Detecting Hidden Chaotic Regions and Complex Dynamics in the Self-Exciting Homopolar Disc Dynamo

2017 ◽  
Vol 27 (02) ◽  
pp. 1730008 ◽  
Author(s):  
Zhouchao Wei ◽  
Irene Moroz ◽  
Julien Clinton Sprott ◽  
Zhen Wang ◽  
Wei Zhang
Keyword(s):  
Vector Fields ◽  
Eddy Currents ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Melnikov Method ◽  
Homoclinic Orbits ◽  
Three Dimensions ◽  
Chaotic Attractors ◽  
Polynomial Vector Fields ◽  
Poincaré Compactification

In 1979, Moffatt pointed out that the conventional treatment of the simplest self-exciting homopolar disc dynamo has inconsistencies because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo. Here we return to the simple dynamo system proposed by Moffatt, and demonstrate previously unknown hidden chaotic attractors. Then we study multistability and coexistence of three types of attractors in the autonomous dynamo system in three dimensions: equilibrium points, limit cycles and hidden chaotic attractors. In addition, the existence of two homoclinic orbits is proved rigorously by the generalized Melnikov method. Finally, by using Poincaré compactification of polynomial vector fields in three dimensions, the dynamics near infinity of singularities is obtained.

Modeling and analysis of dynamics for a 3D mixed Lorenz system with a damped term

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xianyi Li ◽  
Umirzakov Mirjalol
Keyword(s):  
Vector Fields ◽  
Lorenz System ◽  
Global Analysis ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Polynomial Vector Fields ◽  
Poincaré Compactification ◽  
Modeling And Analysis ◽  
Analysis Of Dynamics

Abstract The work in this paper consists of two parts. The one is modelling. After a method of classification for three dimensional (3D) autonomous chaotic systems and a concept of mixed Lorenz system are introduced, a mixed Lorenz system with a damped term is presented. The other is the analysis for dynamical properties of this model. First, its local stability and bifurcation in its parameter space are in detail considered. Then, the existence of its homoclinic and heteroclinic orbits, and the existence of singularly degenerate heteroclinic cycles, are studied by rigorous theoretical analysis. Finally, by using the Poincaré compactification for polynomial vector fields in R 3 ${\mathbb{R}}^{3}$ , a global analysis of this system near and at infinity is presented, including the complete description of its dynamics on the sphere near and at infinity. Simulations corroborate corresponding theoretical results. In particular, a possibly new mechanism for the creation of chaotic attractors is proposed. Some different structure types of chaotic attractors are correspondingly and numerically found.

scholarly journals A Novel 2D – Grid of Scroll Chaotic Attractor Generated by CNN

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 99 ◽  
Author(s):  
Ahmed M. Ali ◽  
Saif M. Ramadhan ◽  
Fadhil R. Tahir
Keyword(s):  
Theoretical Analysis ◽  
Chaotic Attractor ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Electronic Circuits ◽  
Nonlinear Networks ◽  
Design And Implementation ◽  
Simulation Results ◽  
New System

The complex grid of scroll chaotic attractors that are generated through nonlinear electronic circuits have been raised considerably over the last decades. In this paper, it is shown that a subclass of Cellular Nonlinear Networks (CNNs) allows us to generate complex dynamics and chaos in symmetry pattern. A novel grid of scroll chaotic attractor, based on a new system, shows symmetry scrolls about the origin. Also, the equilibrium points are located in a manner such that the symmetry about the line x=y has been achieved. The complex dynamics of system can be generated using CNNs, which in turn are derived from a CNN array (1×3) cells. The paper concerns on the design and implementation of 2×2 and 3×3 2D-grid of scroll via the CNN model. Theoretical analysis and numerical simulations of the derived model are included. The simulation results reveal that the grid of scroll attractors can be successfully reproduced using PSpice.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350074 ◽  
Author(s):  
N. HAN ◽  
Q. J. CAO ◽  
M. WIERCIGROCH
Keyword(s):  
Nonlinear System ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Viscous Damping ◽  
Approximate System ◽  
Smoothness Parameter

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

scholarly journals Periodic perturbations of quadratic planar polynomial vector fields

2002 ◽  
Vol 74 (2) ◽  
pp. 193-198 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Vector Fields ◽  
Dynamical Behavior ◽  
Saddle Points ◽  
Heteroclinic Cycle ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Poincaré Compactification ◽  
Stable And Unstable Manifolds ◽  
Perturbed System ◽  
Periodic Perturbations

In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

scholarly journals Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mingshu Chen ◽  
Zhen Wang ◽  
Xiaojuan Zhang ◽  
Huaigu Tian
Keyword(s):  
Chaotic System ◽  
Vector Fields ◽  
Dynamical Analysis ◽  
Stable Node ◽  
Polynomial Vector Fields ◽  
Coexisting Attractors ◽  
Poincaré Compactification ◽  
Unstable Node ◽  
New Chaotic System ◽  
Multiple Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at infinity for this new chaotic system are studied using Poincaré compactification of polynomial vector fields in R 3 . Meanwhile, the dynamics near the infinity of the singularities are obtained by reducing the system’s dimensions on a Poincaré ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.

scholarly journals Cofactors and equilibria for polynomial vector fields

2014 ◽  
Vol 144 (4) ◽  
pp. 753-760
Author(s):  
Antoni Ferragut ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Equilibrium Points ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Existence Of Equilibrium ◽  
Invariant Algebraic Curves ◽  
Exponential Factors

We present a relationship between the existence of equilibrium points of differential systems and the cofactors of the invariant algebraic curves and the exponential factors of the system.

A MODIFIED GENERALIZED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2009 ◽  
Vol 19 (06) ◽  
pp. 1931-1949 ◽  
Author(s):  
QIGUI YANG ◽  
KANGMING ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Special Form ◽  
Topological Structure ◽  
Complex Dynamics ◽  
Type System ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Compound Structure ◽  
Two Parameters ◽  
First Time ◽  
New System

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

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