A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues

2020 ◽  
Vol 30 (02) ◽  
pp. 2050026 ◽  
Author(s):  
Zahra Faghani ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Julien C. Sprott
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Special Property ◽  
Computer Search ◽  
Chaotic Flows ◽  
Stable Equilibria ◽  
Dynamical Properties ◽  
Simple Systems ◽  
First Time

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems with identical eigenvalues were found. We believe that systems with identical eigenvalues are described here for the first time. These simple systems are listed in this paper, and their dynamical properties are investigated.

GLOBALLY EXPONENTIALLY ATTRACTIVE SETS AND POSITIVE INVARIANT SETS OF THREE-DIMENSIONAL AUTONOMOUS SYSTEMS WITH ONLY CROSS-PRODUCT NONLINEARITIES

2013 ◽  
Vol 23 (01) ◽  
pp. 1350007 ◽  
Author(s):  
XINQUAN ZHAO ◽  
FENG JIANG ◽  
JUNHAO HU
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Invariant Set ◽  
Invariant Sets ◽  
Special Cases ◽  
Attractive Sets ◽  
Positive Invariant Set

In this paper, the existence of globally exponentially attractive sets and positive invariant sets of three-dimensional autonomous systems with only cross-product nonlinearities are considered. Sufficient conditions, which guarantee the existence of globally exponentially attractive set and positive invariant set of the system, are obtained. The results of this paper comprise some existing relative results as in special cases. The approach presented in this paper can be applied to study other chaotic systems.

Multiscroll Chaotic Sea Obtained from a Simple 3D System Without Equilibrium

2016 ◽  
Vol 26 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Tomasz Kapitaniak
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Equilibrium System ◽  
Frequency Spectra ◽  
Chaotic Flows ◽  
New System

Recently, many rare chaotic systems have been found including chaotic systems with no equilibria. However, it is surprising that such a system can exhibit multiscroll chaotic sea. In this paper, a novel no-equilibrium system with multiscroll hidden chaotic sea is introduced. Besides having multiscroll chaotic sea, this system has two more interesting properties. Firstly, it is conservative (which is a rare feature in three-dimensional chaotic flows) but not Hamiltonian. Secondly, it has a coexisting set of nested tori. There is a hidden torus which coexists with the chaotic sea. This new system is investigated through numerical simulations such as phase portraits, Lyapunov exponents, Poincaré map, and frequency spectra. Furthermore, the feasibility of such a system is verified through circuital implementation.

scholarly journals The Related Extension and Application of the Ši'lnikov Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Baoying Chen
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Extended Version

The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

Simple Chaotic Flows with a Curve of Equilibria

2016 ◽  
Vol 26 (12) ◽  
pp. 1630034 ◽  
Author(s):  
Kosar Barati ◽  
Sajad Jafari ◽  
Julien Clinton Sprott ◽  
Viet-Thanh Pham
Keyword(s):  
Chaotic Systems ◽  
Unusual Feature ◽  
Computer Search ◽  
Hidden Attractors ◽  
Engineering Applications ◽  
Chaotic Flows

Using a systematic computer search, four simple chaotic flows with cubic nonlinearities were found that have the unusual feature of having a curve of equilibria. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

2017 ◽  
Vol 27 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

BIFURCATION ANALYSIS OF CHEN'S EQUATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1917-1931 ◽  
Author(s):  
TETSUSHI UETA ◽  
GUANRONG CHEN
Keyword(s):  
Bifurcation Analysis ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Dynamical Properties ◽  
The Lorenz System

Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

A Useful Chaotic Family with Single Linearity and Circuit Implementation Based on FPGA

2016 ◽  
Vol 26 (01) ◽  
pp. 1750017 ◽  
Author(s):  
Zeshi Yuan ◽  
Hongtao Li ◽  
Xiaohua Zhu
Keyword(s):  
Field Programmable Gate Array ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Register Transfer Level ◽  
Chaotic Circuit ◽  
Chaotic Flows ◽  
Chaotic Model ◽  
Field Programmable ◽  
Simulation Results

Recently, a series of typical three-dimensional dissipative chaotic flows where all but one of the nonlinearities are quadratic are studied. Based on this research, a novel chaotic model with only one single linearity is proposed by introducing cubic terms and four new chaotic systems with various characteristics are found. Besides, a chaotic family with a single linearity is constructed with those four chaotic systems and 12 existing systems SL1–SL[Formula: see text] of the chaotic flows. Exploiting the new systems, basic dynamic behaviors are analyzed, including the strange attractors, equilibrium points, Lyapunov exponents as well as the property of multistability. In addition, the corresponding simulation results are illustrated to show those properties expressly. In realizing the chaotic circuit, we utilize the field programmable gate array (FPGA), which is of considerable flexibility, good programmability and stability, instead of analog devices that are easily affected by surroundings. More importantly, the circuit of the proposed chaotic family is realized on a single FPGA over register transfer level (RTL) using 32-bit fixed-point operation. Finally, an experimental FPGA-based circuit is constructed, and the output results are shown on oscilloscope, which agree well with the numerical simulations.

SIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM

2013 ◽  
Vol 23 (11) ◽  
pp. 1350188 ◽  
Author(s):  
MALIHE MOLAIE ◽  
SAJAD JAFARI ◽  
JULIEN CLINTON SPROTT ◽  
S. MOHAMMAD REZA HASHEMI GOLPAYEGANI
Keyword(s):  
Equilibrium Point ◽  
Stability Criterion ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Computer Search ◽  
Stable Equilibrium Point ◽  
Hidden Attractors ◽  
Chaotic Flows ◽  
Hurwitz Stability ◽  
Quadratic Nonlinearities

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

scholarly journals Cobalt-based magnetic Weyl semimetals with high-thermodynamic stabilities

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Wei Luo ◽  
Yuma Nakamura ◽  
Jinseon Park ◽  
Mina Yoon
Keyword(s):  
Tight Binding ◽  
Three Dimensional ◽  
Interlayer Coupling ◽  
First Principles Calculation ◽  
Optimization Approach ◽  
Binding Model ◽  
Nodal Lines ◽  
Weyl Semimetal ◽  
Topological Quantum ◽  
First Time

AbstractRecent experiments identified Co3Sn2S2 as the first magnetic Weyl semimetal (MWSM). Using first-principles calculation with a global optimization approach, we explore the structural stabilities and topological electronic properties of cobalt (Co)-based shandite and alloys, Co3MM’X2 (M/M’ = Ge, Sn, Pb, X = S, Se, Te), and identify stable structures with different Weyl phases. Using a tight-binding model, for the first time, we reveal that the physical origin of the nodal lines of a Co-based shandite structure is the interlayer coupling between Co atoms in different Kagome layers, while the number of Weyl points and their types are mainly governed by the interaction between Co and the metal atoms, Sn, Ge, and Pb. The Co3SnPbS2 alloy exhibits two distinguished topological phases, depending on the relative positions of the Sn and Pb atoms: a three-dimensional quantum anomalous Hall metal, and a MWSM phase with anomalous Hall conductivity (~1290 Ω−1 cm−1) that is larger than that of Co2Sn2S2. Our work reveals the physical mechanism of the origination of Weyl fermions in Co-based shandite structures and proposes topological quantum states with high thermal stability.

scholarly journals Shell-crossing in a ΛCDM Universe

2020 ◽  
Vol 501 (1) ◽  
pp. L71-L75
Author(s):  
Cornelius Rampf ◽  
Oliver Hahn
Keyword(s):  
Dark Matter ◽  
Perturbation Theory ◽  
Large Scale ◽  
Cold Dark Matter ◽  
Three Dimensional ◽  
Random Initial Data ◽  
Recursion Relations ◽  
Indispensable Tool ◽  
First Time ◽  
Very High

ABSTRACT Perturbation theory is an indispensable tool for studying the cosmic large-scale structure, and establishing its limits is therefore of utmost importance. One crucial limitation of perturbation theory is shell-crossing, which is the instance when cold-dark-matter trajectories intersect for the first time. We investigate Lagrangian perturbation theory (LPT) at very high orders in the vicinity of the first shell-crossing for random initial data in a realistic three-dimensional Universe. For this, we have numerically implemented the all-order recursion relations for the matter trajectories, from which the convergence of the LPT series at shell-crossing is established. Convergence studies performed at large orders reveal the nature of the convergence-limiting singularities. These singularities are not the well-known density singularities at shell-crossing but occur at later times when LPT already ceased to provide physically meaningful results.

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