WEIGHTED POWER MEAN DISCRETE DYNAMICAL SYSTEMS: FAST CONVERGENCE PROPERTIES

We studied families of discrete dynamical systems obtained by using iteration functions given by weighted power mean in order to understand the role of hyperrapid convergence in nonlinear maps. Our interest resides in concepts related to the velocity of convergence. We introduce new concepts regarding the time of convergence and we provide an ordering of these families according to their dependence on parameters.


Introduction
The arithmetico-geometrical algorithm is well known for the numerical evaluation of elliptic functions and integrals [8].The algorithm starts with two numbers (x 0 , y 0 ) and successive numbers (x n , y n ), n ∈ N, are calculated from the recurrence formulas x n+1 = (x n + y n )/2 (arithmetic mean) and y n+1 = √ x n y n (geometric mean).Thus, a convergent sequence {(x n , y n )} is generated with a common limit given by an elliptic integral.
The joint iteration of the arithmetic and the geometric means has been explored already by Carl Friedreich Gauss in a related problem regarding secular perturbations of orbital elements [5].The algorithm has found many uses in several disciplines, in areas of mathematics such as numerical analysis [2,3], number theory [1,4], in physics [7,9], in finances in problems related to portfolio market value [6], and so on.In spite of the applicability of this process, there has not been much attention devoted to dynamical processes with different versions of iterated means.In this work we will introduce and explore new dynamical systems with an iteration function given by a generalization of the arithmetic and geometric means.Our interest resides in the study of some dynamical properties such as convergence and velocity.We establish a classification on these new dynamical systems according to the time of convergence and the critical exponent associated concepts that will be introduced afterwards.
There is a huge variety of possible generalizations of the arithmetic and geometric means.One of them is given by a weighted power mean, which is defined as follows: given two positive numbers x and y and a weight w, with 0 < w < 1, their weighted power mean is defined as (wx r + (1 − w)y r ) 1/r , where the power r is a nonzero real number.We are in position to state the next definition.Definition 1.1.A weighted power mean function is defined as the function f : R It is straightforward to show that a power mean function f satisfies the following properties. (1) So we defined f (x, y,w,0) = x w y 1−w .Notice that in particular when w = 1/2 we obtain the geometric mean of x and y, that is f (x, y,1/2,0) = √ xy.
(2) f (x, y,w,r) is an increasing function of r for each fixed w, x, and y.
(3) If x ≤ y, then f (x, y,w,r) is an increasing (decreasing) function of w for each fixed x, y, and r > 0, (r < 0).(4) If x ≤ y, then the harmonic weighted power mean, f (x, y,w,−1), is related to the arithmetic, f (x, y,w,1), and geometric, f (x, y,w,0), weighted power mean as x ≤ f (x, y,w,−1) ≤ f (x, y,w,0) ≤ f (x, y,w,1) ≤ y. (1.3)This paper is organized as follows.In Section 2 we define and give properties of weighted power mean discrete dynamical systems.In Section 3 the dynamic behavior of the dynamical systems is analyzed by reducing their dimension.In Section 4 we introduce the concepts of convergence and critical exponents in a general setting.Numerical experiments between the different systems and conclusions are given in Section 5.

Weighted power mean systems
Let us start with the following definition.Definition 2.1.Given a weighted power mean function f with fixed w i and r i , i = 1,2, a weighted power mean (WPM) discrete dynamical system is defined as with 0 < x 0 and 0 < y 0 .
Francisco J.Solis 3 In the successive sections we study some elementary properties of WPM discrete dynamical systems.

Convergence
Proposition 2.2.The WPM discrete dynamical system (2.1) converges for appropriate initial conditions to a common value.
Proof.Let x 0 and y 0 be nonnegative initial conditions for the system (2.1), without loss in generality we assume that x 0 ≤ y 0 , then we have that x r 0 ≤ w 1 x r 0 + (1 − w 1 )y r 0 .Thus, inductively we obtain that the sequence {x n } ∞ 0 satisfies and similarly Thus {x n } ∞ 0 and {y n } ∞ 0 are two convergent sequences.Let x ∞ and y ∞ be their corresponding limits.Using (2.1) we get that x ri ∞ = y ri ∞ for i = 1,2, which implies that x ∞ = y ∞ .The limit values of WPM discrete dynamical systems can be calculated explicitly for several specific cases which we study in the following sections.

2.2.
Case r 1 = r 2 (linear).Let us consider the case where r 1 and r 2 have the same value which we denote as r.
Proposition 2.3.The WPM discrete dynamical system converges to a common value given by Proof.In this case the discrete dynamical system (2.5) can be written as (2.7) Setting x n = x r n and y n = y r n in (2.7) we get the linear system (2.8) The linear system in x n and y n can be written in the form with (2.10) (2.11) Notice that x n and y n converge to the common limit given by (2.12) Therefore, the common limit of the original system (2.5) is given by (2.13) Notice that the system converges exponentially, that is, as exp(nln(|w 1 − w 2 |)) with n approaching ∞. into the following equality known as Gauss's transformation:

Case with
x 2r 0 cos 2 (θ) + y 2r 0 sin 2 (θ) . (2.18) Taking the limit as n goes to infinity and using the fact that the system (2.14) has a common limit we obtain the desired result.
This system converges faster than the system with r 1 = r 2 .Later on we will show in detail the reason for this behavior.

Decoupling
Using the following transformation x n = ρ n cos(θ n ) and y n = ρ n sin(θ n ) with 0 < θ n < π/2 for all n, the system (2.1) decouples into ) where , we obtain a one-dimensional discrete system z n+1 = H(z n ) given by This system inherits the convergence properties of system (2.1), therefore it converges globally to the fixed point z = 1 for all values of w i and r i , i = 1,2.Notice that |H (1)| = |w 1 − w 2 | < 1 which implies that if w 1 = w 2 , then the system converges exponentially to the fixed point and the error decays as e −n/τ where τ is a constant, see [10].
Assume now that w 1 = w 2 , then |H (1)| = 0 and |H (1 Notice that H (1) = 0 only if r 1 = r 2 and this is the case which was already analyzed in Section 2.2.Therefore we can assume that H (1) = 0. Let us now investigate this case in the next section using a general setting.

Critical exponents and time of convergence
Given a discrete dynamical system of the form x n+1 = H(λ,x n ), where λ is a parameter, assume the existence of an isolated attracting fixed point x H (λ), which may depend on λ.Define the error sequence, { n }, as Let λ be a value of the control parameter satisfying H ( λ,x H ) = 0, and assume that H ( λ,x H ) is not identically equal to zero, then using (4.1) we conclude that | n | decays as exp(−2 n /τ) with τ a constant independent of n.From now on we will refer to the points ( λ,x H ) as points of fast convergence.The constant τ has a particular meaning, which is given in the next definition.Definition 4.1.Define the time of convergence of the system x n+1 = H(λ,x n ) at a point of fast convergence ( λ,x H ) as and we also define the critical exponent, δ, as the smallest power of the nonzero term in the Taylor series of g(λ It is noticeable that both concepts depend on the initial condition.With these definitions, we obtain a classification of discrete dynamical systems at points of fast convergence.For each class, defined by specific values of λ and derivatives of the function ln |( 0 /2)(∂H 2 /∂x 2 )(λ,x H )|, the value of δ is independent of the iteration function.The main ideas to define these new concepts are taken from [10] which is a work regarding slower dynamical systems.
We now return to the analysis of WPM dynamical systems to show the existence of classes of dynamical systems with different associated critical exponent values.
For the system (w,1) is a point of fast convergence for all w ∈ (0,1).So considering a fixed weight w 0 , the Taylor expansion of the function g(w Therefore we obtain that if 0 w 0 (1 − w 0 )|r 1 − r 2 | = 2, the system (4.3) has associated a critical exponent δ = 0.So except for a set of Lebesgue measure zero, that is, when 0 w 0 (1 − w 0 )|r 1 − r 2 | = 2, zero is the typical value of the critical exponent for WPM systems with the same weight.Now assume that 0 w 0 (1 − w 0 )|r 1 − r 2 | = 2, requiring that 0 |r 1 − r 2 | ≥ 8, then the system (4.3) has a critical exponent δ = 1 only if w 0 = 1/2.Finally, the only existing critical exponent is δ = 2 if we have that w 0 = 1/2 and 0 |r 1 − r 2 | = 8.Therefore we have proven the following proposition.

Numerical examples and conclusions
The WPM discrete dynamical system (5.1) (1) converges exponentially if w 1 = w 2 .In Figure 5.1 we show the set of initial conditions {(x 0 , y 0 ) | 0 ≤ x 0 ≤ 1, 0 ≤ y 0 ≤ 1} for a WPM system with r 1 = 2, r 2 = 4, w 1 = 0.5, and w 2 = 0.3.The number of iterations necessary to achieve convergence with a tolerance of 10 −6 is shown in the different colored regions of the unit square.The black portion of Figure 5.1 means that only one iteration is needed to achieve convergence; (2) with w 1 = w 2 , converges with a critical exponent of δ = 0 for all initial conditions except for a set of measure zero.In Figure 5.2, we show the number of iterations to achieve convergence with a tolerance of 10 −6 in the unit square of initial conditions for a system with r 1 = 5, r 2 = 0.5, w 1 = w 2 = 0.3.Notice the notable reduction in the number of iterations from the previous example.