This document discusses period-doubling bifurcations and the route to chaos in an area-preserving discrete dynamical system. It highlights two objectives: 1) Evaluating period-doubling bifurcations using computer programs as the system parameter p is varied, obtaining the Feigenbaum universal constant and accumulation point beyond which chaos occurs; and 2) Confirming periodic behaviors by plotting time series graphs. The document provides background on Feigenbaum universality and describes the specific area-preserving map studied. It outlines the numerical method used to obtain periodic points via Newton's recurrence formula and describes applying this to the map's Jacobian matrix.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Multidimensional integrals may be approximated by weighted averages of integrand values. Quasi-Monte Carlo (QMC) methods are more accurate than simple Monte Carlo methods because they carefully choose where to evaluate the integrand. This tutorial focuses on how quickly QMC methods converge to the correct answer as the number of integrand values increases. The answer may depend on the smoothness of the integrand and the sophistication of the QMC method. QMC error analysis may assumes the integrand belongs to a reproducing kernel Hilbert space or may assume that the integrand is an instance of a stochastic process with known covariance structure. These two approaches have interesting parallels. This tutorial also explores how the computational cost of achieving a good approximation to the integral depends on the dimension of the domain of the integrand. Finally, this tutorial explores methods for determining how many integrand values are needed to satisfy the error tolerance. Relevant software is described.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
37coins: Hubba Bitcoin Intro, Jonathan Zobro37coins
37coins bitcoin meetup in Bangkok
Hubba (the major co-working space in Bangkok) Bitcoin Intro, Jonathan Zobro
37coins facebook page: https://www.facebook.com/37Coins
37coins twitter: https://twitter.com/37Coins
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Multidimensional integrals may be approximated by weighted averages of integrand values. Quasi-Monte Carlo (QMC) methods are more accurate than simple Monte Carlo methods because they carefully choose where to evaluate the integrand. This tutorial focuses on how quickly QMC methods converge to the correct answer as the number of integrand values increases. The answer may depend on the smoothness of the integrand and the sophistication of the QMC method. QMC error analysis may assumes the integrand belongs to a reproducing kernel Hilbert space or may assume that the integrand is an instance of a stochastic process with known covariance structure. These two approaches have interesting parallels. This tutorial also explores how the computational cost of achieving a good approximation to the integral depends on the dimension of the domain of the integrand. Finally, this tutorial explores methods for determining how many integrand values are needed to satisfy the error tolerance. Relevant software is described.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
37coins: Hubba Bitcoin Intro, Jonathan Zobro37coins
37coins bitcoin meetup in Bangkok
Hubba (the major co-working space in Bangkok) Bitcoin Intro, Jonathan Zobro
37coins facebook page: https://www.facebook.com/37Coins
37coins twitter: https://twitter.com/37Coins
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Development of a Cassava Starch Extraction Machineijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Engendering sustainable socio-spatial environment for tourism activities in t...ijceronline
The South Eastern part of Nigeria was a single state of the twelve states of Nigeria before 1976 – East Central State. Presently, the area is made up of five states. This Paper presents a study carried out to assess the possibility of knitting the once-one-state and presently five states, together to form a tourist destination worthy of international importance. Although the unprecedented dearth of infrastructural development in the zone continues to be a subject of discussion, little attention has
been given to investigate the place of synergized intervention. Descriptive analysis was employed, in which the tourism potentials in the zone that are of national recognition were itemized. Data were collected on available amenities, resident population and accessibility of such potentials. Existing literature on various Environmental Management Systems were explored to elicit a system that is
capable of affording a sustainable environment for tourism matters in the zone. The study revealed that the zone exhibits similar socio-economic and physical characteristics that are replete with potentials for improvement for optimum tourism utility in the entire zone. Several measures were advanced in the study to achieve this, chief of which is the adoption of the Environmental Planning and Management (EPM) process
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International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
This paper presents concepts of Bernoulli distribution, and how it can be used as an approximation of Binomial, Poisson, and Gaussian distributions with a different approach from earlier existing literature. Due to discrete nature of the random variable X, a more appropriate method of the principle of mathematical induction (PMI) is used as an alternative approach to limiting behavior of the binomial random variable. The study proved de Moivre–Laplace theorem (convergence of binomial distribution to Gaussian distribution) to all values of p such that p ≠ 0 and p ≠ 1 using a direct approach which opposes the popular and most widely used indirect method of moment generating function.
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and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
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1. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
Route to Chaos in an Area-preserving System
Dr. Nabajyoti Das
Assistant Professor, Depart ment of Mathematics Jawaharlal Nehru Co llege, Bo ko -781123
District : Kamrup, State: Assam, Country: India
Abstract:
Th is paper highlights two important objectives on a two-dimensional area-preserving discrete dynamical system:
E( x, y) ( y px (1 p) x 2 , x y px (1 p) x 2 ) ,
where p is a tunable parameter. Firstly, by adopting suitable computer programs we evaluate period-doubling:
period 1 period 2 period 4 ... period 2 k ... chaos
bifurcations, as a universal route to chaos, for the periodic orbits when the system parameter p varied and obtain the
Feigenbaum universal constant = 8.7210972…, and the accumulation point = 7.533284771388…. beyond which
chaotic region occurs.. Secondly, the periodic behaviors of the system are confirmed by plotting the time series graphs.
Key Words: Period-doubling bifurcations/ Periodic orbits / Feigenbaum universal constant / Accumulation point /
Chaos / Area-preserving system
2010 AMS Classificat ion: 37 G 15, 37 G 35, 37 C 45
1. Introduction
The initial universality discovered by Mitchell J. Feigenbaum in 1975 has successfully led to discover that large
classes of nonlinear systems exhib it transitions to chaos which are universal and quantitatively measurable. If X be a
suitable function space and H, the hypersurface of co-d imension 1 that consists of the maps in X having derivative -1 at
the fixed point, then the Feigenbaum uni versality is closely related to the doubling operator, F acting in X defined by
(F )( x) ( ( 1 x)) X
where = 2.5029078750957… , a universal scaling factor. The principal properties of F that lead to universality are
*
(i) F has a fixed point x ;
(ii) The linearised transformation DF ( x * ) has only one eigenvalue greater than 1 in modulus; here =
4.6692016091029…
(iii) The unstable man ifold corresponding to intersects the surface H transversally; In one dimensional case, thes e
properties have been proved by Lanford [2, 10].
Next, let X be the space of two parameter family of area- preserving maps defined in a domain U 2 , and Y, the
space of two parameter family o f maps defined in the same domain having not necessarily constant Jacobian. Then Y
contains X. In area- preserving case, the Doubling operator F is defined by
F 1 2 ,
where : 2 2 , ( x, y) (x, y) is the scaling transformation. Here and are the scaling factors;
numerically we have 0.248875... and 0.061101... In the area p reserving case, Feigenbaum constant, =
8.721097200….. Furthermore, one of his fascinating discoveries is that if a family presents period doubling
n n1
bifurcations then there is an infinite sequence { n } of bifurcation values such that lim , where
n n1 n
is a universal number already mentioned above. Moreover, his observation suggests that there is a universal size -
d
scaling in the period doubling sequence designated as the Feigenbaum value, lim n 2.5029... where d n
n d n1
is the size of the bifurcation pattern of period 2 n just before it g ives birth to period 2 n1 [1, 6-8].
Issn 2250-3005(online) November| 2012 Page 38
2. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
The birth and flowering of the Feigenbaum universality with numerous non -linear models has motivated me to wrire this
paper.
2. Our Nonlinear Map and the Feigenbaum Universality:
Our concerned map:
E( x, y) ( y px (1 p) x 2 , x y px (1 p) x 2 ) (1.1)
where p is a tunable parameter. The Jacobian of E is the unity, so is area-preserving.
The map has one fixed point other than (0,0) whose coordinates is given by
3 p 6 2 p
x ,y
1 p 1 p
Fro m this one finds that E has no fixed point if p = 1. In this context, we also wish to point out that the stability
theory is intimately connected with the Jacobian matrix of the map, and that the trace of the Jacobian matrix is the sum
of its eigenvalues and the product of the eigenvalues equal the Jacobian determinant. For a particular value of p, the map
E depends on the real parameter p, and so a fixed point x s of this map depends on the parameter value p, i.e.
x s x s ( p) . No w, first consider the set, U (,3) (7,) .
The fixed point x s remains stable for all values of p lying in U and a stable periodic trajectory of period-one appears
around it. This means, the two eigenvalues
5 7 p 3 p (1 p) 6 p p 2
1 ,
2(1 p)
5 7 p 3 p (1 p) 6 p p 2
2
2(1 p)
of the Jacobian mat rix:
p 2(1 p) x 1
J
1 p 2(1 p) x 1
at x s remains less than one in modulus and consequently, all the neighbouring points (that is, points in the
domain of attraction) are attracted towards x s ( p) ), p lying in U. If we now begin to increase the value of p, then it
happens that one of the eigenvalues starts decreas ing through –1 and the other remains less than one in modulus . When
p = 7, one of the eigenvalues becomes –1 and then x s loses its stability, i.e. p1 7 emerg ing as the first bifurcati on
value of p. Again, if we keep on increasing the value of p the point x s ( p) becomes unstable and there arises around it
two points, say, x 21 ( p) and x 22 ( p) forming a stable periodic trajectory of period-two. A ll the neighbouring points
except the stable manifold of x s ( p) are attracted towards these two points . Since the period emerged becomes double,
the previous eigenvalue which was –1 becomes +1 and as we keep increasing p, one of the eigenvalues starts decreasing
fro m +1 to –1.Since the trace is always real, when eigenvalues are complex, they are conjugate to each other moving
n
along the circle of radius pe , where p e = p 2 is the effective Jacobian, in the opposite directions . When we reach a
certain value of p, we find that one of the eigenvalues of the Jacobian of E2 (because of the chain rule of differentiation, it
does not matter at which periodic point one evaluates the eigenvalues) becomes -1, indicating the loss of stability of the
periodic trajectory of period 2. Thus, the second bifurcation takes place at this value p 2 of p. We can then repeat the same
arguments, and find that the periodic trajectory of period 2 becomes unstable and a periodic trajectory of period 4
appears in its neighbourhood. This phenomenon continues upto a particular value o f p say p 3 (p), at wh ich the periodic
trajectory of period 4 losses its stability in such a way that one of the eigenvalues at any of its periodic points become –1,
and thus it gives the third bifurcat ion at p 3 (p). Increasing the value further and fu rther, and repeating the same argu ments
we obtain a sequence {p n (p)} as bifurcation values for the parameter p such that at pn (p) a periodic trajectory of period
2n arises and all periodic trajectories of period 2m(m<n) remain unstable. The sequence {pn (p)} behaves in a universal
manner such that p (p)- p n (p)} c(p )-n , where c(p) is independent of n and and is the Feigenbaum uni versal
constant. Since the map has constant Jacobian 1(unity), we have the conservative case, i.e. the preservation of area and
in this case equals 8.721097200…. Furthermore, the Feigenbaum theory says that the our map E at parameter = p (p)
Issn 2250-3005(online) November| 2012 Page 39
3. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
has an invariant set S of Cantor type encompassed by infinitely many unstable periodic orbits of period 2n (n = 0, 1, 2,…),
and that all the neighbouring points except those belonging to these unstable orbits and their stable manifolds are
attracted to S under the iterations of the map E.
3. Numerical Method For Obtaining Periodic Points [2]:
Although there are so many sophisticated numerical algorithms available, to find a periodic fixed point, we have
found that the Newton Recurrence formula is one of the best numerical methods with neglig ible error for our purpose.
Moreover, it gives fast convergence of a periodic fixed point.
The Newton Recurrence formu la is
x n 1 x n Df (x n ) 1 f (x n ),
where n = 0,1,2,… and Df ( x) is the Jacobian of the map f at the vector x . We see that this map f is equal to E k I in
our case, where k is the appropriate period. The Newton formu la actually gives the zero(s) of a map, and to apply this
numerical tool in our map one needs a number of recurrence formu lae which are given below.
Let the in itial point be ( x0 , y 0 ),
Then,
E( x0 , y 0 ) ( y 0 px0 (1 p) x0 , x0 y 0 px0 (1 p) x0 ) ( x1 , y1 )
2 2
E 2 ( x0 , y 0 ) E ( x1 , y1 ) ( x 2 , y 2 )
Proceeding in this manner the following recurrence formu la fo r our map can be established.
x n y n1 px n1 (1 p) x n1 , and y n y n1 pxn1 (1 p) xn1 ,
2 2
where n = 1,2,3…
Since the Jacobian of Ek ( k times iteration of the map E ) is the product of the Jacobian of each iteration of the map, we
proceed as follows to describe our recurrence mechanism for the Jacobian matrix.
The Jacobian J 1 for the transformation
E( x0 , y0 ) ( y0 px0 (1 p) x0 , x0 y0 px0 (1 p) x0 ) is
2 2
p 2(1 p) x0 1 A1 B1
J1
1 p 2(1 p) x
0 1 C1
D1
where A1 p 2(1 p) x0 , B1 1, C1 1 p 2(1 p) x0 , D1 1.
Next the Jacobian J2 for the transformat ion
E2 ( x0 , y0 ) = ( x2 , y2 ), is the product of the Jacobians for the transformations
E ( x1 , y1 ) ( y1 px1 (1 p) x1 , x1 y1 px1 (1 p) x1 ) and
2 2
E ( x0 , y0 ) ( y0 px0 (1 p) x0 , x0 y0 px0 (1 p) x0 ).
2 2
So, we obtain
p 2(1 p) x1 1 A1 B1 A2 B2
J2
1 p 2(1 p) x ,
1 1 C1
D1 C 2
D2
where A2 [ p 2(1 p) x1 ] A1 C1 , B2 [ p 2(1 p) x1 ]B1 D1 ,
C2 [1 p 2(1 p) x1 ] A C1 , D2 [1 p 2(1 p) x1 ]B1 D1.
1
m
Continuing this process in this way, we have the Jacobian for E as
A Bm
Jm m
C
m Dm
with a set of recursive formula as
Am [ p 2(1 p) xm1 ] Am1 Cm1 , Bm [ p 2(1 p) xm1 ]Bm1 Dm1 ,
Cm [ p 2(1 p) xm1 ] Am1 Cm1 , Dm [ p 2(1 p) xm1 ]Bm1 Dm1 ,
(m = 2, 3, 4, 5…).
Since the fixed point of this map E is a zero of the map
G( x, y) E( x, y) ( x, y),
the Jacobian of G (k ) is given by
Issn 2250-3005(online) November| 2012 Page 40
4. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
A 1 Bk
Jk I k
C
k Dk 1
1 Dk 1 Bk
Its inverse is ( J k I ) 1 ,
Ck
Ak 1
where ( Ak 1)( Dk 1) Bk Ck ,
the Jacobian determinant. Therefo re, Newton‟s method gives the following recurrence formu la in order to yield a
periodic point ofEk
( D 1)( xn xn ) Bk ( y n y n )
ˆ ˆ
xn1 xn k
(C k )( xn xn ) ( Ak 1)( y n y n )
ˆ ˆ
y n1 y n ,
where E k (x n ) ( xn , y n ).
ˆ ˆ
4. Numerical Methods For Finding Bifurcation Values [2, 4 ]:
k
First of all, we recall our recurrence relations for the Jacobian matrix of the map E described in the Newton‟s
method and then the eigenvalue theory gives the relation Ak Dk 1 Det ( J k ) at the bifurcation value. Again the
Feigenbaum theory says that
p pn
p n2 p n1 n1 (1.2)
where n = 1,2,3,… and is the Feigenbaum universal constant.
In the case of our map, the first two bifurcat ion values p 1 and p 2 can be evaluated.
Furthermore, it is easy to find the periodic points for p 1 and p 2. We note that if we put I Ak Dk 1 Det ( J k ) , then
I turns out to be a function of the parameter p . The bifurcation value of p of the period k occurs when I(p) equals zero.
This means, in order to find a bifurcation value of period k, one needs the zero of the function I(p), which is given by the
Secant method,
I ( pn )( pn pn 1 )
pn 1 pn .
I ( pn ) I ( pn1 )
Then using the relation (1.2), an approximate value
p3 of p3 is obtained. Since the Secant method needs two initial
values, we use
p3 and a slightly larger value, say, 4
p3 10 as the two initial values to apply this method and
ultimately obtain p3 . In like manner, the same procedure is emp loyed to obtain the successive bifurcation values
p4 , p5 ,... etc. to our requirement.
For finding periodic points and bifurcation values for the map E, above numerical methods are used and consequently,
the following Period-Doubling Cascade : Table 1.1, showing bifurcation points and corresponding periodic points , are
obtained by using suitable computer programs :
Table 1.1
Period One of the Periodic points Bifurcation Pt.
1 (x=0.666666666666...., y= -1.333333333333....) p1=7.00000000000
2 (x= -0.500000000003..., y= -1.309016994376...) p2=7.47213595500
4 (x= -0.811061640408 ..., y= -1.273315586957...) p3=7.525683372…
8 (x=-0.813878975794 ..., y= -1.275108054848...) p4=7.531826966…
16 (x=-0.460474775277...,y= -1.273990103300...) p5=7.532531327…
32 (x=-0.46055735696 ..., y= -1.274198905689...) p6=7.532612093…
64 (x=-0.54742669886 ..., y= -1.356526357634...) p7=7.532621354…
128 (x=-0.547431479795 ..., y= -1.356530832564...) p8=7.532636823…
… … … … … … ..
Issn 2250-3005(online) November| 2012 Page 41
5. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
For the system (1.1), the values of are calculated as follows:
p p1 p p2
1 2 8.8171563807015..., 2 3 8.715976842041...,
p3 p 2 p 4 p3
p 4 p3 p p4
3 8.722215813427..., 4 5 8.721026780948...,
p5 p 4 p 6 p5
and so on.
The ratios tend to a constant as k tends to infinity: mo re formally
b bk 1
lim k 8.7210972...
k bk 1 bk
And the above table confirms that the „universal‟ Feigenbaum constant δ = 8.7210972…
is also encountered in this area-preserving two-dimensional system.
The accumu lation point p can be calculated by the formula
1
p ( p2 p1 ) p2 ,
1
where δ is Feigenbaum constant, it is found to be 7.533284771388…. ., beyond which the system (1.1) develops chaos.
5. Time Series Graphs [3]
The key theoretical tool used for quantifying chaotic behavior is the notion of a time series of data for the
system [9]. A t ime series is a chronological sequence of observations on a particular variable. Usually the observations
are taken at regular intervals. The system (1.1) giv ing the difference equations:
xn1 y n pxn (1 p) xn , y n1 xn y n pxn (1 p) xn , n 0,1,2,...
2 2
(1.3)
On the horizontal axis the number of iterat ions („time‟) are marked, that on the vertical axis the amp litudes are given for
each iteration. The system (1.3) exhib is the follo wing discrete time series graphs for the values of xn and yn , plotted
together, showing the existences of periodic orb its of periods 2k , k = 0, 1, 2,…, at d ifferent parameter
0.50
0.55
Values xn and yn
0.60
0.65
0.70
0.75
0.80
0 10 20 30 40 50
No. of iterations n
values.
Fig. 1 Showi ng period-1 behavi or, parameter = 1st bifurcati on poi nt
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6. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
0.4
0.5
Values xn and yn
0.6
0.7
0.8
0 10 20
No. of iterat ions
30
n
40 50
Fig. 1.2 Showing period-2 behavior, parameter = 2nd bi furcation point
.4
0
.5
0
Values xn and yn
.6
0
.7
0
.8
0
0 10 20 30
N o . o f i t erat i o n s
n
40
Fig. 1.3 Showing period-4 behavior, parameter = 3rd bifurcation point
50
0.4
0.5
Values xn and yn
0.6
0.7
0.8
0 10 20
No. of iterations
30
n
40 50
Fig. 1.4 Showing period-8 behavior, parameter = 4th bi furcation point
6. Acknowledgement:
I am g rateful to Dr. Tarini Kumar Dutta, Sen ior Professor of the Depart ment of Mathematics, Gauhati Un iversity,
Assam: India, for a valuable suggestion for the improvement of this research paper.
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7. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
7. References:
[1] Deveney R.L. “ An introduction to chaotic dynamical systems ”, Addison-Wiseley Publishing Co mpany,Inc
[2] Davie A. M . and Dutta T. K., “Period-doubling in Two-Parameter Families,” Physica D, 64,345-354, (1993).
[3] Das, N. and Dutta, N., “ Time series analysis and Lyapunov exponents for a fi fth degree chaotic map ”, Far East
Journal of Dynamical Systems Vo l. 14, No. 2, pp 125- 140, 2010
[4] Das, N., Sharmah, R. and Dutta, N., “Period doubling bifurcation and associated universal properties in the
Verhulst population model”, International J. of Math. Sci. & Engg. Appls. , Vo l. 4 No. I (March 2010), pp. 1 -14
[5] Dutta T. K. and Das, N., “Period Doubling Route to Chaos in a Two-Dimensional Discrete Map”, Global
Journal of Dynamical Systems and Applications, Vol.1, No. 1, pp 61 -72, 2001
[6] Feigenbaum .J M, “Quantitative Universality For Nonlinear Transformations” Journal of Statistical Physics, Vol.
19, No. 1,1978.
[7] Chen G. and Dong X., Fro m Chaos to Order: Methodologies, Perspectives and Applications, World Scientific,
1998
[8] Peitgen H.O., Jurgens H. and Saupe D. , “ Chaos and Fractal”, New Frontiers of Science, Springer Verlag, 1992
[9] Hilborn R. C “Chaos and Nonlinear Dynamics”,second edition, Oxford University
Press, 1994
[10] Lanford O.E. III, “A Co mputer-Assisted Proof of the Feigenbaum Conjectures,” Bu ll.A m.Math.Soc. 6, 427 -34
(1982).
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