it’s original point. A periodic orbit of period 2 is one where x0 = x2 = x4 = x6...

and x1 = x3 = x5 = x7... while x0 6= x1.

Definition 1.3.1. An orbit O(x0) of 1.1.3 is periodic is of period p ≥ 1 if xp = x0.

The period of the orbit is the smallest integer p such that xp = x0 [4].

Rewriting this we can say that x0 is periodic of period p if x0 = Fp(x0) = xp,

where Fm(x0) 6= x0 for every m < p. For equation 1.1.3, F (x) = ax(1 − x), it is

5

possible to prove for what a values periodic orbits exist. For example, a point x0 is

periodic of period 2 if x2 = x0 and x1 6= x0. Specifically if x0 = a+1+√

a2−2a−32a

and

x1 =a+1−

√a2−2a−32a

, then they are periodic of period 2 [4]. The periodic orbit (x0, x1)

can be considered different than the period 2 orbit (x1, x0), but for my purposes, I

consider them equivalent. There is no orbit of period 2 for a ≤ 3 and exactly one when

a > 3. Figure 1.3 shows an example of two points of period 2 for F (x) = 2x(1− x),

where F (F (x)) = x and F (x) 6= x. The periodic orbit in Figure 1.3 is given by

(x0, x1) = (.84215, .45196)

Figure 1.3: Points of period 2 of F (x) = 3.4x(1− x)

Table 1.1 summarizes the range of a values where equations 1.1.2 and 1.1.3 have

a particular periodic orbit, regardless of their initial x values.

Values of a Resulting Period[0, 3) 1

(3, 1 +√

6) 2

(1 +√

6,3.54) 4(3.54,3.59946) 8,16,32,...,2n

(3.569946,4] aperiodic orbits

Table 1.1: Periodic behavior of F (x) = ax(1− x)

6

The closer a is to 4, the more variation there is in F (x). When a ∈ [3.54, 3.59946]

there are periodic orbits larger than 32, however the subintervals where they exist

are very small. When a=3.83 there exists a periodic orbit of period 3, which means

there exists an orbit of every integer period, due to the theorem of Li and Yorke (see

theorem 1.4.1). However when a ∈ [3.569946, 4] most of the interval displays chaotic

behavior.

It is possible to see the different periodic orbits and their ranges by examining a

bifurcation diagram. Bifurcation diagrams plot all the branches of fixed and periodic

points.

Figure 1.4: Bifurcation diagram of F (x) = ax(1− x)

From Figure 1.4 we can visually examine the stability of various orbits and fixed

points. Instead of just using the definitions of stable and unstable I use a more specific

definition.

Definition 1.3.2. Let I be an open interval and (x0, ..., xp−1, ...) be a periodic orbit

of period p, (x0, x1, ..., xp−1) of a continuous function F : I → I. Let d > 0 be

7

such that F is differentiable on (xi − d, xi + d) with a continuous derivative at xi

for all i = 0, 1, ..., p − 1. Then a periodic orbit of period p is a sink, or attractor, if

|(F p)′(x0)| < 1. If |(F p)′(x0)| > 1 then the periodic orbit is a source , or repeller, [4].

Section 1.2 discusses the stability of the fixed points. The bifurcation diagram

shows that the fixed point xs1 = 1− 1a loses stability for a > 3 and the periodic orbit

of period 2 becomes stable until a = 1 +√

6, at which point the periodic orbit of

period 4 becomes stable.

Proposition 1.3.1. The periodic orbit of period 2 given by x0 =a+1+

√a2−2a−32a

and

x1 =a+1−

√a2−2a−32a

is a sink for a ∈ (3, 1 +√

6).

Proof. Given F (x) = ax(1− x), F ′(x) = a− 2ax.

F ′(x0) = a− (a + 1 +√

a2 − 2a− 3) = −1−√

a2 − 2a− 3

F ′(x1) = a− (a + 1−√

a2 − 2a− 3) = −1 +√

a2 − 2a− 3

ddx

F 2(x0) = (−1−√

a2 − 2a− 3)(−1 +√

a2 − 2a− 3) = −a2 + 2a + 4

dda

(−a2 + 2a + 4) = 2− 2a, 2− 2a < 0 for all a > 3

When a = 1 +√

6, ddx

F 2(x0) = −1, so ddx |F2(x0)| < 1 for all a ∈ (3, 1 +

√6). So the

periodic orbit (x0, x1) is a sink and thus stable for a ∈ (3, 1 +√

6).

Similarly, it can be shown that the period 4 orbit is a sink for a ∈ (1+√

6, 3.54409)

[4]. While these orbits are stable, there exist other period 2 and period 4 orbits in

[3.57, 4] that may be unstable.

8

1.4 Aperiodic and Chaotic Behavior

Every orbit is either asymptotically periodic or aperiodic. In order to explain these

terms it is necessary to define a limit point and a limit set.

Definition 1.4.1. A point p is a limit point of an orbit O(x0) if there exists a

subsequence xnk : k = 0, 1, ... of O(x0) such that ‖xnk − p‖ → 0 as k → ∞. A limit

set L(x0) of O(x0) is the set of all the limit points of the orbit [4].

If a limit set is finite, then the orbit is O(x0) is asymptotically periodic. If a limit

set is infinite, then the orbit O(x0) is aperiodic which almost implies it is chaotic.

There is more than one definition of chaos and chaotic behavior. Chaos in the Li-

Yorke sense is that if there is a period 3 orbit, then there is an orbit of every period,

as well as aperiodic orbits. As I mentioned before, when a = 3.83 there is an orbit of

period 3.

Theorem 1.4.1. Let I be an interval and F : I → I be continuous. If F has a

periodic orbit of period 3, then F has a periodic orbit of every period and there exists

an infinite set S contained in I, such that every orbit starting from a point in S is

aperiodic [4].

This requires continuity and is limited to R, not Rn. An alternative definition is:

Definition 1.4.2. Let I be an interval and F : I → I. Then chaotic behavior exists

if there exists some x0 ∈ I such that the orbit is aperiodic and unstable.

9

This definition is good for Rn but one can use the Li-Yorke definition for F (a, x) =

ax(1− x) or a third option: if an orbit is dense and unstable, then it is chaotic [4].

In Table 1.1 we said that in the range [3.560046, 4] there are aperiodic orbits and

chaotic behavior. For example when a = 3.81 there exists chaotic behavior.

Figure 1.5: Chaotic behavior of F (x) = 3.81x(1− x)

Since the graph on the left is filled solidly the orbit is dense. The graph on the

right shows that F (x) is aperiodic since the points do not fall into distinct lines and

thus periods. The first graph shows that the orbit is dense; the second one shows it

is aperiodic.

1.5 Conclusion

The study of the logistic map F (x) = ax(1 − x) with constant a values is well

researched and documented. However, the properties of F (x) with oscillating pa-

rameters are not as well known. The rest of this thesis is original research when the

parameter a of F (x) oscillates between 2 and 3 parameters.

10

Chapter 2

Oscillations Between Two a Values

2.1 Introduction

Periodic changes occur often in population modeling due to daily, monthly, or annual

fluctuations in the physical environment [3]. There is very little data that specifically

deals with the effects of periodic fluctuating environments on population density, let

alone rigorous mathematical models in population dynamics that could be used to

explain such data [3]. It is logical to conclude that the fluctuations in nature are so

small that the resulting changes in population are not noticed by scientists or that

another equation is chosen to model the population when F (x) = ax(1 − x) could

still apply. Thus it is important to examine one of the models used in for studying

population and see what information can be gathered from it.

11

In order to study what happens to equation 1.1.3, F (x) = ax(1−x) when it alternates

between two a values, a1 and a2, it was necessary to consider the two following

equations,

f1(x) = a1x(1− x) (2.1.1)

f2(x) = a2x(1− x) (2.1.2)

where 0 < a1 < a2 < 4 and x ∈ [0, 1]. From there another equation was needed,

F2(x) =1

2(a1(1− (−1)t)x(1− x) + a2(1− (−1)t+1)x(1− x)) (2.1.3)

This allows for the numerical study of what happens when F (x) alternates a1a2a1a2a1a2.

In equation 2.1.3, t is a positive integer and gives the proper parameter to iterate

with; for example, when t = 3, F2(x) = a1x(1 − x) which is the equation that is

applied for the third iterate. In order to research other patterns of iteration, such as

a1a1a1a2a1a1a1a2, I needed to create another F (x), specifically:

Fn(x) =1

2(a1(1− (−1)floor(

tn

))x(1− x) + a2(1− (−1)floor(tn

+1))x(1− x)) (2.1.4)

In equation 2.1.4, n is a positive integer and determines on what iteration the pa-

rameter changes. For example, when n = 3, the parameters would be applied in

the following order: a1a1a1a2. The results of iterating these equations are generally

periodic (see section 2.3).

12

2.2 Fixed Points

F2(x), equation 2.1.1, has only 1 fixed point, xs1 = 0 regardless of the a values chosen.

It is stable when a1 ≤ 1a2 . This means that 0 < a1 < 1 and 0 < a2 < 4 must also

hold. The other fixed for F (x), xs2 = 1 − 1a , is not a fixed point of F2(x) because it

requires a1 = a2 which, by definition, are not equal.

Proof. Assume xs2 is a fixed point of F2(x). Then by definition f1(f2(x)) = x and

f2(x) = x. Composing f1 and f2 gives f1(f2(x)) = a1a2x(1− x)(1− a2x(1− x))

We know x = a1a2x(1− x)(1− a2x(1− x))

Simplifying gives 1 = a1a2(1− x)(1− a2x(1− x))

1 = a1a2(1− (1− 1a2 ))(1− a2(1−1a2

)(1− (1− 1a2

)))

1 = a1a2(1a2

)(1− a2(1− 1a2 )(1a2

))

1 = a1(1− (1− 1a2 )) =a1a2

a2 = a1

Therefore, xs = 0 is the only fixed point of f1(f2(x)) and thus F2(x).

2.3 Periodic Orbits

As I just showed, when a1 = a2 and a1, a2 ∈ [1, 3], this is the only case where a period

1 orbit for F2(x) exists. This is true for all Fn(x).

13

Theorem 2.3.1. Given F1(x) = a1x(1 − x), F2(x) = 12(a1(1 − (−1)t)x(1 − x) +

a2(1 − (−1)t+1)x(1 − x)), and Fn(x) = 12(a1(1 − (−1)floor( t

n))x(1 − x) + a2(1 −

(−1)floor( tn+1))x(1 − x)) where a1, ..., an ∈ (1, 3),n ∈ (3,∞), n and t positive inte-

gers. If a1 = a2 = ... = an then Fn(x) = F2(x) = F (x).

Proof. Given Fn(x) as defined above. Let a1 = a2 then Fn(x) =12a1x(1 − x)((1 −

(−1)floor( tn ) + 1− (−1)floor( tn+1))) which after some simplification implies that

Fn(x) = 212(a1x(1 − x)) = F1(x). Given F2(x). Let a1 = a2 then F2(x) = 12a1x(1 −

x)((1− (−1)t) + (1− (−1)t+1)) = 12(a1x(1− x))2 = F1(x)

So Fn(x) = F2(x) = F1(x).

While there are no fixed points when a1 6= a2, there almost always exists an orbit

of period 2. One example of an orbit of period 2 is given by (x0, x1) = (1 − 1a2 ,1a2

).

The figure below shows the case where a1 =32

and a2 = 3.

Figure 2.1: Stable Periodic behavior of (x0, x1) = (1− 1a2 ,1a2

) = (23, 1

3)

14

Theorem 2.3.2. Given f1(x) = a1x(1− x), f2(x) = a2x(1− x), 0 < a1 < a2 < 4 and

1a1

+ 1a2

= 1, then the periodic orbit of period 2 is given by (x0, x1) = (1− 1a2 ,1a2

).

Proof. By definition, f2(x0) = 1 − 1a2 = x0 because x0 is a fixed point for f2(x).

Substituting x0 into f1 gives: f1(x0) = a1(1 − 1a2 )(1 − (1 −1a2

) = a1(1 − 1a2 )(1a2

). If

1a1

+ 1a2

= 1 then f1(x0) =1a2

= x1. f2(x1) = a2(1a2

)(1− 1a2

) = 1− 1a2

= x0, so (x0, x1)

is a periodic orbit of period 2.

1a1

+ 1a2

= 1 implies that 2 < a2 < 4 and 0 < a1 < 2. Consider the case when a2 < 2

then f2(x0) = x0 and f1(x0) = a1(1− 1a2 )(1− (1−1a2

)). Let f1(x0) = x0 and solve for

x0 this results in 1− 1a2 = a1(1−1a2

)( 1a2

), so 1 = a1a2

which means a1 = a2 = 2 which

is a contradiction, since a1 < a2. So there are no period 2 orbits when a2 < 2.

If a2 = 2 then the fixed points of f(x) = 2x(1 − x) are xs1 = 0 and xs2 = 12 .

f1(12) = a1(

12)(1

2) = a1

4. Well, a1

4= 1

2when a1 = 2, which means a1 = a2 = 2. So, the

period 2 orbit can only exist when 2 < a2 < 4.

This is not to say that there are not other orbits of period 2. In fact, with the

help of Mario Martelli, I found a range of values for a1 and a2 where we know a

periodic orbit of period 2 will exist. Assume 0 < a1 < a2 < 3. If G(x) = a1a2x(1 −

x)(1− a2x(1− x)) = x has a real solution then there exists a periodic orbit of period

2. There is always a non-zero solution to G(x) = x because by a corollary to the

15

Intermediate Value Theorem, which says that since F is continuous and there exists

an interval [a,b] such that G(a) and G(b) ∈ [a, b] there exists a fixed point [4]. (

One example is a = .1, b = .8.) Solving the cubic resulting from G(x) = x gives one

real and two complex roots, for my purposes I only consider the real solution. The

following graph shows the intersection of the real solutions with the x = 0 and y = z

planes to help separate positive solutions as I need 0 < a1 < a2 < 3.

Figure 2.2: Values that result in period 2 orbits of G(x) = a1a2x(1−x)(1−a2x(1−x))

Recreating this figure into two dimensions results in the following graph:

Figure 2.3: Possible Period 2 Orbits of G(x) =1 a2x(1− x)(1− a2x(1− x))

16

The possible choices for a1 and a2 so that there exists an orbit of period 2 are in

the enclosed region of this graph. When a2 = 3, a1 =13. As a1 increases the range of

values for a2 begins the horizontal line a2 = 3 before the graph meets the line a1 = a2.

After that, the range of values for a1 is from a2 to 3.

Numerical investigations of the periodic orbits of F2(x) led to the following table:

Period a 2.8 3.2 3.5 3.561 2.4 2 2 4 81 2.8 1 2 4 82 3.2 2 2 4 84 3.5 4 4 4 88 3.56 8 8 8 8

Table 2.1: Periodic behavior of F2(x)

Table 2.1 shows the period of F2(x) when iterated with two a values, one from

the column of a values and one from the top row of a values. The only case where

there is a period 1 result was when a1 = a2, a ∈ (0, 3). Also, all cases where a1 = a2

maintain the original period of a1under F (x). The examination Table 2.1 led me to

five major conclusions : Theorem 2.3.1 ( at the beginning of this section), Proposition

2.3.3, Corollary 2.3.4, Proposition 2.3.5 and Proposition 3.2.1.

This table (2.1) shows that when F2(x) is iterated the original period of the larger

a values becomes the resulting period of F2(x). For example, let a1 = 3 and a2 = 3.56,

then the period of F2(x) is 8, which is the same as the period of a = 3.56 under F (x).

17

Proposition 2.3.3. Given equations f1(x) = a1x(1−x) and f2(x) = a2x(1−x), if f1

has a periodic orbit of period p and f2 has a periodic orbit of period q and 1 < p < q

then F2(x) has periodic orbit of period q.

All of the cases with distinct values of athe resulting period of F2(x) is even, thus

divisible by two which is the same as the number of equations. This property also

holds for the cases where there are more than two a values.

Corollary 2.3.4. Given equations f1(x) = a1x(1− x) and f2(x) = a2x(1− x) (equa-

tions 2.1.1 and 2.1.2), if a1 6= a2 then all the resulting periods of F2(x) are divisible

by 2, both the number of equations and the number of a values.

After examining what happens when the two equations, f1(x) and f2(x), alternate

every iterate, I considered cases where the change of a values from a1 to a2 did not

occur on every iteration, for example, when the change occurred on the 5th iteration.

The results are:

Period a 2.8 3.2 3.5 3.561 2.4 10 10 10 101 2.8 1 10 10 102 3.2 10 2 20 204 3.5 10 20 4 208 3.56 10 20 20 8

Table 2.2: Periodic behavior of F5(x)

Table 2.2 shows that, again, the only period 1 point is when a1 = a2, a ∈ (0, 3).

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Also the other cases where a1 = a2 retain their original periods from F (x). However,

Proposition 2.3.3 does not hold for Fn(x), as the resulting periods are no longer the

original periods, p and q. Interestingly enough, the examination of F5(x) shows that

when a1 6= a2 all the resulting periods are divisible by 10. So, Corollary 2.3.4 holds

and F5(x) is divisible by 5. In fact, all periodic orbits of Fn(x) are divisible by n.

(See appendix A for more numerical results.)

Proposition 2.3.5. Given 2 equations of the form xn+1 = axn(1− xn) with

distinct real 1 < a < 3.7 of a known period, the periodic orbits of Fn(x) =12(a1(1 −

(−1)floor( tn )x(1− x)) + a2(1− (−1)floor(tn

+1))x(1− x)) are divisible by n.

Propositions 2.3.5 only considers one of the possible iteration schemes. Other

iteration schemes could have completely different results, in some of them it may be

easier to consider them as separate parameters, for example F5(x) can be considered

to have 6 parameters, but 5 of them are equal.

2.4 Aperiodic Orbits and Chaotic Behavior

There are many cases where periodic orbits exist when using a values that under

F (x) are periodic. From there, I began to explore what happens when one of the a

values causes chaotic behavior or a period 3 orbit under F (x). If there exists dense

and unstable orbits, then Fn(x) is still chaotic. However, what about the theorem

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of Li-Yorke? Does it still hold? Numerical evidence shows that as long as a1, a2 are

distinct, every periodic orbit of Fn(x) is a multiple of 2. So there may be a period 6

orbit, but it is unlikely that an orbit of period 3 exists. If there are no orbits with odd

numbered periods then the theorem of Li-Yorke would not hold or at least have the

same consequences. Extending Table 2.1 shows that there exists aperiodic behavior

for F2(x). (See the appendix for other Fn.) When a = 3.81, there exists F (x) has

Period a value a=3.81 a=3.831 2.4 4 41 2.8 6 62 3 aperiodic aperiodic2 3.2 aperiodic aperiodic4 3.5 4 48 3.56 4 4

Table 2.3: Periodic behavior using with an aperiodic a

a period 3 orbit, thus it is very interesting to see that when a1 = 3 and a2 = 3.81,

the orbit appears to be aperiodic. Also, it is surprising that the aperiodic behavior

seems to occur only when a1 ∈ (3, 1 +√

6), which is periodic of period 2 under F (x).

In addition, the same behavior occurs when a2 = 3.83 which is chaotic under F (x).

This is an area that demands further investigation.

20

Chapter 3

Three and Beyond

3.1 Introduction

Periodic changes aren’t necessarily a simple variance between two numbers. An en-

vironment could change periodically between 3, 4, or even 20 different situations.

Looking at these cases theoretically is much more difficult than with only two pa-

rameters. So I only present numerical evidence in this chapter for further study and

discussion.

3.2 Three and Four Equations

When considering 3 equations: f1, f2 and f3 of the form fi(x) = aix(1− x), there are

two main situations that have to be considered. What happens when all the a values

are distinct? What happens when two of the a values are equal? Table 3.1 suggests

many areas for further investigation.

21

Original Period Points Resulting Period1,1,1 2.4,2.4,2.4 11,1,1 2.2,2.4,2.8 31,1,2 2.2,2.4, 3.2 31,1,4 2.2,2.4,3.5 31,1,8 2.2,2.4,3.56 31,2,2 2.8,3.1,3.2 31,2,2 2.9, 3.1,3.2 61,2,4 2.8,3.1,3.2 31,2,4 2.9,3.1,3.2 61,2,8 2.5,3.2,3.56 31,2,8 2.8,3.2,3.56 61,4,4 2.4,3.5,3.51 31,4,4 2.8,3.5,3.51 61,4,4 2.9,3.5,3.51 121,4,8 2.4,3.5,3.56 31,4,8 2.4,3.5,3.56 61,4,8 2.9,3.5,3.56 122,2,2 3.1,3.2,3.3 62,2,4 3.1,3.2,3.5 122,2,8 3.1,3.2,3.56 aperiodic2,2,8 3,3.3,3.56 62,4,4 3,3.5,3.51 122,4,8 3,3.5,3.56 122,4,8 3.2,3.5,3.56 aperiodic4,4,4 3.5,3.51,3.52 124,4,8 3.5,3.51,3.56 18

Table 3.1: Periodic behavior of F(x) when it oscillates between 3 values for a

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Remark 3.2.1. Table 3.1 is set up similarly to Tables 2.2 and 2.3, however the left

column lists the periods of F (x) with the given a values in column two. The third

column lists the resulting period when F (x) oscillates between 3 a values.

As I showed in the last chapter, the only case where there is a fixed point or orbit

of period 1 is when a1 = a2 = a3 and a ∈ (0, 3). Numerically, Table 3.1 shows that

in all of the cases where there exists a periodic orbit (besides 1), the number of the

period is divisible by 3. This implies that there are orbits of period 2,4,5,7... which

means that the theorem of Li-Yorke does not hold as there are not orbits of every

period. However, it is possible that variations that do not conclude that there exists

an orbit of every period, may work.

Unlike the cases with 1 or 2 equations, the intervals that are periodic for F (x) =

ax(1−x) do not hold in the same way. When only considering F (x) there are clearly

defined ranges with defined periods. With two a values, the ranges don’t hold in the

same way, but given that a1 < a2 the resulting period is the same as the period of F (x)

with a2 as it’s a value. However, with three a values there seem to be subintervals

that cause the resulting period to shift sooner than expected. For example, when

using a values 2.8, 3.1 and 3.2, you get a period 3 orbit; but 2.9, 3.1 and 3.2 results

in a period 6 orbit. One of the places that it seems to shift (in certain combinations)

is when a1 = 2.658. When a1 < 2.658, it is period 3 (when paired with 3.2 and 3.56).

When a1 = 2.658, it’s almost a period 3, but closer to a periodic orbit of period 6.

23

When a1 > 2.658 it has a period 6 orbit.

It is surprising to note that some cases, like 3.1, 3.2 and 3.56, appear to exhibit

aperiodic behavior, yet none of their a values originally tend to chaotic behavior.

These complications caused me to limit my research on 4 parameters to variations of

three, in other words, two of the a values are equal.

Original Period a values Resulting Period1,1,2,4 2.4,2.4,3,3.5 41,1,2,8 2.4,2.4,3,3.56 41,2,2,4 2.8,3,3,3.5 81,2,2,8 2.4,3,3,3.56 81,2,4,4 2.4,3,3.5,3.5 81,2,8,8 2.4,3,3.56,3.56 81,4,4,8 2.4,3.5,3.5,3.56 aperiodic1,4,8,8 2.4,3,3.5.56,3.56 82,2,4,8 3,3,3.5,3.56 42,4,4,8 3,3.5,3.5,3.56 aperiodic2,4,8,8 3,3.5,3.56,3.56 16

Table 3.2: Periodic behavior of F(x) when it oscillates between 4 values for a

Tables 3.1 and 3.2 both show that when at least three of the a values are distinct

that the iterations are all divisible by the number of a values. This leads to the

following proposition:

Proposition 3.2.1. Given t equations of the form xn+1 = axn(1− xn), with distinct

real a values, 1 < a1 < a2 < ... < at

It would be interesting to explore this, particularly when the iterations do not

occur in increasing order of the a value.

3.3 Conclusions & Directions for Further Research

While answering many questions about the behavior of F (x) = ax(1− x) with oscil-

lating parameters, this thesis revealed many questions to be answered in with further

study the future. I have indicated in some parts of this thesis some areas that I

believe require more investigation. I discuss some of these ideas more in this section.

When a oscillates between two parameters, I proved that there does not exist

a fixed point. This proof can be extended to n parameters. I would recommend

examining what happens numerically when the difference between the parameters is

very small and comparing it to what happens when a is constant. Are the differences

actually distinguishable?

I also found a specific stable period 2 orbit, and proved analytically when a period

2 orbit can exist when F (x) oscillates between 2 parameters. The same should be done

for at least periods 3 and 4. Proving this for general periods would be interesting, or

at least showing that there exists an orbit of period p when F (x) oscillates between

p parameters.

25

The study of aperiodic and chaotic behavior of F (x) when it oscillates between

any number of parameters, has not been studied in depth yet to my knowledge. More

numerical investigation is a key place to start for this topic. From there, I recommend

comparing all the definitions of chaotic behavior mentioned in this thesis and other

theorems about the existence of chaos and how they apply to the cases where the

parameters oscillate and certain periods do not exist. Before doing that, one needs

to verify that for 2 parameters no orbits of odd periods actually exist and that for 3

parameters no periods of 2n exist, and so forth for other numbers of parameters.

There are many directions that others can go to continue my research. The study

of this equation is just the beginning, there are many other models where similar

examinations can be done. I hope that people do continue researching this area, from

undergraduate students to professionals, both from a mathematical standpoint and

from a biological one, in the classroom and outside in a field with the sun shining

down.

26

Appendix A

Resulting Periodic Behavior Data

Remark A.0.1. These tables are of the same format as Table 2.2. (See Page 18 for theexplanation.) Also, When an entry in the tables below is aperiodic then numericallyit appears that there is chaos or at least aperiodic behavior. However, it is possiblethat an orbit of a very large period may exist.

Period a 2.8 3.2 3.5 3.56 3.81 3.831 2.4 12 12 12 24 6 61 2.8 1 6 12 24 6 122 3.2 6 2 6 6 Aperiodic Aperiodic4 3.5 12 6 4 12 Aperiodic Aperiodic8 3.56 24 6 12 8 Aperiodic Aperiodic

Table A.1: Periodic behavior of F3(x)

F3(a, x) = 12 (a1(1− (−1)(floor t3 ))x(1− x) + a2(1− (−1)floor

t3+1)x(1− x)))

Period a 2.8 3.2 3.5 3.56 3.81 3.831 2.4 8 8 8 8 8 81 2.8 1 8 8 8 8/16 8/162 3.2 8 2 8 8 Aperiodic Aperiodic4 3.5 8 8 4 16 Aperiodic Aperiodic8 3.56 8 8 16 8 Aperiodic Aperiodic



t4+1)x(1− x))

27

Period a 2.8 3.2 3.5 3.56 3.81 3.831 2.4 10 10 10 10 10 101 2.8 1 10 10 10 10 102 3.2 10 2 20 20 10 104 3.5 10 20 4 20 Aperiodic 208 3.56 10 20 20 8 10 Aperiodic



t5+1)x(1− x))

Period a 2.8 3.2 3.5 3.56 3.81 3.831 2.4 12 12 12 12 12 121 2.8 1 12 12 12 Aperiodic 122 3.2 12 2 12 36 Aperiodic Aperiodic4 3.5 12 12 4 12 Aperiodic Aperiodic8 3.56 12 36 12 8 Aperiodic Aperiodic


Where F6(a, x) = 12 (a1(1− (−1)(floor t6 ))x(1− x) + a2(1− (−1)floor

t6+1)x(1− x))

Period a 2.8 3.2 3.5 3.56 3.81 3.831 2.4 14 14 14 14 14 141 2.8 1 14 14 14 14 Aperiodic2 3.2 14 2 14 14 Aperiodic Aperiodic4 3.5 14 14 4 42 Aperiodic 288 3.56 14 14 42 8 Aperiodic Aperiodic



t7+1)x(1− x))

28

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[2] J. M. Cushing, Shandelle M. Henson, The Effect of Periodic Habitat Fluctuations

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[3] J. M. Cushing, Shandelle M. Henson, Global Dynamics of Some Periodically

Forced, Monotone Difference Equations, J. Difference Equations and Applica-

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[4] Mario Martelli, Introduction to Discrete Dynamical Systems and Chaos, John

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