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Nonlinear Analysis 68 (2008) 1868–1873

www.elsevier.com/locate/na

The relaxed Newton method derivative: Its dynamics and

non-linear properties

Mehmet¨Ozera, Yasar Polatoglub, G¨ ursel Hacibekirogloua, Antonios Valaristosc,

Amalia N. Miliouc,∗, Antonios N. Anagnostopoulosd, AntanasˇCenyse

aDepartment of Physics, Istanbul Kultur University, TR-34191, Turkey

bDepartment of Mathematics, Istanbul Kultur University, TR-34191, Turkey

cDepartment of Informatics, Aristotle University of Thessaloniki, GR-54124, Greece

dDepartment of Physics, Aristotle University of Thessaloniki, GR-54124, Greece

eInformation 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) = c2[(a − 1)x + c1]−λ/(α−1)is examined, for

representative values of the control parameters a,c1, c2and 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 xnversus 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,c1and 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 c1between

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

theirderivativeshavebeenappliedtospecificmapsintherealandcomplexplane[2,11].Nevertheless,itissurprisingly

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

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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

Nf(x) = x −

f (x)

f?(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

Nλ, f = x −

f (x)

1

λ

(1/λ) f (x)

1

λ−1f?(x)

= x − λ ·

f (x)

f?(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

?

and

?

respectively.

In previous works [7,8], we have examined the dynamic behaviour of the corresponding non-relaxed maps

satisfying N?

been considered [9]. We now generalize the above investigation to maps satisfying

N?

f(x) =

x −

f (x)

f?(x)

??

=

f (x) f??(x)

( f?(x))2

(3)

N?

λ, f(x) =

x − λ ·

f (x)

f?(x)

??

= 1 − λ + λ ·f (x) f??(x)

( f?(x))2

(4)

f(x) = 2, N?

f(x) = a, for 0 < a < 2 and a ?= 1. Also the behaviour of the relaxed case N?

λ, f(x) = 2 has

N?

λ, f(x) = a.

(5)

Simple calculations based on successive integrations show that N?

λ, f= a if and only if f is of the form

f (x) =

c2

[(a − 1)x + c1]

λ

a−1

.

(6)

On the other hand,

N?

λ, f(x) =

(1 − λ)f?(x)

?

f (x)+ λf??(x)

f?(x)

f (x)

x − λ ·

f (x)

f?(x)

??

= a ⇒ (1 − λ)f?(x)

= 1 − λ + λ ·f (x) f??(x)

( f?(x))2

f (x)+ λf??(x)

= a

⇒

f?(x)

f?(x)= af?(x)

f (x)

which can also be written as

f??(x)

f?(x)=

?a + λ − 1

λ

?f?(x)

f (x).

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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?(x) − β [ f (x)]

The integral parameter β is calculated by substitution of (6) in (7):

?a+λ−1

λ

?

= 0.

(7)

−λc2[(a − 1)x + c1]−a+λ−1

α−1 = β

⇒ β = −λc1−a+λ−1

?

2

c2[(a − 1)x + c1]−

λ

α−1

?a+λ−1

λ

⇒ −λc2= β c

Therefore

a+λ−1

λ

2

λ

= −λc

1−a

λ

2

.

β =

−λ

c

2

a−1

λ

.

(8)

Finally the differential equation given in (7) becomes

f?(x) +

λ

a−1

λ

2

c

[ f (x)]

α+λ−1

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

Sf =

f???

f?−3

2

?f??

f?

?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 Sf. Using Eq. (6) and after some elementary calculations we obtain the explicit

expression for Sf:

f?(x) = −λc2[(a − 1)x + c1]

f??(x) = −λc2(−λ + 1 − a)[(a − 1)x + c1]

f???(x) = −λc2(−λ + 1 − a)(−λ + 2 − 2a)[(a − 1)x + c1]

Finally,

−λ+1−a

a−1

(11)

−λ+2−2a

a−1

(12)

−λ+3−3a

a−1

.

(13)

Sf =1

2[(a − 1)2− λ2] [(a − 1)x + c1]−2.

Obviously, the sign of Sf depends on the values that λ takes on. In the following, we restrict our attention to the case

λ2> (a − 1)2, where Sf < 0.

(14)

3. Exhibition of the dynamics — results

We examine the behaviour of the family of maps of Eq. (6):

c2

λ

a−1

In a first step we fix the values of a and c2and we vary the parameter λ, in the range (0,30).

To do so, we iterate the recurrent form of (6):

f (x) =

[(a − 1)x + c1]

.

xn=

c2

?(a − 1)xn−1+ c1

? λ

a−1

(15)

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Fig. 1. (a)–(i) The curves obtained after iterating Eq. (15) for a = 3,c2= 1 and c1between 0.90 and 1.20.

using the representative values a = 3 and c2= 1. We plot the maps of xnversus λ for different values of the parameter

c1. In this way the curves of Fig. 1 are obtained. To avoid initial fluctuations we performed the averaging over the last

100 values of 10000 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 c1< 1, a bifurcation occurs around λ = 5

with the two branches diverging exponentially with increasing λ. For c1= 1 the two branches become parallel, while

for c1> 1 they converge, forming a bleb. The size of these blebs decreases with increasing c1and finally disappears

for c1 = 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 Sf < 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

?

n=1

ln

?????

dxn+1

dxn

????

?

.

(16)

In the case of our map Eq. (16) becomes

Λ ≈

1

N

N

?

n=1

ln

????−λc2[(a − 1)xn+ c1]

−λ+1−a

a−1

???

?

.

(17)

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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, c1= 1.17, c2= 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 10000 iterations.

In Fig. 2, we show the bifurcation diagram obtained for a = 3, c1 = 1.17 and c2 = 1 and the corresponding

behaviour of the Lyapunov exponent Λ. Apparently, at the critical points at which the initial and reverse bifurcations

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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) =

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 RLC

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.

c2

[(a−1)x+c1]

λ

a−1

are discussed and its bifurcating behaviour is realized for specific

Acknowledgments

This work was supported by NATO, ICS.EAP.CLG 981947.

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