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.

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 Basin of Attraction Analysis of New Memristor-Based Fractional-Order Chaotic System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Long Ding ◽  
Li Cui ◽  
Fei Yu ◽  
Jie Jin
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Fractional Order ◽  
Chaotic System ◽  
Exponential Function ◽  
Local Minimum ◽  
System Analysis ◽  
Basin Of Attraction ◽  
Stable Point ◽  
The Basin Of Attraction

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a chaotic phase diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system.

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 An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

2020 ◽  
Vol 30 (05) ◽  
pp. 2030012 ◽  
Author(s):  
Víctor J. García-Garrido
Keyword(s):  
Phase Space ◽  
Invariant Manifolds ◽  
Initial Conditions ◽  
Escape Time ◽  
Discrete Time System ◽  
Classical Analysis ◽  
Time System ◽  
Working Definition ◽  
Original Approach ◽  
Lagrangian Descriptors

In this paper, we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that is built on the original approach introduced in [ Lopesino et al., 2015a ], and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete-time system, the Hénon map.

scholarly journals Strategies for an Efficient Official Publicity Campaign

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Juan Neirotti
Keyword(s):  
Fixed Point ◽  
Public Opinion ◽  
Phase Space ◽  
Fixed Points ◽  
Basin Of Attraction ◽  
Opinion Formation ◽  
Publicity Campaign ◽  
The Basin Of Attraction

AbstractWe consider the process of opinion formation, in a society where there is a set of rules B that indicates whether a social instance is acceptable. Public opinion is formed by the integration of the voters’ attitudes which can be either conservative (mostly in agreement with B) or liberal (mostly in disagreement with B and in agreement with peer voters). These attitudes are represented by stable fixed points in the phase space of the system. In this article we study the properties of a perturbative term, mimicking the effects of a publicity campaign, that pushes the system from the basin of attraction of the liberal fixed point into the basin of the conservative point, when both fixed points are equally likely.

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.

scholarly journals On embedding of invariant manifolds of the simplest Morse-Smale flows with heteroclinical intersections

2018 ◽  
Vol 20 (4) ◽  
pp. 378-383
Author(s):  
E. Ya. Gurevich ◽  
D. A. Pavlova
Keyword(s):  
Phase Space ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Classification Problem ◽  
Closed Curve ◽  
Topological Classification ◽  
Dimensional Phase Space ◽  
Smale Flows ◽  
Source Sink ◽  
The Basin Of Attraction

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. More precisely, we consider a class G(S4) of Morse-Smale flows on the sphere S4 such that for any flow f∈G(S4) its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves that belong to intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of saddle equilibria’s invariant manifolds; that is the first step in the solution of topological classification problem. In particular, we prove that the closures of invariant manifolds of saddle equlibria that do not contain heteroclinical curves are locally flat 2-sphere and closed curve. These manifolds are attractor and repeller of the flow. In set of orbits that belong to the basin of attraction or repulsion we construct a section that is homeomoprhic to the direct product S2×S1. We study a topology of intersection of saddle equlibria’s invariant manifolds with this section.

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.

scholarly journals A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Qingdu Li ◽  
Xiao-Song Yang
Keyword(s):  
Discrete Time ◽  
Interval Analysis ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Discrete Time System ◽  
Topological Horseshoe ◽  
Time System ◽  
Henon Map ◽  
Smale Horseshoe ◽  
Generalized Hénon Map

This paper presents a three-dimensional topological horseshoe in the hyperchaotic generalized Hénon map. It looks like a planar Smale horseshoe with an additional vertical expansion, so we call it 3D Smale horseshoe. In this way, a computer assisted verification of existence of hyperchaos is provided by means of interval analysis.

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