Investigation of Transition to Chaos for a Lotka Volterra System … · 2015-09-18 ·...

Applied Mathematical Sciences, Vol. 9, 2015, no. 117, 5801 - 5837

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.57506

Investigation of Transition to Chaos for a Lotka–

Volterra System with the Seasonality Factor Using

the Dissipative Henon Map

Yu. V. Bibik

Federal Research Center, “Computer Science and Control”

of Russian Academy of Sciences, Vavilov str. 38/40, 119333, Moscow, Russia

Copyright © 2015 Yu. V. Bibik. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

Conditions under which the classical Lotka–Volterra system with a seasonality

factor exhibits chaotic behavior are investigated. The seasonality factor is

introduced in such a way that the system can be investigated using the Henon

map. The reduction of the system to Henon’s map allows one to calculate the

period doubling bifurcations and determine the point of the transition to chaos.

Keywords: Lotka–Volterra system, seasonality, Henon map, bifurcations,

transition to chaos

1. Introduction

Conditions of the emergence of bifurcations and conditions for the transition to

chaos in a classical mathematical biology problem—the Lotka–Volterra model to

which the author added a seasonality factor are investigated. This factor is

represented by a dependence of the system coefficients on time and makes it

possible account for the influence of seasonal temperature variations on the

species population. It turned out that the seasonality factor changes the behavior

of the classical Lotka–Volterra system, which acquires the fundamental properties

of the universal transition to chaos. The system with the seasonality factor is

reduced to the dissipative Henon map. A renormalization of the dissipative Henon

map using an analog of Hellemann's method [1] makes it possible to find the con-

5802 Yu. V. Bibik

ditions under which period doubling bifurcations occur and determine the point of

the transition to chaos.

Historical background

The classical Lotka–Volterra system of equations was first proposed by the

American mathematician, statistician, and demographer Lotka [15] (1925) and by

the Italian mathematician Volterra [23] (1926), [24] (1931). This system gives an

adequate description of the dynamics of two-species biological systems (predator

and prey) when the specious population is not too large. It can be investigated

analytically. The classical Lotka–Volterra system is Hamiltonian with one degree

of freedom, it is integrable by quadratures, and therefore exhibits no chaotic

behavior.

Attempts to make mathematical biology models more realistic by taking into

account additional features and phenomena result in complications in the

analytical investigation and to the loss of integrability.

To analyze the model, we will employ modern methods that are used for the

study of chaotic dynamics.

Beginning in the 1960–1970s, researchers (physicists, mathematicians, biologists,

and ecologists) paid attention to chaotic phenomena and discovered certain order

and regularities in them. An important conclusion made in the theory of chaos is

the fact that even insignificant variations in any part of such a system can result in

a radically different development of the system. Chaotic phenomena were studied

by many researchers.

Here, I want to mark out the researchers who can be considered to be

pioneers in the theory of chaos.

It is remarkable that their brilliant discoveries were made literally with the end of

a pen based on the intuition and using only simple calculators and primitive

computers.

One of those researchers was the American meteorologist and mathematician E.

Lorenz. While studying weather prediction, he proposed and analyzed a system of

three coupled differential equations that specify a flux in the three-dimensional

space. This system could not be analyzed using known attractors, i.e.. geometric

figures that describe the behavior of the system in the phase space. The new

attractor found by Lorenz, which was later called by his name, provided an

example of a chaotic or strange attractor that has a more complex structure than

the attractors known earlier. The computer model of the atmosphere proposed by

Lorenz showed that even insignificant variations in the atmosphere can result in

radical and unexpected consequences. It was Lorenz who revealed in [13, 14] the

main features of the system that provide a key to the understanding of chaotic

behavior.

Investigation of transition to chaos for a Lotka–Volterra system 5803

The French mathematician Henon wanted to construct a simpler map than

Lorenz's system. He proposed a two-dimensional attractor with the same

properties as Lorenz's attractor. Henon's model was easier to be analyzed

mathematically, and numerical computations were faster and more accurate.

Henon's map is a reference two-dimensional map [9].

A considerable contribution to the theory of chaos was made by the brilliant

works of the American physicist М. Feigenbaum. In particular, he investigated the

logistic map. Feigenbaum showed that chaos can emerge via bifurcations. He

revealed universal laws of transition to chaos in the process of period doubling.

Using the renormalization group method, he created a theory that explained the

universality of period doubling. He also discovered a new mathematical

constant—Feigenbaum's constant that describes period doubling bifurcations for

one-dimensional maps. He found out that the points of period doubling

bifurcations accumulate near a certain point—the threshold of transition to

chaos—by a geometric progression with the ratio 4.669. This ratio turned out to

be universal and valid for other maps and various nonlinear dissipative systems

[3, 4, 5].

Feigenbaum's discovery was confirmed experimentally by the French physicist А.

Libchaber, who found a cascade of bifurcations resulting in chaos in dynamical

systems [11, 12].

The conservative Henon map was renormalized by the Dutch physicist

Hellemann. The renormalization made it possible to determine the sequence of

period doubling bifurcations and obtain a new constant for the two-dimensional

case with the ratio 9.09 (8.72109) (the number in parentheses is the best value of

the constant obtained so far). This constant is an analog of Feigenbaum's constant

for the one-dimensional case [7, 8].

Renormalizations of other area preserving maps were described in the works of

the American physicist MacKay [16].

The French and American mathematician Mandelbrot is the founder of fractal

geometry. While studying various phenomena, such as variations in cotton prices

and noises in electronic circuits, he noticed that absolutely random processes bear

indications of similarity. It was Mandelbrot who defined the concept of fractal.

This concept became a global concept in the physics of chaos, and it made the

picture of reality more clear [17].

The Russian physicist Chirikov published the paper [2] in 1979, where he

proposed a novel approach to the investigation of chaos in Hamiltonian systems

using the resonance overlap method.

The modern view of the theory of deterministic chaos is presented in the works

by the German physicist Shuster [18, 19], American physicist Tabor [20],

Russian and American physicist Zaslavsky [25], British physicist Thompson [21,

22], and the American researchers Fishman and Egolf [6].

The theory of strange attractors is presented in the collection of papers [10] edited

by Hunt, Li, Kennedy, and Nusse.

5804 Yu. V. Bibik

The research presented in this paper is important because a realistic system of

equations is investigated. It makes it possible to take into account the influence of

seasonality on the dynamics. The modern techniques of the theory of chaos allow

us to find the whole chain of period doubling bifurcations and the point of the

transition to chaos.

Investigation techniques:

- we introduce into the classical Lotka–Volterra system the seasonality factor in

such a way that the continuous map could be transformed into a discrete one,

which facilitates the investigation;

- the second step is to eliminate one of the variables from the system of discrete

equations in two variables to simplify the problem;

-the third step is to reduce the resultant discrete map with a single variable to the

dissipative Henon map. This map is renormalized using a generalized Hellemann's

method. The map thus obtained is used to analyze conditions for the emergence of

period doubling bifurcations and conditions for the transition to chaos.

The paper is organized as follows:

1. Introduction.

2. Transformation of the classical Lotka–Volterra system to a two-dimensional

discrete map by adding a seasonality factor.

3. Transformation of the system of discrete equations in two variables to a discrete

equation in one variable.

4. Reduction of the discrete equation in one variable to the dissipative Henon

map.

5. Renormalization of the dissipative Henon map.

6. Conditions for the emergence of the first and second period doubling

bifurcations for the one-dimensional discrete map.

7. Conditions for the emergence of the next period doubling bifurcations and

conditions for the transition to chaos for the generalization of the Lotka–Volterra

equations with a seasonality factor.

8. Description and analysis of figures.

9. Conclusions.

2. Transformation of the classical Lotka–Volterra system to a two-

dimensional discrete map by adding a seasonality factor

Let us add a seasonality factor to the classical Lotka–Volterra system (2.1), (2.2).

To this end, we introduce into the equations coefficients that depend on time, and

then transform the resulting equations to a two-dimensional discrete map. In the

general case, the introduction of time-dependent coefficients considerably

complicates the system's behavior and analysis. However, we introduce the

seasonality factor in such a way that even though the behavior of the Lotka–

Volterra system becomes more complex, its analysis becomes simpler. The differ-


ential equations are reduced to discrete equations. This is achieved due to the use

of delta functions for modeling the dependence of the system's coefficients on

time. The effectiveness of this method is largely explained by the specific

algebraic structure of the Lotka–Volterra system. After discarding the linear terms

containing the delta functions, the system is easily integrated. This is the first step

in the proposed method for obtaining discrete equations. At the second step, the

influence of the delta functions on the system's dynamics is taken into account.

The nonlinear terms do not play any role in this case. Thus, the problem is solved

in two steps. To obtain difference equations, the classical Lotka–Volterra

equations are used as the original ones. They have the form

,xyxdt

dx (2.1)

.xyydt

dy (2.2)

Here, the variable x is the prey population and y is the predator population. The

coefficients and determine the intensity of the species interaction with the

environment, and and determine the intensity of interaction between the

species. Below we assume that and depend on time. The coefficients and

can be made equal to unity by an appropriate change of variables.

If we assume that the time-dependent coefficients and have a constant value

in winter and in summer, the system still remains too complicated to be analyzed

analytically. For that reason, we make further simplifications. We assume that the

increase of the prey biomass x due to external factors occurs at a point in time in

the beginning of summer, and the decrease in the predator biomass due to external

factors occurs at the same time, which can be considered as the end of winter. We

denote this time by nTt , where n is the integer number indicating the number

of cycles of increase and decrease of the prey and predator biomasses, and T is the

year duration.

Then, the time-dependent and can be written as

n

nTt 1)( , (2.3)

n

nTt .)( 2 (2.4)

Here 1 and 2 are the amplitudes of the corresponding delta functions. Formulas

(2.3) and (2.4) show the way in which the seasonality factor is represented in the

present paper. They represent the influence of seasonal temperature variations on

the dynamics of the two-species interaction. They are a mathematical reflection of

the fact that the increase in the biomass of prey and the decrease in the biomass of

5806 Yu. V. Bibik

predators due to external factors occur at the times nTt . The coefficients

and are represented by sums of delta functions. These coefficients vanish

everywhere except for nTt . These are the points in time when by our

assumption the winter is replaced by summer. Therefore, the equations are

simplified everywhere except for these points. In the simplified form, these

equations are valid everywhere on the time interval from to T , where is

an infinitesimal quantity.

The first step to deriving difference equations. At the first step, we simplify the system of equations (2.1), (2.2) with the

coefficients and having the form (2.3) and (2.4). To implement this step, we

neglect the sum of the delta functions in (2.3) and (2.4) on the interval from to

T . This simplifies the system integration. Let us make the change of variables

q =ln(x), p=ln(y) to reduce Eqs. (2.1), (2.2) to the form

n

p

t enTtq ,)( 1 (2.5)

n

q

t enTtp .)( 2 (2.6)

The coefficients and defined by (2.3) and (2.4), respectively, appear in these

equations linearly. To obtain difference equations, we should integrate Eqs. (2.5),

(2.6) on the interval from to T . We solve this problem in two steps. First,

we integrate the differential equations (2.5), (2.6) on the interval from to T

and then on the interval from T to T . (Here, is a small parameter that

allows us to decompose the integration of Eqs. (2.5), (2.6) into two steps. At the

first step, the influence of the delta functions is neglected).

At the first step, we have the equations

p

t eq , (2.7)

q

t ep . (2.8)

To solve these equations, it is convenient to return to the original variables x and

y. We have

xyxt , (2.9)

xyyt . (2.10)

Next, we reduce the two equations (2.9) and (2.10) in two variables to one

equation in one variable. For this purpose, we use a conservation law. It is implied

by Eqs. (2.9), (2.10) that the variable equal to the sum of species populations is

preserved:


0)( tt yx . (2.11)

This conservation law enables us to replace the investigation of the population of

two species to the investigation of the population of one species, which is

considerably simpler. We have

xy . (2.12)

As a result, only the variable x remains in Eq. (2.9):

22222 )()( aXaaxxxxxxt . (2.13)

For the convenience of calculations, we introduce the new variable X defined by

axX , (2.14)

2

a . (2.15)

In terms of the new variables, Eq. (2.9) becomes very simple:

22 aXX t . (2.16)

This equation is solved by separation of variables as

dtaX

aXd

aaX

dX

ln

2

122

. (2.17)

The left-hand side is integrable in elementary functions. Upon the integration of

the left-hand side we obtain

aTaX

aX

aX

aX2lnln

0

0

. (2.18)

Let us transform Eq. (2.18) to an equation in x , y , 0x , and 0y . To this end, we

use the relationships between the variables 0X , 0x , 0y , a , and :

00 yaX , (2.19)

00 xaX , (2.20)

xaX , (2.21)

5808 Yu. V. Bibik

xaX . (2.22)

Using these relations, we obtain from (2.18) an equation connecting the variables

x , y , 0x , and 0y

Tx

y

x

x

0

0lnln . (2.23)

To simplify Eq. (2.23), we get rid of the logarithm:

Te

x

yxx

0

0)( . (2.24)

Now we can easily find equations for x and y , which are the populations of the

species at the time Tt

Teyx

xx

00

0

, (2.25)

xy . (2.26)

Thus, we have transformed the original differential equations into the preliminary

difference equations (2.25), (2.26). The delta functions were not used for this

purpose. Formulas (2.25) and (2.26) relate the values of the variables at the time

with their values at Tt . Therefore, we have integrated the simplified

system of equations on the time interval from to T . At the second step, in

order to obtain the final difference equations, we take into account the delta

functions while integrating on the time interval from T to T , which yields

difference equations instead of Eqs. (2.25), (2.26).

Second step: Derivation of the ultimate difference equations.

To obtain the final form of the difference equations, we integrate Eqs. (2.5), (2.6)

on the time interval from T to T . Up to , we obtain

)()()( 1 OTqTq , (2.27)

)()()( 2 OTpTp . (2.28)

The preliminary difference equations are written for the variables x and y. For

that reason, upon deriving the relations between the variables )( Tq and

)( Tp with the variables )( Tq , )( Tp , we return to the variables x, y .


Calculate the exponential function of the left and right-hand sides of Eqs. (2.27)

and (2.28) to obtain

)()()( 1 OTxTx , (2.29)

)()()( 2 OTyTy . (2.30)

Here, 1 and 2 are the exponential functions of the delta function amplitudes:

1

1

e , (2.31)

2

2

e . (2.32)

By combining (2.25), (2.26) and (2.29), (2.30), we obtain the final form of the

difference equations:

]))exp((

)([11

Tyxyx

xyxx

nnnn

nnn

n

, (2.33)

]))exp((

)()[(21

Tyxyx

xyxyxy

nnnn

nnn

nnn

. (2.34)

Here, 1nx and 1ny are the populations of species at the time Tnt )1( , and T

is the year duration. Thus, we have reduced the differential equations (2.5), (2.6)

to the difference equations (2.33), (2.34). Therefore, the classical Lotka–Volterra

system has been reduced to a two-dimensional discrete map by adding the

seasonality factor.

3. Transformation of the system of discrete equations in two

variables to a discrete equation in one variable

The difference equations (2.33), (2.34) are still too complicated to be analyzed

analytically. We simplify them while preserving their basic properties. Introduce

the new variables

nn Txx , (3.1)

nn Tyy . (3.2)

In terms of these variables, Eqs. (2.33) and (2.34) take the form

5810 Yu. V. Bibik

])exp(

)([11

nnnn

nnn

nyxyx

xyxx

, (3.3)

])exp(

)()[(21

nnnn

nnn

nnnyxyx

xyxyxy

. (3.4)

Now they are independent of the year duration T and can be reduced to form (3.3),

(3.4) for any T. Next, we transform (3.3), (3.4) to make them more convenient for

the analysis. To this end, we introduce the variables

n

n

ny

xu , (3.5)

nnn yxz . (3.6)

Then, Eqs. (3.3), (3.4) take the form

])exp(

[11

nn

nn

nzu

uzx

, (3.7)

or

]))(exp(1

)exp([11

nn

nnn

nzu

zuzx

. (3.8)

For 1ny we have

]))(exp(1

)exp([21

nn

nnn

nnzu

zuzzy

(3.9)

or

].))(exp(1

[21

nn

n

nzu

zy

(3.10)

Let us eliminate the variables 1nx , 1ny on the left-hand sides of Eqs. (3.8) and

(3.10). To this end, we divide Eq. (3.8) by (3.10) to obtain

)).(exp(2

11 nnn zuu

(3.11)


By summing Eqs. (3.8) and (3.10) we obtain

])(exp1

)exp()( 2121

nn

nnn

nnzu

zuzzz

. (3.12)

To simplify the dependence on the variables nu and nz , we take the logarithm of

both sides of Eq. (3.11):

nnn zuu

)ln()ln()ln(

2

11 . (3.13)

Now, the dependence of the logarithm of 1nu on the logarithm of nu and on nz is

simple.

To simplify Eq. (3.13) even further, we introduce the new variables

)ln( nn uq , (3.14)

nnn zqa

)ln(

2

1 . (3.15)

Then, Eq. (3.13) takes the form

nnn aqu 11)ln( . (3.16)

Using (3.15), we obtain for nz the formula

nnn aaz

1

2

1 )ln( . (3.17)

Finally, using (3.16) and (3.17), we can obtain an equation in one variable na . To

this end, we plug (3.16) and (3.17) into (3.12) to obtain

n

n

a

a

nn

nnnn

e

eaa

aaaa

2

1

2

11

2

1

211

2

121

2

1

1

))(ln(

)())(ln()ln(

.

(3.18)

Rearrange the terms and reverse the signs on the right- and left-hand sides to

obtain

5812 Yu. V. Bibik

1

))(ln(

)-(--)1( )ln()1(

1

2

1

21

2

1

21122

2

121

n

n

a

a

nn

nnn

e

eaa

aaa

.

(3.19)

Thus, the system of two discrete equations (2.33), (2.34) in two unknowns has

been transformed to the discrete equation (3.19) in a single unknown. In the next

section, we continue the transformation of Eq. (3.19) to reduce it to the dissipative

Henon map. This will enable us to determine conditions for the transition to

chaos.

4. Reduction of the discrete equation in one variable to the

dissipative Henon map

To find the conditions under which the Lotka–Volterra system with the

seasonality factor begins to exhibit chaotic behavior, we reduce Eq. (3.19) to the

dissipative Henon map. The Henon map is a typical system in which it can be

seen how deterministic chaos emerges. This map is given by the equations

nnn yaxx

2

1 1, (4.1a)

nn xy 1 . (4.1b)

Let us introduce the notation

;2

nn xa

x ;0122 a aA , (4.2)

Using notation (4.2), we transform Eqs. (4.1a), (4.1b) to the form

nnnn yxAxx

2

1 22, (4.3)

nn xy 1 . (4.4)

Next, Eqs. (4.3), (4.4) in two variables are transformed to an equation in one

variable:

2

11 22 nnnn xAxxx . (4.5)

To simplify Eq. (3.19) and reduce it to Henon's equation (4.5) we should

overcome some difficulties.


The first difficulty is that the right-hand side of Eq. (3.19) includes a stronger

linearity than the quadratic nonlinearity in Henon's map. To overcome this

difficulty, we approximate this nonlinearity by a quadratic term in the vicinity of

the fixed point of period one. Map (3.19) produces a sequence of points.

Typically, it converges to one or several points. When it converges to a single

point, we deal with a fixed point of period one.

The second difficulty is that, in addition to a stronger nonlinearity, the right-hand

side includes the term 1na along with the term na . The term 1na appears on the

right-hand side of Eq. (3.19) as the difference with na . We should eliminate it

because it complicates the transition to Henon's map. The idea of replacing 1na

with the fixed point of period one seems to be quite reasonable. The first step in

the construction of the simplified map is as follows: replace the term

n

n

a

a

nn

e

eaa

F

1

2

1

21

2

1

21

1

))(ln(

)(

(4.6)

in Eq. (3.19) with the simplified term

n

n

a

a

n

e

eaaw

F

1

2

1

2*21

2

1

21*

1

)])(,()[ln(

)(

. (4.7)

Here *a is the fixed point of period one and F is the nonlinear part of map

(3.19). In turn, *F is the simplified nonlinear part map (3.19). The choice of *F

makes the fixed point of period one invariant. The next step is to reduce Eq. (3.19)

with the term *F to Henon's map. To this end, we change to the variable nx :

nn xaa * . (4.8)

Now, the nonlinear term *F takes the form

n

n

xa

xa

n

ee

eexw

F*

*

1

2

1

221

2

1

21*

1

])(,()[ln(

)(

. (4.9)

5814 Yu. V. Bibik

Here, w is replaced with w for convenience.

The new term *F contains the product of the linear term in nx with a nonlinear

term, which is denoted by Q . The factor Q has the form

n

n

n

n

xa

xa

a

a

ee

ee

e

e

Q*

*

1

2

1

2

1

2

1

2

11

. (4.10)

To make further simplifications, we retain in Q the degrees of nx not greater than

two. The resulting nonlinear factor will be denoted by *Q :

])2

1

2

3()([ 2232

* nn xxQ ; (4.11)

here,

*

*

1

2

1

2

1a

a

e

e

. (4.12)

Now, we can replace *F with the additionally simplified term 1F

*21

2

1211 ]),())[ln(( QxwF n

. (4.13)

To make further simplifications, we find the fixed point *a of map (3.19). For this

purpose, we substitute in Eq. (3.19) *a for the terms 1na , na , and 1na . Then, we

obtain the equation

*

*

1

2

1

2

2

112

2

12

1

)ln()()ln()1(a

a

e

e

. (4.14)


Next, it is convenient to calculate because the linear terms in na cancel one

another in (3.19).

We have

)(

)1(

12

2

. (4.15)

Now, it remains to calculate 1F :

])2

1

2

3()(][),())[ln((

2232

21

2

1211 nnn xxxwF

. (4.16)

Neglecting the cubic terms in (4.16), we obtain

].))()2

1

2

3)((ln(

)))((ln()ln()[(

2223

2

1

2

2

1

2

1211

n

n

xw

xwF

(4.17)

Having in (4.17) the nonlinear term 1F , which is quadratic in nx , we can

transform (3.19) to Henon's map (4.5). In terms of the new notation, it has the

form

].))()2

1

2

3)((ln(

)))()[(ln(()1(

2223

2

1

2

2

1121221

n

nnnn

xw

xwxxx

(4.18)

By rearranging and renaming the terms, we obtain

2

121 22 nnnn PxAxxx ; (4.19)

here

)))()(ln(()1(2 2

2

1122 wA

(4.20)

or, taking into account formula (4.15) for ,

5816 Yu. V. Bibik

,)1()1

)(1)(ln()1(2 2

12

12

2

12 wA

(4.21)

)).())2

1

2

3)()(ln((2 223

2

1

12

wP (4.22)

We reduce Eq. (4.19) to a more usual form. Let us transform the variable :nx by

multiplying it by the calibrating factor P :

.nn Pxx (4.23)

This yields the equation

.22 2

121 nnnn xAxxx (4.24)

Upon deriving Eq. (4.24), the reduction of Eq. (3.19) in one variable to the

dissipative Henon map is almost completed. It only remains to determine the

unknown function w . In Eq. (4.24) it appears in the parameter A . The same

function also appears on the right-hand side of Eq. (4.21). It is a fitting function.

Using it, we will be able not only to reveal bifurcations in the system under

examination but also to choose the value of w such that the first two bifurcations

of Henon's map correspond to the first two bifurcations of Eq. (3.19). This enables

us to obtain a good approximation of Eq. (3.19) by Eq. (4.24) upon which the

approximation of Eq. (3.19) will be completed.

Note that Eq. (4.21) is actually a link between Eq. (3.19) in one variable and the

dissipative Henon map (4.24). The left-hand side of this equation contains the

parameter A , which also appears in (4.24). The right-hand side of Eq. (4.21)

contains the parameters 1 and 2 . When a bifurcation occurs, these parameters

become related by a functional relationship )()( 221 i , where i is the

bifurcation index. These parameters 1 and 2 also appear in (3.19). We will use

Eq. (4.21) to determine bifurcations of Eq. (3.19) given the bifurcations of Eq.

(4.24).

The function w will be determined in the next section in terms of the functions

1w and 2w . Formulas for determining the fitting functions 1w and 2w are derived

in the next section (formulas (5.39) and (5.40)). These functions are determined in

terms of the functions 1 and 2 , which are obtained in Section 6 (formulas (6.30)

and (6.59)).

5. Renormalization of the dissipative Henon map

Let us briefly discuss the purpose of renormalization. The introduction of the

seasonality factor into the original equations (2.1), (2.2) results in the emergence


of chaos in the Lotka–Volterra system. In this paper, the renormalization method

is used to detect the chaotic behavior. It was shown that the transition of Henon's

map to chaos occurs through a chain of.

Since Eq. (3.19) is reduced to the dissipative Henon map (4.24), it also has the

basic properties of Henon's map. To determine the points of period doubling

bifurcations, fixed points of the corresponding periods should be determined and

then analyzed for stability. In the best case, this yields polynomial equations of

large degrees, which are difficult to solve. The renormalization procedure makes

it possible to avoid these operations. It relates the form of equations on different

scales and thus provides recurrent formulas for relating bifurcation values.

Renormalizations have specific features depending on the equations being

renormalized. The renormalization of the dissipative Henon map (4.24) also has

some specific features compared with the renormalization of the conservative

Henon map performed using Hellemann's technique [7, 8].

Consider the basic idea underlying the renormalization. The renormalization of

the dissipative Henon map performed in [1] is based on a simple idea that the

assumption of the possibility of renormalization makes it possible to actually

perform the renormalization. Let us explain this in more detail. Assume that Eq.

(4.24) is already renormalized. For convenience, we change the notation in Eq.

(4.24) by replacing x with X . Then, Eq. (4.24) takes the form

2

11 22 nnnn XAXXX . (5.1)

It has two fixed points of period one, which we denote by *

1x and *

2x , and two

fixed points of period two, which we denote by *

3x and *

4x .

Let

nn xxX 2

*

32 , (5.2)

12

*

412 nn xxX . (5.3)

The renormalizability of a system implies that the following equation is satisfied:

2

22 22 nnnn PxСxxx . (5.4)

It must follow from Eq. (5.1). On the other hand, Eq. (5.1) can be expanded in the

vicinity of the fixed points *

3x and *

4x . Then, we have the equations

2

111 2 nnnn xxxx , (5.5)

2

1122 2 nnnn xxxx . (5.6)

5818 Yu. V. Bibik

They are consequences of Eq. (5.1).

*

31 42 xA , (5.7)

*

42 42 xA . (5.8)

By combining them, we obtain

2

1

2

1121222 22)1( nnnnnnn xxxxxxx , (5.9)

22

2121213 22)1( nnnnnnn xxxxxxx . (5.10)

The left-hand sides of Eqs. (5.4) and (5.9) are identical. It is clear that we can

derive from them equations that relate 1nx , nx , and 1nx . However, we already

have other equations relating these variables—these are Eqs. (5.5) and (5.6). The

combined use of Eqs. (5.4), (5.9), (5.5), and (5.6) makes it possible to eliminate

one of the variables 1nx , nx , or 1nx and establish a relationship between the two

other variables. For definiteness, we eliminate 1nx (and 2nx in (5.10)) using Eqs.

(5.4), (5.5), (5.9). As a result, we obtain

)(11 nn xfx , (5.11)

)( 12 nn xfx . (5.12)

Below we will consider the cases where nx are small; for that reason, we use the

approximations

2

111 nnn xxx , (5.13)

2

1212 nnn xxx . (5.14)

Equations (5.13) and (5.14) are consequences of the assumption that Eq. (4.24)

can be renormalized. They allow us to restore the symmetry of Eq. (4.24) when

1n is replaced with 1n and then use Hellemann's renormalization technique.

Using the results obtained above, we proceed to the renormalization of Eq. (4.24).

Define the parameter 1 . Equation (4.24) is then written as

2

111 22 nnnnn XAXXXX . (5.15)

We expand this equation about the points of period two, which are denoted by *

3x

and *

4x . This yields


2

2

*

3121212 2)42( nnnnn xxxAxxx , (5.16)

2

1212

*

42222 2)42( nnnnn xxxAxxx . (5.17)

To make the renormalization, we use relations (5.13) and (5.14):

2

212112 nnn xxx , (5.18)

2

1221222 nnn xxx . (5.19)

Using these formulas, we reduce Eqs. (5.16) and (5.17) to the form

2

21211212 nnnn xxxx , (5.20)

2

122122222 nnnn xxxx , (5.21)

where

)())42(( 11

*

311 xA , (5.22)

)())42(( 221

*

422 xA , (5.23)

)2( 11 , (5.24)

)2( 22 . (5.25)

Equations (5.20) and (5.21) have the same form as the expansion of the

conservative Henon map about the fixed points of period two; therefore, they can

be renormalized using Hellemann's technique. To this end, we combine Eqs.

(5.20) and (5.21) for 12 n and 12 n :

][][22

12

2

1221212222222 nnnnnnn xxxxxxx . (5.26)

Next, we eliminate the sum of 12 nx and 12 nx and the sum of 2

12 nx and 2

12 nx

from Eq. (5.26). We make this using Eq. (5.20) for n2 :

2

2

2

2

2

1212212

2

2

2

2

2

12

2

2121222222

)]([]2[

][][2

nn

nnnnnn

xx

xxxxxx

. (5.27)

5820 Yu. V. Bibik

Upon these transformations, Eq. (5.27) became similar to Eq. (4.24). It remains to

transform the conservative equation (5.27) into a dissipative equation. To this end,

we rewrite it as

2

2

2

2

2

1212212

2

2

2

2

2

12

2

212122222222

)]([]2[

][][2)(

nn

nnnnnnn

xx

xxxxxxx

.

(5.28)

Now, we represent 22 nx in terms of nx2 using (5.13) and (5.14). We obtain

2

2

2

1221

2

2

2

121222112

2

2

2

2

2

12

2

212122222222

)]()([)](2[

][][2

nn

nnnnnnn

xx

xxxxxxx

. (5.29)

It remains to calculate the constants 1 , 2 and 1 , 2 . Equation (5.29) has the

same form as Eq. (5.5). Repeating the reasoning used after formula (5.5), we

obtain equations for 1 , 2 and 1 , 2 . This completes the renormalization of

Eq. (4.24). Next, we use this renormalization to derive a recurrence relating the

conditions for the emergence of period doubling bifurcations (the quantities iA ).

Since Eq. (5.29) on the doubled scale has the same form as Eq. (4.24), the role of

the constant A in (4.24) is now played by the expression

)(2 2112 .

Therefore, if the preceding period doubling bifurcation occurred at nA , the next

period doubling bifurcation will occur at 1nA determined from the equation

nnnnn AAAAA 2)()(2)()( 12111211 . (5.30)

After writing the expressions for 1 and 2 explicitly, we finally obtain the

recurrence for determining the period doubling bifurcations for Eq. (4.24):

2

2

14

]])2([2[5(

q

AAqqA

nn

n

. (5.31)

In this case, qA 1 , ,2

1 q and .1


The final recurrence (5.31) allows us to determine period doubling bifurcations

for the original equation (3.19) with the seasonality factor using Eqs. (4.24) and

(4.21).

To ensure the best match between Eqs. (3.19) and (4.24), we should determine the

fitting parameter w from Eq. (4.21). Formula (5.31) determines the left-hand side

of Eq. (4.21). Therefore, knowing recurrence (5.31), we can find w from (4.21).

To this end, we proceed as follows.

On the one hand, the renormalization allows us to calculate for Henon's map

(4.24) the parameters )( 21 A and ),( 22 A which appear on the left-hand side of

Eq. (4.21). At these parameters, we have the first two period doublings.

On the other hand, we can define the functions )( 21 and )( 22 , which

appear on the right-hand side of Eq. (4.21). We define them in such a way as to

ensure that the first period doubling in (3.19) occurs at )( 211 , and the

second period doubling occurs at )( 221 .

Next, we choose w such that the first period doubling for (3.19) corresponds to

the first period doubling for Henon's map (4.24). )( 21 A on the left-hand side of

Eq. (4.21) is associated with the function )( 21 on the right-hand side of Eq.

(4.21).

For the second period doubling, we choose w so as to ensure that the second

period doubling for (3.19) corresponds to the second period doubling for Henon's

map (4.24). In other words, the parameter )( 22 A on the left-hand side of Eq.

(4.21) is associated with the function )( 22 on the right-hand side of Eq. (4.21).

Hence, we obtain the equations

,)1())(

)(1)(1)(

)(ln()1(2 2

212

212

2

2121 wA

(5.32)

.)1())(

)(1)(1)(

)(ln()1(2 2

222

222

2

2222 wA

(5.33)

We rewrite these equations as

),()),(( 21221 Pw (5.34)

),()),(( 22222 Pw (5.35)

where

5822 Yu. V. Bibik

),)(

)(1)(

)(ln(

)1(

))1(2(

212

21

2

21

2

211

AP (5.36)

).)(2

)(1)(

)(ln(

)1(

))1(2(

22

22

2

22

2

222

AP (5.37)

To determine w , we will seek it as a combination of two unknown simpler

functions

).())](()([)( 2221121 wffww (5.38)

Then, for 1w and 2w , we obtain the equations

121 )( Pw , (5.39)

.))](())(([

)]([)(

2122

21222

ff

wPw (5.40)

In this paper, the function f is taken in the form ).ln()( xxf Formulas (5.39) and

(5.40) include the unknown functions )( 21 and ).( 22 They will be found

using a computer (formulas (6.30) and (6.59)). Next, the final values of the fitting

coefficients 1w and 2w (5.39) and (5.40) will be obtained. Substitute them into

(4.21) to obtain a ready-to-use formula for determining the conditions for the

emergence of period doubling bifurcations and transition to chaos.

6. Conditions for the emergence of the first and second period

doubling bifurcations for the one-dimensional discrete map (3.19)

Since the conditions of the emergence of period doubling bifurcations for Henon's

map are already found (formula (5.31)), it remains to determine the conditions for

the emergence of period doubling bifurcations for Eq. (3.19). This also yields the

unknown functions )( 21 and ).( 22

To find the conditions for the first and second period doubling for Eq. (3.19), we

rewrite it in a slightly different form.

Make the change of variables

.)1)(ln(2

1nn aa

(6.1)

Then,


],1

)[-)(1-(--)1( )1( 1-n2112221n

n

a

a

nnnne

eaaaaa

(6.2)

).ln(2

1

(6.3)

Equation (6.2) is equivalent to a two-dimensional difference equation that

involves only one time step. This equation has the form

]1

)[-)(1-(--)1( )1( n212221n

n

a

a

nnnne

eabbaa

, (6.4)

nn ab 1 . (6.5)

Equation (6.4) has a fixed point of period one ),(),( aaba nn and two fixed

points of period two— ),( za and ),( az .

First, we determine the fixed points of period two for Eq. (6.4), (6.5). Then, we

find the fixed point of period one as a special case for za . To determine the

fixed points of period two, we should find a and z . Equation (6.2) implies that

],1

)[-)(1-(--)1( )1( 21222 z

z

e

ezaaza

(6.6)

]1

)[-)(1-(--)1( )1( 21222 a

a

e

eazzaz

. (6.7)

We rearrange these equations as follows:

],1

[)( )1[(

]z1

[)( )1[(]1

[)( )1[(

212

212212

z

z

z

z

z

z

e

e

e

ea

e

e

(6.8)

].1

[)( )1[(

]1

[)( )1[(]z1

[)( )1[(

212

212212

a

a

a

a

a

a

e

e

ae

e

e

e

(6.9)

Using (6.8) and (6.9), we find a as

5824 Yu. V. Bibik

]]1

[)( )1[(

]]1

[)( )1[(

212

212

z

z

z

z

e

e

e

e

za

. (6.10)

Multiply the numerator and denominator by aze1 to obtain

][

][

43

21

z

z

e

eza

. (6.11)

In (6.11), 1 , 2 , 3 , and 4 have the values

21 1 , (6.12)

12 1 , (6.13)

23 1 , (6.14)

14 1 , (6.15)

Similarly, we find z in the form

][

][

43

21

a

a

e

eaz

. (6.16)

Equations (6.9) and (6.14) include two variables a and z . We simplify (6.9) and

(6.14) by introducing a new variable P . This allows us to reduce the equations in

two unknowns to an equation with one unknown P . Define

][

][

43

21

a

a

e

eP

. (6.17)

Using (6.11) and (6.17), we obtain

.0][

][

43

21

Pa

Pa

ee

eeP

(6.18)

It is clear that


.][

][

42

31

P

Pee Pa

(6.19)

On the other hand, (6.17) implies

.][

][

42

31

P

Pe a

(6.20)

Therefore,

.]][[

]][[

4231

4231

PP

PPe P

(6.21)

Thus, we have obtained formula (6.19), which contains only one variable P .

After calculating its value using (6.17), we obtain a . Now we use (6.16) to find

z . Let us analyze the stability of the fixed points of period one and two. The loss

of stability of these fixed points indicates the emergence of period doubling

bifurcations. The stability of fixed points is determined by the eigenvalues of the

Jacobian matrix. First we analyze the stability of the fixed points of period one.

The corresponding Jacobian matrix is

]]1

[]1

)[[1( )(]1

[)( )1( 2

12212 a

a

a

a

a

a

xxe

e

e

ez

e

eL

,

(6.22)

]1

[)( 122 a

a

xye

eL

, (6.23)

1xyL , (6.24)

0yyL . (6.25)

It is convenient to rewrite these formulas in terms of the parameter P because it

is this parameter that is found from Eq. (6.19). This allows us to skip the

intermediate calculations needed for finding the parameters a and z . Formula

(6.17) implies

].)1(

)1([

)(

1

)1)((

][-

])()[(

][-]

1[

2

12

12

31

4312

31

P

P

P

P

P

P

e

ea

a

(6.26)

5826 Yu. V. Bibik

The determinant D of the Jacobian matrix (6.22)–(6.25) is equal to xyL . First,

we calculate this determinant. Using (6.26), we obtain

.)1(

)1(]

)1(

)1([]

)1(

)1([

)(

1)( 222

12

122P

P

P

P

P

PLxy

(6.27)

Formula (6.16) implies that the fixed point ),( aa of period one corresponds to

0P ; therefore the absolute value of the matrix determinant is 1D at this

point, hence, this point is elliptic. This considerably simplifies the analysis of

stability of this fixed point because the eigenvalues of the Jacobian matrix in this

case are

].4[2

1 2

2,1 spLspL (6.28)

Next, we find the trace of the Jacobian matrix (6.22)–(6.25) , which appears in

(6.28):

).1]()(

)1([2

]])(

)1(

)(

)1([[)(

)(

)1()[()1(

2

12

1

2

12

2

12

212

12

2212

xxLspL

(6.29)

Formula (6.28) implies that the elliptic fixed point becomes unstable under the

condition

.2spL (6.30)

This is an equation for determining the first period doubling. The point ),( aa

becomes unstable exactly under this condition. Equations (6.29) and (6.30) yield

the function )( 21 , which was earlier denoted by )( 21 .

Proceed to the analysis of stability of the fixed points ),( za and ),( az of period

two. For this purpose, we find the determinant of the Jacobian matrix for the two-

step map (twice consistently applied map (6.4), (6.5)). It equals the product of the

determinants of one-step maps (6.4), (6.5):

)()()()(

)2( PDPDDDD za . (6.31)


The fixed point ),( za is associated with P ; therefore, the fixed point ),( az is

associated with P . Hence,

.1))1(

)1()(

)1(

)1(()2(

P

P

P

PD (6.32)

The fixed points of the two-step map are elliptic, and they are found by the

formula

].4)([2

1 2)2()2()2(

2,1 spLspL (6.33)

These elliptic fixed points become unstable under the condition

.2)2( spL (6.34)

This equation determines the second period doubling. Let us write out this

equation in more detail. The Jacobian matrix of the two-step map is

),()()()2( PLPLPLL xyxxxxxx (6.35)

),()()2( PLPLL xyxxxy (6.36)

),()2( PLL xxyx (6.37)

).()2( PLL xyyy (6.38)

Its trace has the form

).()()()()2( PLPLPLPLspL xyxyxxxx (6.39)

The equation determining the conditions for the second period doubling is written

as

.2)()()()( PLPLPLPL xyxyxxxx (6.40)

Let us find )(PLxx :

5828 Yu. V. Bibik

].)1)((

][1[

)1)((

][)1()(

)1)((

][)()1(

]]1

[]1

)[[1()(

]1

)[()1()(

12

31

12

31

12

12

31

212

2

12

212

P

P

P

PP

P

P

e

e

e

eaz

e

ePL

a

a

a

a

a

a

xx

(6.41)

Upon collecting similar terms, we obtain

.)1)((

]])([)]()(2[[

][)1)((

][

)1(

2)(

12

3241

2

432112

42

12

31

P

PP

PP

P

PPLxx

(6.42)

Now we can find the product )()( PLPLxx :

.)1()(

])()]()(2[[

)()(

22

12

22

3241

222

432112

P

PP

PLPL xxxx

(6.43)

We use the fact

,)1)(1()1)(1()( 2

2121

2

4321 PP (6.44)

).1(2)( 213241 (6.45)

Then, formula (6.43) takes the form

.)1(()(

1])1(4

)])1)(1()1)(1(()(2[[)()(

22

12

22

21

2

22

212112

PP

PPLPL xxxx

(6.46)

Now we can find )(PLxy :


.)1(

][

)1()(

][)(]

1)[()(

31

2

2

12

31

122122

P

P

P

P

e

ePL

a

a

xy

(6.47)

Finally, we find the sum )()( PLPL xyxн :

.]1[

]1[2

)1(

]22[2

)1(

][][2

)1(

][

)1(

][2)()(

2

2

2

2

31

2

2

1

2

3311

2

331

2

3131

2

P

P

P

P

P

PPPPPP

P

P

P

PPLPL xyxy

(6.48)

Using these results, we rewrite (6.40) as

.2]1[

]1[2

)1(()(

1])1(4)])1)(1(

)1)(1(()(2[[)()()()(

2

2

22

12

22

21

222

21

2112

P

P

PPP

PLPLPLPL xyxyxxxx

(6.49)

This formula determines the conditions for the second period doubling. Multiply

the left- and right-hand sides of (6.49) by )1()( 22

12 P to obtain

.0]1[)(2)](1[2])1(4

)])1)(1()1)(1(()(2[[

22

12

2

12

222

21

2

22

212112

PPP

P

(6.50)

This equation can be written as

03

2

2

4

1 aPaPa , (6.51)

,)1()1( 2

2

2

1

2

1 a (6.52)

5830 Yu. V. Bibik

,)(2)(2)1(4

)]1)(1()(2[)1)(1(2

2

12

2

12

2

21

2

2112212

a

(6.53)

.)(2)(2

)]1)(1()(2[

2

12

2

12

2

21123

a (6.54)

For 2P we have

].4[2

131

2

22

1

2

4,3,2,1 aaaaa

P (6.55)

Introduce the notation

,)1()1(

1])(2)(2

)1(4[)1()1(

)]1)(1()(2[2

2

2

2

1

2

2

12

2

12

2

21

2

2

2

2

1

2

2112

1

21

a

a

(6.56)

,)1()1(

1])(2)()(2

)]1)(1()(2[[

2

2

2

1

2

2

12

2

12

2

2112

1

3

2

a

a

(6.57)

].4[2

12

2

11

2

4,3,2,1 P (6.58)

Now we can plug (6.58) into Eq. (6.21), which determines P , and obtain

conditions for the emergence of the second period doubling:

.]][[

]][[

4,3,2,1424,3,2,131

4,3,2,1424,3,2,1314,3,2,1

PP

PPe

P

(6.59)

Thus, formulas (6.59) and (6.30) determine conditions for the first and second

period doubling for the one-dimensional discrete map. Equations (6.57) and (6.59)

can be used to find the function )( 21 , which was earlier denoted by )( 22 .

Among the solutions to Eq. (6.59), the one that is the closest to the solution to Eq.

(6.30) should be chosen. Having formulas determining the conditions for the first


and second period doubling, we will determine the next period doubling in the

following section.

7. Conditions for the emergence of the next period doubling

bifurcations and conditions for the transition to chaos for the

generalization of the Lotka–Volterra equations with a seasonality

factor

In the preceding sections, we considered individual parts of the study, which we

will combine into the whole in this section. These partial investigations provided

us with the data that now enable us to analyze the generalization of the Lotka–

Volterra system with the seasonality factor (3.19) from the viewpoint of the

emergence of bifurcations and conditions for the transition to chaos.

In Section 3, the original Lotka–Volterra equation with seasonality factor (2.5),

(2.6) in two variables was reduced to Eq. (3.19) in one variable.

In Section 4, the Lotka–Volterra equation (3.19) with seasonality factor was

basically approximated by the dissipative Henon map and transformed to the

approximating equation (4.24). This enabled us to conclude that Eq. (3.19) has the

properties of Henon's map, including the conditions for the emergence of

bifurcations and transition to chaos.

In Section 5, the dissipative Henon map was renormalized, and a recurrence for

determining the period doubling bifurcations of Henon's map (4.24) was obtained.

This recurrence has the form

2

2

14

]])2([2[5(

q

AAqqA

nn

n

. (7.1)

Here, qA 1 , ,2

1 q and .1

In Section 4 Eq. (3.19) was approximated only accurate to an arbitrary function

w , and this function was then found using the results obtained in Sections 5 and

6. The function w provides a link between the approximating equation (4.24) and

the basic Lotka–Volterra equation (3.19) with the seasonality factor. Using this

function and formula (4.21), we can establish a relationship between the

parameter A of Henon's map (4.24) and the parameters 1 and 2 of the Lotka–

Volterra equation (3.19) with the seasonality factor. Therefore, given the

conditions for the emergence of bifurcations for the parameter A of Henon's map (4.24), we can find conditions for the emergence of bifurcations for the parameters

5832 Yu. V. Bibik

1 and 2 , which appear in the Lotka–Volterra equation (3.19) with the

seasonality factor.

In Section 6, conditions for the emergence of the two first period doubling

bifurcations for the Lotka–Volterra equation (3.19) with the seasonality factor

were found.

In the present section, we find conditions for the next period doubling bifurcations

for Eq. (3.19).

The values of the parameter iA at which the period doubling bifurcations of

Henon's map (4.24) occur were found in Section 5 (formulas (5.31) and (7.1)). In

addition iA can be represented in terms of the parameters 1 and 2 in (4.21).

The left-hand side of this formula includes the known parameter iA , and the right-

hand side includes the known function w and the parameters 1 and 2 . After iA

determining the emergence of period doubling bifurcations of Henon's map using

formula (4.21) has been found, the relationship between the parameters 1 and 2

can be established. It is clear that period doubling bifurcations for Eq. (3.19) occur

at certain combinations of the parameters 1 and 2 . The relations between those

parameters at which such bifurcations occur determines the conditions for the

emergence of the third and all subsequent bifurcations for the Lotka–Volterra

equations with the seasonality factor (3.19). The corresponding formula is

wAi

ii

i )1()])(

)(1)[(1)(

)(ln()1(2 2

22

2

2

2

2

2

. (7.2)

Denote the relation )( 21 by the function )( 21 . For each individual

bifurcation, this function will be denoted by )( 2i . Formulas (4.21) and (7.2)

can be used to find )()( 212 i . This yields the functions )( 2i for the third

and the subsequent period doubling bifurcations for Eq. (3.19), starting from

3i . The first two functions 1 and 2 for the first two period doubling

bifurcations were earlier found in Section 6 (formulas (6.30 ) and (6.59) .

Due to the complexity of formulas (4.21) and (7.2), the values of )( 2i were

obtained on a computer. The plots of 1 as a function of 2 are depicted in Fig. 1.

To determine the point of the transition to chaos for the Lotka–Volterra equations

with the seasonality factor , Eq. (7.2) is also used. By plugging the parameter A

into this equation, we find the function . This function determines the

conditions for the transition to chaos of system (3.19). Thus, the analysis of the

conditions for the emergence of period doubling bifurcations and the transition to

chaos for the Lotka–Volterra equations with the seasonality factor (3.19) is

completed.


8. Description and analysis of figures

Figure 1 shows the curves of the first three period doubling bifurcations. The

parameter 2

2

e , where 2 is the amplitude of seasonal variations in the

predator population decrease coefficient, is plotted on the horizontal axis. The

parameter 12 )1( , where 1 is the amplitude of seasonal variations in the prey

population increase coefficient, is plotted on the vertical axis.

These parameters indirectly determine the population of predators and prey. The

three bifurcation curves are constructed based on the data obtained using formula

(7.2). It is seen in Fig. 1 that the curve corresponding to the next bifurcation lies

above the curve corresponding to the preceding bifurcation. Therefore, for each

fixed 2 , the transition to the next bifurcation requires the parameter 1 to be

increased. The parameter 1 indirectly affects the population of prey. The inflow

of biomass into the system in the form of prey population results in the system

5834 Yu. V. Bibik

excitation, the emergence of new bifurcations, and ultimately in the transition to

chaos.

On the other hand, the curves representing the bifurcations increase with

increasing 2 . The increase in 2 indirectly results in the growth of predator

population. Therefore, as the number of predators increases, bifurcations can

emerge only if the population of prey increases. It is seen from Fig. 1 that, in

order for bifurcations in the vicinity of 12 to appear, the parameter 1 must

grow as 221

constconst.

Figure 2 depicts the curve of transition to chaos for the original system. The

parameter 2 is plotted on the horizontal axis, while 12 )1( is plotted on the

vertical axis. It is seen from Fig. 2 that the curve of transition to chaos lies above

all the period doubling bifurcation curves shown in Fig. 1. Therefore, at a fixed

2 , the curve of the transition to chaos is attained at the maximum value of the

parameter 1 . Furthermore, as in Fig. 1, the increase in 2 is associated with the

increase in 1 . As 2 increases, additional inflow of biomass in the form of prey

population is needed for the transition to chaos. As 12 , we have

22

11

constconst

.

9. Conclusions

Recent studies of nonlinear systems showed that even a small modification of

simple models towards more realistic ones results in the emergence of chaos and

complex dynamical behavior, which are characteristic of real life. The Lotka–

Volterra system with a seasonality factor studied in the present paper confirms

this fact. Introduction of the seasonality factor resulted in the emergence of chaos

in the Lotka–Volterra system at certain values of the parameters. Taking into

account the complexity of the problem, the analysis was performed in several

phases. First, the original system of differential equations (2.5), (2.6) was replaced

with the discrete map (2.33), (2.34). This complicated the problem due to the

introduction of the seasonality factor. However strange it may seem, this

simplified the investigation method because the difference equations (2.33), (2.34)

were obtained. Next, these equations were reduced to a form that is more

convenient for analysis; more precisely, Eqs. (2.33), (2.34) were reduced to a

simpler equation (3.19). Next, Eq. (3.19) was approximated by the dissipative

Henon map (4.24). This suggested the conclusion that the Lotka–Volterra system

with the seasonality factor has the basic properties of Henon's map, including

period doubling bifurcations and transition to chaos. Then, the renormalization

group technique was applied to the dissipative Henon map (4.24), which enabled

us to obtain a recurrence for determining the conditions for the emergence of


period doubling bifurcations. The renormalization of the dissipative Henon map

using an analog of Hellemann's method [1] enabled us to find period doubling

bifurcations and determine the point of the transition to chaos for system (3.19).

Thus, the multistep analysis and modern techniques of the theory of chaos enabled

us to find the entire chain of period doubling points and the point of transition to

chaos for the system under examination.

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Received: August 12, 2015; Published: September 18, 2015

http://dx.doi.org/10.1016/0375-9601%2882%2990248-1

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