The relaxed Newton method derivative: Its dynamics and non-linear properties

Article (PDF Available)inNonlinear Analysis 68(7):1868-1873 · April 2008with9 Reads
DOI: 10.1016/j.na.2007.01.020
Abstract
The dynamic behaviour of the one-dimensional family of maps f(x)=c2[(a−1)x+c1]−λ/(α−1) is examined, for representative values of the control parameters a,c1, c2 and λ. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a. The maps f(x) are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an xn versus λ plot, an initial exponential decay followed by a bifurcation. The value of λ at which this bifurcation takes place depends on the values of the parameters a,c1 and c2. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x) undergoing a period doubling. For values of a higher than 1 and at higher values of λ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c1 between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.
Nonlinear Analysis 68 (2008) 1868–1873
www.elsevier.com/locate/na
The relaxed Newton method derivative: Its dynamics and
non-linear properties
Mehmet
¨
Ozer
a
, Yasar Polatoglu
b
, G
¨
ursel Hacibekiroglou
a
, Antonios Valaristos
c
,
Amalia N. Miliou
c,
, Antonios N. Anagnostopoulos
d
, Antanas
ˇ
Cenys
e
a
Department of Physics, Istanbul Kultur University, TR-34191, Turkey
b
Department of Mathematics, Istanbul Kultur University, TR-34191, Turkey
c
Department of Informatics, Aristotle University of Thessaloniki, GR-54124, Greece
d
Department of Physics, Aristotle University of Thessaloniki, GR-54124, Greece
e
Information Systems Department, Vilnius Gediminas Technical University, LT-10223, Lithuania
Received 22 December 2006; accepted 9 January 2007
Abstract
The dynamic behaviour of the one-dimensional family of maps f (x) = c
2
[(a 1)x + c
1
]
λ/(α1)
is examined, for
representative values of the control parameters a, c
1
, c
2
and λ. The maps under consideration are of special interest, since they
are solutions of the relaxed Newton method derivative being equal to a constant a. The maps f (x) are also proved to be solutions
of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these
maps, after excessive iterations, shows, in an x
n
versus λ plot, an initial exponential decay followed by a bifurcation. The value
of λ at which this bifurcation takes place depends on the values of the parameters a, c
1
and c
2
. This corresponds to a switch to an
oscillatory behaviour with amplitudes of f (x) undergoing a period doubling. For values of a higher than 1 and at higher values of
λ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c
1
between
1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.
c
2007 Elsevier Ltd. All rights reserved.
Keywords: Relaxed Newton method; Bifurcation
1. Introduction
The study of the derivatives of the Newton method and relaxed Newton method has attracted the attention of several
groups involved in dynamical systems over the last few years. In fact, due to iterative schemes, discrete dynamics
techniques have disclosed striking results in the area of chaotic dynamics [1,3,5]. In most cases, these methods and
their derivatives have been applied to specific maps in the real and complex plane [2,11]. Nevertheless, it is surprisingly
interesting to view such iterative schemes as serving as dynamical systems by themselves.
In previous reports [7–9], we have studied the cases where the derivative of the Newton method or relaxed Newton
method takes on specific values, namely in (0, 2). The choice of these values was motivated both by preliminary
Corresponding author. Tel.: +30 2310 998407; fax: +30 2310 998419.
E-mail address: amiliou@csd.auth.gr (A.N. Miliou).
0362-546X/$ - see front matter
c
2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2007.01.020
M.
¨
Ozer et al. / Nonlinear Analysis 68 (2008) 1868–1873 1869
numerical results on electronic circuits and the incorporation of the Schwarzian derivative in our studies. The
Schwarzian derivative, which was defined by the German mathematician Hermann Schwarz in 1869 for studying
complex valued functions and has been essentially used in dynamics since 1978 [10], offers conditions under which
such systems can be driven to chaos via bifurcations.
2. Relaxed Newton method derivative
For any real or complex valued function f , we define the Newton method on f by
N
f
(x) = x
f (x)
f
0
(x)
. (1)
Because the Newton method is only linearly convergent at multiple roots, various modifications have been
suggested for improving the convergence. When we know that f has a multiple root of order λ, we can apply the
Newton method to f (x )
1
λ
, to obtain
N
λ, f
= x
f (x)
1
λ
(1/λ) f (x)
1
λ
1
f
0
(x)
= x λ ·
f (x)
f
0
(x)
. (2)
This is called the Relaxed Newton Method or the Newton method for a root of order λ. This method converges
quadratically to a root of order exactly λ.
One can define the first derivative of the Newton method and the relaxed Newton method, by
N
0
f
(x) =
x
f (x)
f
0
(x)
0
=
f (x) f
00
(x)
(
f
0
(x)
)
2
(3)
and
N
0
λ, f
(x) =
x λ ·
f (x)
f
0
(x)
0
= 1 λ + λ ·
f (x) f
00
(x)
(
f
0
(x)
)
2
(4)
respectively.
In previous works [7,8], we have examined the dynamic behaviour of the corresponding non-relaxed maps
satisfying N
0
f
(x) = 2, N
0
f
(x) = a, for 0 < a < 2 and a 6= 1. Also the behaviour of the relaxed case N
0
λ, f
(x) = 2 has
been considered [9]. We now generalize the above investigation to maps satisfying
N
0
λ, f
(x) = a. (5)
Simple calculations based on successive integrations show that N
0
λ, f
= a if and only if f is of the form
f (x) =
c
2
[
(a 1)x + c
1
]
λ
a1
. (6)
On the other hand,
N
0
λ, f
(x) =
x λ ·
f (x)
f
0
(x)
0
= 1 λ + λ ·
f (x) f
00
(x)
(
f
0
(x)
)
2
= a
(1 λ)
f
0
(x )
f (x)
+ λ
f
00
(x )
f
0
(x )
f
0
(x )
f (x)
= a
(
1 λ
)
f
0
(
x
)
f
(
x
)
+ λ
f
00
(
x
)
f
0
(
x
)
= a
f
0
(
x
)
f
(
x
)
which can also be written as
f
00
(
x
)
f
0
(
x
)
=
a + λ 1
λ
f
0
(
x
)
f
(
x
)
.
1870 M.
¨
Ozer et al. / Nonlinear Analysis 68 (2008) 1868–1873
If both sides of this equation are integrated, then we obtain the following equation:
f
0
(
x
)
β
[
f
(
x
)
]
a+λ1
λ
= 0. (7)
The integral parameter β is calculated by substitution of (6) in (7):
λc
2
[
(a 1)x + c
1
]
a+λ1
α1
= β
c
2
[
(a 1)x + c
1
]
λ
α1
a+λ1
λ
λc
2
= β c
a+λ1
λ
2
β = λ c
1
a+λ1
λ
2
= λc
1a
λ
2
.
Therefore
β =
λ
c
a1
λ
2
. (8)
Finally the differential equation given in (7) becomes
f
0
(
x
)
+
λ
c
a1
λ
2
[
f
(
x
)
]
α+λ1
λ
= 0. (9)
At this point, we incorporate in our discussion the idea of the Schwarzian derivative. Recall that the Schwarzian
derivative of f is
S
f
=
f
000
f
0
3
2
f
00
f
0
2
. (10)
It has to be mentioned here that the existence of period doublings is allowed but not guaranteed by the possession
of a negative Schwarzian derivative S
f
. Using Eq.
(6) and after some elementary calculations we obtain the explicit
expression for S
f
:
f
0
(
x
)
= λc
2
[
(a 1)x + c
1
]
λ+1a
a1
(11)
f
00
(
x
)
= λc
2
(λ + 1 a)
[
(a 1)x + c
1
]
λ+22a
a1
(12)
f
000
(
x
)
= λc
2
(λ + 1 a)(λ + 2 2a)
[
(a 1)x + c
1
]
λ+33a
a1
. (13)
Finally,
S
f
=
1
2
[(a 1)
2
λ
2
]
[
(a 1)x + c
1
]
2
. (14)
Obviously, the sign of S
f
depends on the values that λ takes on. In the following, we restrict our attention to the case
λ
2
> (a 1)
2
, where S
f
< 0.
3. Exhibition of the dynamics results
We examine the behaviour of the family of maps of Eq. (6):
f (x) =
c
2
[
(a 1)x + c
1
]
λ
a1
.
In a first step we fix the values of a and c
2
and we vary the parameter λ, in the range (0, 30).
To do so, we iterate the recurrent form of (6):
x
n
=
c
2
(a 1)x
n1
+ c
1
λ
a1
(15)
M.
¨
Ozer et al. / Nonlinear Analysis 68 (2008) 1868–1873 1871
Fig. 1. (a)–(i) The curves obtained after iterating Eq. (15) for a = 3, c
2
= 1 and c
1
between 0.90 and 1.20.
using the representative values a = 3 and c
2
= 1. We plot the maps of x
n
versus λ for different values of the parameter
c
1
. In this way the curves of
Fig. 1 are obtained. To avoid initial fluctuations we performed the averaging over the last
100 values of 10 000 iterations. For this purpose we have used Mathcad [4,6] to calculate the bifurcation diagrams.
As is evident from the curves of this figure, for values of the parameter c
1
< 1, a bifurcation occurs around λ = 5
with the two branches diverging exponentially with increasing λ. For c
1
= 1 the two branches become parallel, while
for c
1
> 1 they converge, forming a bleb. The size of these blebs decreases with increasing c
1
and finally disappears
for c
1
= 1.20. It should be mentioned here that the initial bifurcation at λ 5 as well as the initial and reverse
bifurcations forming the blebs occur at higher values of λ, in accordance with the restriction λ > 2, where S
f
< 0.
To confirm these transitions we have calculated numerically the corresponding Lyapunov exponents Λ of the maps.
For this purpose we have used Mathcad. Doing so, we took into account that the Lyapunov exponent can be estimated
using the formula
Λ = lim
N →∞
1
N
N
X
n=1
ln
d x
n+1
d x
n
. (16)
In the case of our map Eq. (16) becomes
Λ
1
N
N
X
n=1
ln
λc
2
[
(a 1)x
n
+ c
1
]
λ+1a
a1
. (17)
1872 M.
¨
Ozer et al. / Nonlinear Analysis 68 (2008) 1868–1873
Fig. 1. (continued)
Fig. 2. The curve obtained after iterating Eq. (15) for: a = 3, c
1
= 1.17, c
2
= 1 (upper curve) and the corresponding plot of the Lyapunov
exponent (lower curve).
For the calculations of the Lyapunov exponent Λ we have used instead of the general Eq. (16) the more specific
Eq. (17). To avoid initial fluctuations we performed the averaging over the last 100 values of 10 000 iterations.
In Fig. 2, we show the bifurcation diagram obtained for a = 3, c
1
= 1.17 and c
2
= 1 and the corresponding
behaviour of the Lyapunov exponent Λ. Apparently, at the critical points at which the initial and reverse bifurcations
M.
¨
Ozer et al. / Nonlinear Analysis 68 (2008) 1868–1873 1873
occur, a nullification of Λ takes place. In all other regions, the Lyapunov exponent remains negative, indicating a
periodic oscillation of the solutions of Eq. (9).
4. Comments
The dynamics of the map f (x ) =
c
2
[
(a1)x+c
1
]
λ
a1
are discussed and its bifurcating behaviour is realized for specific
values of the control parameter λ. This family of maps is viewed as solution of a differential equation as well as
solution of the relaxed Newton derivative being equal to a.
The differential equation (9) could be implemented by constructing a non-linear electronic circuit, namely an R LC
circuit driven by a sinusoidal voltage. The non-linear component of such a circuit should be a capacitor (varactor),
whose capacitance varies as a function of the voltage across it, whose form is conjugate to f (x). It would be very
interesting if gradual increase of the driving voltage results in a non-linear behaviour.
Acknowledgments
This work was supported by NATO, ICS.EAP.CLG 981947.
References
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[2] L. Billings, J.H. Curry, V. Robins, Chaos in relaxed Newton’s method: The quadratic case, Contemporary Mathematics 252 (1999).
[3] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989.
[4] D. DiFranco, Discovering chaotic iterations, http://www.mathcad.co.uk/mcadlib/apps/chaotic.mcd, 2002.
[5] G. Fulford, P. Forrester, A. Jones, Modelling with Differential and Difference Equations, Cambridge University Press, New York, 1997.
[6] J. Hayward, Chaotic iteration with MathCad, http://www.bham.ac.uk/ctimath/reviews/aug93/mathcad.pdf, 1993.
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