SINGULAR ORBITS AND DYNAMICS AT INFINITY OF A CONJUGATE LORENZ-LIKE SYSTEM

A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.

Download Full-text

More Dynamical Properties Revealed from a 3D Lorenz-like System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414501338 ◽

2014 ◽

Vol 24 (10) ◽

pp. 1450133 ◽

Cited By ~ 9

Author(s):

Haijun Wang ◽

Xianyi Li

Keyword(s):

Numerical Simulations ◽

Local Stability ◽

Equilibrium Points ◽

Heteroclinic Orbits ◽

Mathematical Work ◽

Chaotic Attractors ◽

Heteroclinic Cycles ◽

Homoclinic And Heteroclinic Orbits ◽

Dynamical Properties ◽

Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

Download Full-text

New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4030028 ◽

2015 ◽

Vol 10 (6) ◽

Cited By ~ 1

Author(s):

Haijun Wang ◽

Xianyi Li

Keyword(s):

Nonlinear Dynamics ◽

Chaotic System ◽

Three Dimensional ◽

Heteroclinic Orbits ◽

Chaotic Attractors ◽

Heteroclinic Cycle ◽

Homoclinic And Heteroclinic Orbits ◽

Singular Orbits ◽

The Stability ◽

Singularly Degenerate Heteroclinic Cycle

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22 by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

Download Full-text

Multistability in a 3D Autonomous System with Different Types of Chaotic Attractors

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421500280 ◽

2021 ◽

Vol 31 (02) ◽

pp. 2150028

Author(s):

Ting Yang

Keyword(s):

Bifurcation Diagram ◽

Lyapunov Exponents ◽

Autonomous System ◽

Heteroclinic Orbits ◽

Chaotic Attractors ◽

Different Types ◽

Lyapunov Coefficient ◽

Weak Saddle

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil’nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors.

Download Full-text

A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

Mathematical Problems in Engineering ◽

10.1155/2016/8585290 ◽

2016 ◽

Vol 2016 ◽

pp. 1-10

Author(s):

Jingjing Feng ◽

Qichang Zhang ◽

Wei Wang ◽

Shuying Hao

Keyword(s):

Dynamic Systems ◽

Approximation Method ◽

Padé Approximation ◽

Global Bifurcation ◽

Heteroclinic Orbits ◽

Symmetric System ◽

Pade Approximation ◽

New Approach ◽

Homoclinic And Heteroclinic Orbits ◽

Single Orbit

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

Download Full-text

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

Symmetry ◽

10.3390/sym11010099 ◽

2019 ◽

Vol 11 (1) ◽

pp. 99 ◽

Cited By ~ 1

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.

Download Full-text

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500740 ◽

2013 ◽

Vol 23 (04) ◽

pp. 1350074 ◽

Cited By ~ 11

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.

Download Full-text

The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo

E3S Web of Conferences ◽

10.1051/e3sconf/20186202017 ◽

2018 ◽

Vol 62 ◽

pp. 02017 ◽

Cited By ~ 1

Author(s):

Olga Lipko ◽

Roman Parovik

Keyword(s):

Mathematical Model ◽

Lyapunov Exponents ◽

Nerve Impulse ◽

Chaotic Attractors ◽

Control Parameters ◽

Schmidt Orthogonalization ◽

Non Local ◽

Explicit Finite Difference ◽

Maximum Lyapunov Exponents ◽

Orthogonalization Procedure

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure and the spectra of the maximum Lyapunov exponents were also constructed depending on the values of the control parameters of the model of the FHN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

Download Full-text