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-
Investigation of transition to chaos for a Lotka–Volterra system 5805
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:
Investigation of transition to chaos for a Lotka–Volterra system 5807
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 .
Investigation of transition to chaos for a Lotka–Volterra system 5809
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)
Investigation of transition to chaos for a Lotka–Volterra system 5811
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.
Investigation of transition to chaos for a Lotka–Volterra system 5813
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)
Investigation of transition to chaos for a Lotka–Volterra system 5815
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
Investigation of transition to chaos for a Lotka–Volterra system 5817
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
Investigation of transition to chaos for a Lotka–Volterra system 5819
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
Investigation of transition to chaos for a Lotka–Volterra system 5821
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,
Investigation of transition to chaos for a Lotka–Volterra system 5823
],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
Investigation of transition to chaos for a Lotka–Volterra system 5825
.][
][
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)
Investigation of transition to chaos for a Lotka–Volterra system 5827
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 :
Investigation of transition to chaos for a Lotka–Volterra system 5829
.)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
Investigation of transition to chaos for a Lotka–Volterra system 5831
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.
Investigation of transition to chaos for a Lotka–Volterra system 5833
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
Investigation of transition to chaos for a Lotka–Volterra system 5835
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