Fixed-point attractor

2012 ◽  
Vol 6 (6) ◽  
pp. 342-342
Author(s):  
Rachel Won
Keyword(s):  
Fixed Point ◽  
Fixed Point Attractor ◽  
Point Attractor

scholarly journals Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 24
Author(s):  
Alexey N. Pyrkov ◽  
Tim Byrnes ◽  
Valentin V. Cherny
Keyword(s):  
Fixed Point ◽  
Landau Equation ◽  
Photon Absorption ◽  
Ginzburg Landau ◽  
Two Photon Absorption ◽  
Two Photon ◽  
Solitonic Solution ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Gain Loss

It was recently shown that the nonlinear Schrodinger equation with a simplified dissipative perturbation features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield’s associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields the complex Ginzburg–Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation, define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show, in a similar way to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case, the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.

Fixed-Point Attractor Analysis for a Class of Neurodynamics

1998 ◽  
Vol 10 (1) ◽  
pp. 189-213 ◽  
Author(s):  
Jianfeng Feng ◽  
David Brown
Keyword(s):  
Fixed Point ◽  
Value Theory ◽  
Competitive Learning ◽  
Output Characteristic ◽  
Sigmoidal Function ◽  
Input Output ◽  
Parameter Region ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Necessary And Sufficient

Nearly all models in neural networks start from the assumption that the input-output characteristic is a sigmoidal function. On parameter space, we present a systematic and feasible method for analyzing the whole spectrum of attractors—all-saturated, all-but-one-saturated, all-but-twosaturated, and so on—of a neurodynamical system with a saturated sigmoidal function as its input-output characteristic. We present an argument that claims, under a mild condition, that only all-saturated or all but-one-saturated attractors are observable for the neurodynamics. For any given all-saturated configuration [Formula: see text] (all-but-one-saturated configuration [Formula: see text]) the article shows how to construct an exact parameter region R([Formula: see text])([Formula: see text]([Formula: see text])) such that if and only if the parameters fall within R([Formula: see text])([Formula: see text]([Formula: see text])), then [Formula: see text]([Formula: see text]) is an attractor (a fixed point) of the dynamics. The parameter region for an all-saturated fixed-point attractor is independent of the specific choice of a saturated sigmoidal function, whereas for an all-but-one-saturated fixed point, it is sensitive to the input-output characteristic. Based on a similar idea, the role of weight normalization realized by a saturated sigmoidal function in competitive learning is discussed. A necessary and sufficient condition is provided to distinguish two kinds of competitive learning: stable competitive learning with the weight vectors representing extremes of input space and being fixed-point attractors, and unstable competitive learning. We apply our results to Linsker's model and (using extreme value theory in statistics) the Hopfield model and obtain some novel results on these two models.

scholarly journals Fixed-point attractor for chirp in nonlinear waveguide arrays

2012 ◽  
Vol 85 (3) ◽  
Author(s):  
Darren D. Hudson ◽  
J. Nathan Kutz ◽  
Thomas R. Schibli ◽  
Qing Chao ◽  
Demetrios N. Christodoulides ◽  
...  
Keyword(s):  
Fixed Point ◽  
Waveguide Arrays ◽  
Nonlinear Waveguide ◽  
Fixed Point Attractor ◽  
Point Attractor

BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS

1995 ◽  
Vol 05 (01) ◽  
pp. 189-207 ◽  
Author(s):  
HELENA E. NUSSE ◽  
JAMES A. YORKE
Keyword(s):  
Fixed Point ◽  
Chaotic Attractor ◽  
Region Of Interest ◽  
Piecewise Smooth ◽  
One Dimensional ◽  
Border Collision Bifurcations ◽  
Bifurcation Phenomena ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
One Dimensional Maps

We examine bifurcation phenomena for continuous one-dimensional maps that are piecewise smooth and depend on a parameter μ. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is smooth. As the parameter μ is varied, a fixed point (or periodic point) Eμ may cross the point c, and we may assume that this crossing occurs at μ=0. The investigation of what bifurcations occur at μ=0 reduces to a study of a map fμ depending linearly on μ and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Eμ may cross the point c, and for μ<0 there can be a fixed point attractor, and for μ>0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as μ→0. More generally, for every integer m≥2, bifurcations from a fixed point attractor to a period-m attractor, a 2m-piece chaotic attractor, an m-piece chaotic attractor, or a one-piece chaotic attractor can occur for piecewise smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a, b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.

Accessible saddles on fractal basin boundaries

1992 ◽  
Vol 12 (3) ◽  
pp. 377-400 ◽  
Author(s):  
Kathleen T. Alligood ◽  
James A. Yorke
Keyword(s):  
Fixed Point ◽  
Basin Of Attraction ◽  
Open Disk ◽  
Fractal Basin Boundaries ◽  
Basin Boundary ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Simply Connected ◽  
Basin Boundaries ◽  
The Basin Of Attraction

AbstractFor a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.

scholarly journals Equivalent Electronic Circuit of a System of Oscillators Connected With Periodically Variable Stiffness

2022 ◽  
Author(s):  
Soumyajit Seth ◽  
Grzegorz Kudra ◽  
Krzysztof Witkowski ◽  
Jan Awrejcewicz
Keyword(s):  
Mechanical System ◽  
Electronic Circuit ◽  
Mechanical Design ◽  
Variable Stiffness ◽  
State Variables ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Magnetic Resistance ◽  
Nonlinear Behaviors ◽  
Resistance Forces

In this paper, we have shown the electronic circuit equivalence of a mechanical system consists of two oscillators coupled with each other. The mechanical design has the effects of the magnetic, resistance forces and the spring constant of the system is periodically varying. We have shown that the system&rsquo;s state variables, such as the displacements and the velocities, under the effects of different forces, lead to some nonlinear behaviors, like a transition from the fixed point attractor to the chaotic attractor through the periodic and quasi-periodic attractors. We have constructed the equivalent electronic circuit of this mechanical system and have verified the numerically obtained behaviors using the electronic circuit.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

scholarly journals Size segregation of intruders in perpetual granular avalanches

2017 ◽  
Vol 825 ◽  
pp. 502-514 ◽  
Author(s):  
Benjy Marks ◽  
Jon Alm Eriksen ◽  
Guillaume Dumazer ◽  
Bjørnar Sandnes ◽  
Knut Jørgen Måløy
Keyword(s):  
Fixed Point ◽  
Continuum Theory ◽  
Particle Sizes ◽  
Size Segregation ◽  
Uniform Size ◽  
Single Location ◽  
Two Dimensional Flow ◽  
Fixed Point Attractor ◽  
Experimental Findings ◽  
The Continuum

Granular flows such as landslides, debris flows and avalanches are systems of particles with a large range of particle sizes that typically segregate while flowing. The physical mechanisms responsible for this process, however, are still poorly understood, and there is no predictive framework for ascertaining the segregation behaviour of a given system of particles. Here, we provide experimental evidence of individual large intruder particles being attracted to a fixed point in a dry two-dimensional flow of particles of otherwise uniform size. A continuum theory is proposed which captures this effect using only a single fitting parameter that describes the rate of segregation, given knowledge of the bulk flow field. Predictions of the continuum theory are compared with the experimental findings, both for the typical location and velocity field of a range of intruder sizes. For large intruder particle sizes, the continuum model successfully predicts that a fixed point attractor will form, where intruders are drawn to a single location.

scholarly journals Permitted and Forbidden Sets in Symmetric Threshold-Linear Networks

2003 ◽  
Vol 15 (3) ◽  
pp. 621-638 ◽  
Author(s):  
Richard H. R. Hahnloser ◽  
H. Sebastian Seung ◽  
Jean-Jacques Slotine
Keyword(s):  
Fixed Points ◽  
Positive Semidefinite ◽  
Volterra Equations ◽  
Stable Steady State ◽  
Long Term Memory ◽  
Linear Networks ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
High Level ◽  
Biological Computation

The richness and complexity of recurrent cortical circuits is an inexhaustible source of inspiration for thinking about high-level biological computation. In past theoretical studies, constraints on the synaptic connection patterns of threshold-linear networks were found that guaranteed bounded network dynamics, convergence to attractive fixed points, and multistability, all fundamental aspects of cortical information processing. However, these conditions were only sufficient, and it remained unclear which were the minimal (necessary) conditions for convergence and multistability. We show that symmetric threshold-linear networks converge to a set of attractive fixed points if and only if the network matrix is copositive. Furthermore, the set of attractive fixed points is nonconnected (the network is multiattractive) if and only if the network matrix is not positive semidefinite. There are permitted sets of neurons that can be coactive at a stable steady state and forbidden sets that cannot. Permitted sets are clustered in the sense that subsets of permitted sets are permitted and supersets of forbidden sets are forbidden. By viewing permitted sets as memories stored in the synaptic connections, we provide a formulation of long-term memory that is more general than the traditional perspective of fixed-point attractor networks. There is a close correspondence between threshold-linear networks and networks defined by the generalized Lotka-Volterra equations.

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