Three Types of Chaotic Attractors in 3 D Maps

1993 ◽  
Vol 48 (5-6) ◽  
pp. 666-668 ◽  
Author(s):  
Michael Klein ◽  
Achim Kittel ◽  
Gerold Baier
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Three Dimensions ◽  
Chaotic Attractors ◽  
Stable Fixed Point ◽  
Fractal Structures ◽  
One Dimensional ◽  
Different Types

Abstract Coupling a one-dimensional chaotic forcing to a stable fixed point in the plane may generate different fractal attractors embedded in three dimensions. The system with real eigenvalues of the fixed point gives rise to simple chaotic attractors with three different types of fractal structures. We show that the competition of local exponents provides a generic criterion for the classification of the fractal structures in dynamical systems.

DIFFUSION, INTERMITTENCY, AND NOISE-SUSTAINED METASTABLE CHAOS IN THE LORENZ EQUATIONS: EFFECTS OF NOISE ON MULTISTABILITY

2008 ◽  
Vol 18 (06) ◽  
pp. 1749-1758 ◽  
Author(s):  
WEN-WEN TUNG ◽  
JING HU ◽  
JIANBO GAO ◽  
VINCENT A. BILLOCK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Stable Fixed Point ◽  
Lorenz Equations ◽  
Periodic Forcing ◽  
Parameter Region ◽  
Interesting Phenomenon ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

2021 ◽  
Vol 31 (03) ◽  
pp. 2150047
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Qinsheng Bi
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Complex Dynamics ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Hidden Attractor ◽  
Hidden Attractors

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

NONLINEAR DYNAMICS AND CHAOS PART I: A GEOMETRICAL APPROACH

1998 ◽  
Vol 2 (4) ◽  
pp. 505-532 ◽  
Author(s):  
Alfredo Medio
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Characteristic Exponent ◽  
Chaotic Attractors ◽  
Modern Theory ◽  
Geometrical Approach ◽  
Lyapunov Characteristic Exponent ◽  
Geometrical Properties ◽  
Nonlinear Dynamics And Chaos ◽  
Different Types

This paper is the first part of a two-part survey reviewing some basic concepts and methods of the modern theory of dynamical systems. The survey is introduced by a preliminary discussion of the relevance of nonlinear dynamics and chaos for economics. We then discuss the dynamic behavior of nonlinear systems of difference and differential equations such as those commonly employed in the analysis of economically motivated models. Part I of the survey focuses on the geometrical properties of orbits. In particular, we discuss the notion of attractor and the different types of attractors generated by discrete- and continuous-time dynamical systems, such as fixed and periodic points, limit cycles, quasiperiodic and chaotic attractors. The notions of (noninteger) fractal dimension and Lyapunov characteristic exponent also are explained, as well as the main routes to chaos.

Critical Exponents for the Model with Unique Stable Fixed Point From Three-Loop RG Expansions

1998 ◽  
Vol 12 (12n13) ◽  
pp. 1365-1377 ◽  
Author(s):  
A. I. Sokolov ◽  
K. B. Varnashev ◽  
A. I. Mudrov
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Critical Exponents ◽  
Structural Transition ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Flow Diagram ◽  
Stable Fixed Point ◽  
Loop Order ◽  
Symmetry Properties

The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.

On detection of multi-band chaotic attractors

2007 ◽  
Vol 463 (2081) ◽  
pp. 1339-1358 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Dynamical Systems ◽  
Numerical Methods ◽  
Piecewise Linear ◽  
Continuous System ◽  
Chaotic Attractors ◽  
System Function ◽  
Discontinuous System ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Multi Band

In this work, we present two numerical methods for the detection of the number of bands of a multi-band chaotic attractor. The first method is more efficient but can be applied only for dynamical systems with a continuous system function, whereas the second one is applicable for dynamical systems with a discontinuous system function as well. Using the developed methods, we investigate a one-dimensional piecewise-linear map and report for both cases of a continuous and a discontinuous system functions some new bifurcation scenarios involving multi-band chaotic attractors.

A note on the comparative anatomy of the clamps in the superfamily Diclidophoroedea (Trematoda: Monogenea)

Parasitology ◽  
1945 ◽  
Vol 36 (3-4) ◽  
pp. 191-194 ◽  
Author(s):  
Nora G. Sproston
Keyword(s):  
Comparative Anatomy ◽  
Three Dimensions ◽  
Diagnostic Characters ◽  
The Family ◽  
Different Types ◽  
Different Shapes

A re-examination of the haptoral sclerites forming the clamps, or ‘armoured suckers’, in Diclidophoroidea shows that they can be grouped into four main types, representing lines of evolution from what is here regarded as the primitive, or more generalized, form of clamp, that of the family Mazocraeidae. It has been shown that the form of the clamp skeleton must be regarded as of primary importance in the classification of the superfamily, since not only is it correlated with certain diagnostic characters of the soft parts of these worms, but it is usually fully developed some time before these parts make their appearance. Due attention has not always been paid to the detail of the clamps in systematic works, so that the position of some forms must remain uncertain until they can be redescribed. Owing to the sclerites often being twisted bars, curving through three dimensions, with the primary bars often jointed or fused, the appearance of different types of clamp is at first sight highly complex. The difficulty of interpreting them is increased by their being semi-transparent, and appearing of different shapes when viewed from various angles. These obscurities disappear when they are regarded as modifications of the fundamental plan found in Mazocraës and its allies.

scholarly journals Impulse signals classification using one dimensional convolutional neural network

2020 ◽  
Vol 71 (6) ◽  
pp. 397-405
Author(s):  
Mikhail Olkhovskiy ◽  
Eva Müllerová ◽  
Petr Martínek
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
Short Range ◽  
Signal Generation ◽  
Surface Discharge ◽  
One Dimensional ◽  
Optimal Sample ◽  
Power Spectral ◽  
Different Types

AbstractThe main purpose of this work is to propose a modern one-dimensional convolutional neural network (1 D CNN) configurations for distinguishing separate PD impulses from different types of PD sources while the parameters of these sources are changed. Three PD sources were built for signal generation: corona discharge, discharge in a void, and surface discharge. The reason for using separate PD impulses for classification is to develop a universal tool with the ability to recognize an insulation defects by analysing very few events in the insulation in a short range of time. Additionally, we found the optimal sample rates for the data acquisition for these network configurations. The necessity of signal filtering was also tested. The following configurations of a neural network were proposed: configuration for classification raw PD impulses; configuration for classification of PD impulses represented by power spectral density, for both filtered and unfiltered variants.

scholarly journals OCTONIONIC REALIZATIONS OF ONE-DIMENSIONAL EXTENDED SUPERSYMMETRIES: A CLASSIFICATION

2003 ◽  
Vol 18 (11) ◽  
pp. 787-798 ◽  
Author(s):  
H. L. CARRION ◽  
M. ROJAS ◽  
F. TOPPAN
Keyword(s):  
Dynamical Systems ◽  
Dimensional Reduction ◽  
Spinning Particles ◽  
One Dimensional ◽  
The One ◽  
M Theory

The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already classified associative representations for 1D extended supersymmetries. Examples of dynamical systems invariant under octonionic realizations of the extended supersymmetries are given. We cite among the others the octonionic spinning particles, the N = 8 KdV , etc. Possible applications to supersymmetric systems arising from dimensional reduction of the octonionic superstring and M-theory are mentioned.

scholarly journals Neural-Impulsive Pinning Control for Complex Networks Based on V-Stability

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1388
Author(s):  
Daniel Ríos-Rivera ◽  
Alma Y. Alanis ◽  
Edgar N. Sanchez
Keyword(s):  
Neural Network ◽  
Pinning Control ◽  
Three Dimensions ◽  
Chaotic Attractors ◽  
Control Law ◽  
Different Types ◽  
Stability Method ◽  
Network States ◽  
High Order Neural Network ◽  
Discrete Complex

In this work, a neural impulsive pinning controller for a twenty-node dynamical discrete complex network is presented. The node dynamics of the network are all different types of discrete versions of chaotic attractors of three dimensions. Using the V-stability method, we propose a criterion for selecting nodes to design pinning control, in which only a small fraction of the nodes is locally controlled in order to stabilize the network states at zero. A discrete recurrent high order neural network (RHONN) trained with extended Kalman filter (EKF) is used to identify the dynamics of controlled nodes and synthesize the control law.

scholarly journals On the optimality of double-bracket flows

2004 ◽  
Vol 2004 (62) ◽  
pp. 3301-3319 ◽  
Author(s):  
Anthony M. Bloch ◽  
Arieh Iserles
Keyword(s):  
Fixed Point ◽  
Global Optimality ◽  
Stable Fixed Point ◽  
Different Types

We analyze the optimality of the stable fixed point of the double-bracket equations. We introduce different types of optimality and prove local and global optimality results with respect to the Schattenp-norms.

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