FINITENESS OF THE AREA OF BASINS OF ATTRACTION OF RELAXED NEWTON METHOD FOR CERTAIN HOLOMORPHIC FUNCTIONS

2004 ◽  
Vol 14 (12) ◽  
pp. 4177-4190 ◽  
Author(s):  
FİGEN ÇİLİNGİR
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basin Of Attraction ◽  
Complex Function ◽  
Holomorphic Functions ◽  
Basins Of Attraction ◽  
Rational Map ◽  
Finite Area ◽  
The Basin Of Attraction ◽  
Parabolic Fixed Point

For a nonconstant function F and a real number h∈]0, 1] the relaxed Newton's method NF,h of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z)=P(z)eQ(z), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that NF,h is a rational map with a parabolic fixed point at infinity.

Tail-Enabled Spiraling Maneuver for Gliding Robotic Fish

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Feitian Zhang ◽  
Fumin Zhang ◽  
Xiaobo Tan
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Three Dimensional ◽  
Relative Equilibria ◽  
Basins Of Attraction ◽  
Robotic Fish ◽  
Experimental Results ◽  
Underwater Gliders ◽  
Local Asymptotic Stability ◽  
Turning Radius

Gliding robotic fish, a new type of underwater robot, combines both strengths of underwater gliders and robotic fish, featuring long operation duration and high maneuverability. In this paper, we present both analytical and experimental results on a novel gliding motion, tail-enabled three-dimensional (3D) spiraling, which is well suited for sampling a water column. A dynamic model of a gliding robotic fish with a deflected tail is first established. The equations for the relative equilibria corresponding to steady-state spiraling are derived and then solved recursively using Newton's method. The region of convergence for Newton's method is examined numerically. We then establish the local asymptotic stability of the computed equilibria through Jacobian analysis and further numerically explore the basins of attraction. Experiments have been conducted on a fish-shaped miniature underwater glider with a deflected tail, where a gliding-induced 3D spiraling maneuver is confirmed. Furthermore, consistent with model predictions, experimental results have shown that the achievable turning radius of the spiraling can be as small as less than 0.4 m, demonstrating the high maneuverability.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

1998 ◽  
Vol 01 (02n03) ◽  
pp. 161-180 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
Yu. L. Maistrenko ◽  
V. L. Maistrenko
Keyword(s):  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Chaotic State ◽  
One Dimensional ◽  
Type Iii ◽  
Different Types ◽  
One Dimensional Maps ◽  
The Basin Of Attraction

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

scholarly journals Basins of Attraction for Various Steffensen-Type Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa ◽  
Stanford Shateyi
Keyword(s):  
Computer Algebra System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Numerical Tests ◽  
Derivative Free ◽  
Iteration Functions ◽  
The Stability ◽  
Programming Package ◽  
The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

scholarly journals Adaptive multimode signal reconstruction from time–frequency representations

2016 ◽  
Vol 374 (2065) ◽  
pp. 20150205 ◽  
Author(s):  
Sylvain Meignen ◽  
Thomas Oberlin ◽  
Philippe Depalle ◽  
Patrick Flandrin ◽  
Stephen McLaughlin
Keyword(s):  
Signal Reconstruction ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Alternative View ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Fm Signals ◽  
Adaptive Reconstruction ◽  
Short Time ◽  
The Basin Of Attraction

This paper discusses methods for the adaptive reconstruction of the modes of multicomponent AM–FM signals by their time–frequency (TF) representation derived from their short-time Fourier transform (STFT). The STFT of an AM–FM component or mode spreads the information relative to that mode in the TF plane around curves commonly called ridges . An alternative view is to consider a mode as a particular TF domain termed a basin of attraction . Here we discuss two new approaches to mode reconstruction. The first determines the ridge associated with a mode by considering the location where the direction of the reassignment vector sharply changes, the technique used to determine the basin of attraction being directly derived from that used for ridge extraction. A second uses the fact that the STFT of a signal is fully characterized by its zeros (and then the particular distribution of these zeros for Gaussian noise) to deduce an algorithm to compute the mode domains. For both techniques, mode reconstruction is then carried out by simply integrating the information inside these basins of attraction or domains.

scholarly journals On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction ◽  
Picard Iteration ◽  
Mann Iteration ◽  
Iteration Parameter ◽  
Root Finding ◽  
Base Method ◽  
Fabric Design ◽  
Root Finding Method

AbstractThere are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

JULIA SETS OF GENERALIZED NEWTON'S METHOD

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 323-336 ◽  
Author(s):  
XINGYUAN WANG ◽  
TINGTING WANG
Keyword(s):  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Basins Of Attraction ◽  
Principal Value ◽  
Extraneous Fixed Points ◽  
Sets Theory ◽  
Steffensen Method ◽  
Generalized Newton’S Method

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

scholarly journals Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 78
Author(s):  
Ankush Aggarwal ◽  
Sanjay Pant
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Short Note ◽  
Basin Of Attraction ◽  
Computational Cost ◽  
Initial Guess ◽  
Mathematical Functions ◽  
New Class ◽  
Root Finding ◽  
Roots Of Equations

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton’s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton’s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton’s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley’s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problems.

Effect of Delay on the Boundary of the Basin of Attraction in a Self-Excited Single Graded-Response Neuron

1997 ◽  
Vol 9 (2) ◽  
pp. 319-336 ◽  
Author(s):  
K. Pakdaman ◽  
C. P. Malta ◽  
C. Grotta-Ragazzo ◽  
J.-F. Vibert
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Transmission Delays ◽  
The Past ◽  
Graded Response ◽  
The Stability ◽  
Delay Dependent ◽  
The Basin Of Attraction

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

2013 ◽  
Vol 66 (3) ◽  
pp. 431-455 ◽  
Author(s):  
José M. Gutiérrez ◽  
Luis J. Hernández-Paricio ◽  
Miguel Marañón-Grandes ◽  
M. Teresa Rivas-Rodríguez
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction

A COMPUTER-ASSISTED STUDY OF GLOBAL DYNAMIC TRANSITIONS FOR A NONINVERTIBLE SYSTEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1305-1321 ◽  
Author(s):  
RAYMOND A. ADOMAITIS ◽  
IOANNIS G. KEVREKIDIS ◽  
RAFAEL DE LA LLAVE
Keyword(s):  
Phase Space ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Computer Assisted ◽  
Discrete Time System ◽  
Time System ◽  
Invariant Circle ◽  
Global Dynamic ◽  
Assisted Analysis ◽  
The Basin Of Attraction

We present a computer-assisted analysis of the phase space features and bifurcations of a noninvertible, discrete-time system. Our focus is on the role played by noninvertibility in generating disconnected basins of attraction and the breakup of invariant circle solutions. Transitions between basin of attraction structures are identified and organized according to "levels of complexity," a term we define in this paper. In particular, we present an algorithm that provides a computational approximation to the boundary (in phase space) separating points with different preimage behavior. The interplay between this boundary and other phase space features is shown to be crucial in understanding global bifurcations and transitions in the structure of the basin of attraction.

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