Biparametric Family Research Articles

Lyapunov functions for a non-linear model of the X-ray bursting of the microquasar GRS 1915+105

This paper introduces a biparametric family of Lyapunov functions for a non-linear mathematical model based on the FitzHugh-Nagumo equations able to reproduce some main features of the X-ray bursting behaviour exhibited by the microquasar GRS 1915+105. These functions are useful to investigate the properties of equilibrium points and allow us to demonstrate a theorem on the global stability. The transition between bursting and stable behaviour is also analyzed.

A survey of different classes of Earth-to-Moon trajectories in the patched three-body approach

This paper deals with Earth-to-Moon transfers in the patched three-body approach, in which the Sun–Earth–Moon–Spacecraft four-body system is approximated by two coupled Circular Restricted Three-Body Problems (CR3BP). This approach provides preliminary solutions that can be numerically refined into full four-body solutions. The standard transfers in this approach are low-energy manifold guided solutions with long transfer time which connect transit and non-transit orbits of each three-body system. Besides the standard transit-non-transit connections, there are alternative solutions involving a bi-parametric family of quasi-periodic orbits around the Earth. These solutions connect quasi-periodic orbits on two-dimensional tori of the Sun–Earth–Spacecraft system with L1 or L2 transit solutions of the Earth–Moon–Spacecraft system to provide transfers with lunar ballistic capture and short flight time. We review the dynamical elements employed to obtain the different classes of transfers and give examples of solutions obtained from sets of initial conditions around the Earth that are consistent with current infrastructure for space exploration.

A new bi-parametric family of temporal transformations to improve the integration algorithms in the study of the orbital motion

One of the fundamental problems in celestial mechanics is the study of the orbital motion of the bodies in the solar system. This study can be performed through analytical and numerical methods. Analytical methods are based on the well-known two-body problem; it is an integrable problem and its solution can be related to six constants called orbital elements. To obtain the solution of the perturbed problem, we can replace the constants of the two-body problem with the osculating elements given by the Lagrange planetary equations. Numerical methods are based on the direct integration of the motion equations. To test these methods we use the model of the two-body problem with high eccentricity.In this paper we define a new family of anomalies depending on two parameters that includes the most common anomalies. This family allows one to obtain more compact developments to be used in analytical series and also to improve the efficiency of numerical methods because it defines a more suitable point distribution with the dynamics of the two-body problem.

Stability of a fourth order bi-parametric family of iterative methods

In this paper we present a dynamical study of the Ostrowski–Chun family of iterative methods on quadratic polynomials. We will use dynamical tools such as the analysis of fixed and critical points, and the calculation of parameter planes, to find the most stable members of the family. These results have been checked on the unidimensional Bratu’s problem.

Construction of fourth-order optimal families of iterative methods and their dynamics

In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f(xn), f(yn) and one of its first-order derivative f ′(xn)) per iteration. Further, we show that the well-known Ostrowski’s method and King’s family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations.

Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure

In this paper, from Traub’s method and by applying weight function technique, a bi-parametric family of predictor–corrector iterative schemes with optimal fourth-order of convergence, for solving nonlinear equations, is presented. By using some algebraic manipulations and a divided difference operator, we extend this family to the multidimensional case, preserving its order of convergence. Some numerical test are made in order to confirm the theoretical results and to compare the new methods with other known ones.

Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

We consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated with discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter.We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schrödinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1D-Schrödinger map equation.

Transparent lattices and their solitary waves

We provide a family of transparent tight-binding models with nontrivial potentials and site-dependent hopping parameters. Their feasibility is discussed in electromagnetic resonators, dielectric slabs, and quantum-mechanical traps. In the second part of the paper, the arrays are obtained through a generalization of supersymmetric quantum mechanics in discrete variables. The formalism includes a finite-difference Darboux transformation applied to the scattering matrix of a periodic array. A procedure for constructing a hierarchy of discrete Hamiltonians is indicated and a particular biparametric family is given. The corresponding potentials and hopping functions are identified as solitary waves, pointing to a discrete spinorial generalization of the Korteweg-deVries family.

An efficient two-parametric family with memory for nonlinear equations

A new two-parametric family of derivative-free iterative methods for solving nonlinear equations is presented. First, a new biparametric family without memory of optimal order four is proposed. The improvement of the convergence rate of this family is obtained by using two self-accelerating parameters. These varying parameters are calculated in each iterative step employing only information from the current and the previous iteration. The corresponding R-order is 7 and the efficiency index 71/3 = 1.913. Numerical examples and comparison with some existing derivative-free optimal eighth-order schemes are included to confirm the theoretical results. In addition, the dynamical behavior of the designed method is analyzed and shows the stability of the scheme.

Study of a Biparametric Family of Iterative Methods

The dynamics of a biparametric family for solving nonlinear equations is studied on quadratic polynomials. This biparametric family includes thec-iterative methods and the well-known Chebyshev-Halley family. We find the analytical expressions for the fixed and critical points by solving 6-degree polynomials. We use the free critical points to get the parameter planes and, by observing them, we specify some values of(α, c)with clear stable and unstable behaviors.

A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

Double Generalized Spatial Econometric Models

The authors offer a unified method extending traditional spatial dependence with normally distributed error terms to a new class of spatial models based on the biparametric exponential family of distributions. Joint modeling of the mean and variance (or precision) parameters is proposed in this family of distributions, including spatial correlation. The proposed models are applied for analyzing Colombian land concentration, assuming that the variable of interest follows normal, gamma, and beta distributions. In all cases, the models were fitted using Bayesian methodology with the Markov Chain Monte Carlo (MCMC) algorithm for sampling from joint posterior distribution of the model parameters.

Mixture of Distributions in the Biparametric Exponential Family: A Bayesian Approach

In this article we propose mixture of distributions belonging to the biparametric exponential family, considering joint modeling of the mean and variance (or dispersion) parameters. As special cases we consider mixtures of normal and gamma distributions. A novel Bayesian methodology, using Markov Chain Monte Carlo (MCMC) methods, is proposed to obtain the posterior summaries of interest. We include simulations and real data examples to illustrate de performance of the proposal.

Probabilistic comparison of weighted majority rules

This paper studies a bi-parametric family of decision rules, so-called <i>restricted distinguished chairman rules</i>, which contains several one-parameter classes of rules considered previously in the literature. Roughly speaking, these rules apply t

A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index

A biparametric family of four-step multipoint iterative methods of order sixteen to solve nonlinear equations are developed and their convergence properties are established. The optimal efficiency indices are all found to be 1 6 1 / 5 ≈ 1.741101 . Numerical examples as well as comparison with existing methods are demonstrated to verify the developed theory.

A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function

A biparametric family of four-step multipoint iterative methods of order sixteen to numerically solve nonlinear equations are developed and their convergence properties are investigated. The efficiency indices of these methods are all found to be 16 1/5≈1.741101, being optimally consistent with the conjecture of Kung–Traub. Numerical examples as well as comparison with existing methods developed by Kung–Traub and Neta are demonstrated to confirm the developed theory in this paper.

A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function

This paper proposes a biparametric family of three-step eighth-order multipoint iterative methods with optimal efficiency index in the sense of Kung-Traub for simple roots of nonlinear equations. We employ their third-step weighting function decomposed into King’s linear fractional function and a two-variable function to construct such a family of optimal methods. Development and convergence analysis on the proposed methods is fully described in addition to numerical experiments including comparison with existing methods.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua's oscillator.

TOPOLOGY AND PERIODIC ORBITS OF RING-SHAPED POTENTIALS AS A GENERALIZED 4-D ISOTROPIC OSCILLATOR

In this work we study a generalized integrable biparametric family of 4-D isotropic oscillators. This family allows to treat, in a unified way, oscillators defined by the potentials given by Hartmann and Quesne and other ring-shaped systems. Using the Liouville–Arnold theorem and the analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. By this topological study and the calculation of the action-angle variables we obtain the full classification of periodic and quasiperiodic orbits for this system.

Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Periodic orbits

A generalized integrable biparametric family of 4-D isotropic oscillators is proposed. It allows to treat in a unified way, Pöschl-Teller, Hartmann and other ring-shaped systems. This approach, based in the use of two canonical extensions, helps to simplify the studies of classical aspects of those systems. As an illustration, an analysis of the periodic solutions of those system is presented.