Multi-spin spacecraft and gyrostats as dynamical systems with multiscroll chaotic attractors

2014 ◽  
Author(s):  
Anton V. Doroshin
Keyword(s):  
Dynamical Systems ◽  
Chaotic Attractors

Generating Fully Bounded Chaotic Attractors

2013 ◽  
pp. 148-153
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

On different families of hidden chaotic attractors in fractional order dynamical systems

2018 ◽  
Author(s):  
Samuel Giovanni Hernandez-Orbe ◽  
Jess Manuel Muoz-Pacheco ◽  
German Ardl Munoz-Hernandez ◽  
Ernesto Zambrano-Serrano
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
Chaotic Attractors

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 885-895 ◽  
Author(s):  
WEN-ZHI HUANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Robustness Analysis ◽  
Original System ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Computer Assisted ◽  
New Class

Chaos, bifurcation and robustness of a new class of Hopfield neural networks are investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well. By virtue of horseshoes theory in dynamical systems, rigorous computer-assisted verifications for chaotic behavior of the system with certain parameters are given, and here also presents a discussion on the robustness of the original system. Besides this, quantitative descriptions of the complexity of these systems are also given, and a robustness analysis of the system is presented too.

BEHAVIOR PATTERNS IN MULTIPARAMETRIC DYNAMICAL SYSTEMS: LORENZ MODEL

2012 ◽  
Vol 22 (06) ◽  
pp. 1230019 ◽  
Author(s):  
ROBERTO BARRIO ◽  
FERNANDO BLESA ◽  
SERGIO SERRANO
Keyword(s):  
Dynamical Systems ◽  
Chaotic Attractors ◽  
Behavior Patterns ◽  
Theoretical Studies ◽  
Careful Selection ◽  
Lorenz Model ◽  
Straight Line ◽  
Global Study ◽  
Experimental And Theoretical Studies ◽  
Selection Of

In experimental and theoretical studies of Dynamical Systems, there are usually several parameters that govern the models. Thus, a detailed study of the global parametric phase space is not easy and normally unachievable. In this paper, we show that a careful selection of one straight line (or other 1D manifold) permits us to obtain a global idea of the evolution of the system in some circumstances. We illustrate this fact with the paradigmatic example of the Lorenz model, based on a global study, changing all three parameters. Besides, searching in other regions, for all the detected behavior patterns in one straight line, we have been able to see that missing topological structures of the chaotic attractors may be found on the chaotic-saddles.

Generating Fully Bounded Chaotic Attractors

2011 ◽  
Vol 2 (3) ◽  
pp. 36-42
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

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.

scholarly journals Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models

2020 ◽  
Vol 25 (4) ◽  
pp. 78
Author(s):  
Anouk F. G. Pelzer ◽  
Alef E. Sterk
Keyword(s):  
Dynamical Systems ◽  
Scaling Law ◽  
Chaotic Attractors ◽  
Bifurcation Parameter ◽  
Coexisting Attractors ◽  
Pitchfork Bifurcations ◽  
Primary Focus ◽  
Divisibility Properties ◽  
Lorenz 96 ◽  
Parameter Values

In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium.

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