Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

2014 ◽  
Vol 24 (11) ◽  
pp. 1450136 ◽  
Author(s):  
Alexander P. Krishchenko ◽  
Konstantin E. Starkov
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Dynamical Analysis ◽  
Two Dimensional ◽  
Main Attention ◽  
Raychaudhuri Equations ◽  
Omega Limit Set

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

scholarly journals DETERMINISTIC CHAOS VERSUS STOCHASTIC OSCILLATION IN A PREY-PREDATOR-TOP PREDATOR MODEL

2011 ◽  
Vol 16 (3) ◽  
pp. 343-364 ◽  
Author(s):  
Ranjit Kumar Upadhyay ◽  
Malay Banerjee ◽  
Rana Parshad ◽  
Sharada Nandan Raw
Keyword(s):  
Deterministic Model ◽  
Driving Forces ◽  
Equilibrium Points ◽  
Chaotic Oscillation ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Model System ◽  
Dynamical Analysis ◽  
Interior Equilibrium ◽  
Predator Model

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

scholarly journals Author’s reply to: Comments on “Asymptotically stable equilibrium points in new chaotic systems”

Nova Scientia ◽  
2017 ◽  
Vol 9 (19) ◽  
pp. 906-909
Author(s):  
K. Casas-García ◽  
L. A. Quezada-Téllez ◽  
S. Carrillo-Moreno ◽  
J. J. Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Systems ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Linear Quadratic ◽  
The Monte Carlo Method

Since theorem 1 of (Elhadj and Sprott, 2012) is incorrect, some of the systems found in the article (Casas-García et al. 2016) may have homoclinic or heteroclinic orbits and may seem chaos in the Shilnikov sense. However, the fundamental contribution of our paper was to find ten simple, three-dimensional dynamic systems with non-linear quadratic terms that have an asymptotically stable equilibrium point and are chaotic, which was achieved. These were obtained using the Monte Carlo method applied specifically for the search of these systems.

scholarly journals KINEMATICS OF FLOWS ON CURVED, DEFORMABLE MEDIA

2009 ◽  
Vol 06 (04) ◽  
pp. 645-666 ◽  
Author(s):  
ANIRVAN DASGUPTA ◽  
HEMWATI NANDAN ◽  
SAYAN KAR
Keyword(s):  
Hyperbolic Space ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Flat Space ◽  
Singular Solutions ◽  
Geodesic Flows ◽  
Two Dimensional ◽  
Singular Behavior ◽  
Deformable Media ◽  
Raychaudhuri Equations

Kinematics of geodesic flows on specific, two-dimensional, curved surfaces (the sphere, hyperbolic space and the torus) are investigated by explicitly solving the evolution (Raychaudhuri) equations for the expansion, shear and rotation, for a variety of initial conditions. For flows on the sphere and on hyperbolic space, we show the existence of singular (within a finite value of the time parameter) as well as non-singular solutions. We illustrate our results through a phase diagram which demonstrates under which initial conditions (or combinations thereof) we end up with a singularity in the congruence and when, if at all, we can obtain non-singular solutions for the kinematic variables. Our analysis portrays the differences which arise due to positive or negative curvature and also explores the role of rotation in controlling singular behavior. Subsequently, we move on to geodesic flows on two-dimensional spaces with varying curvature. As an example, we discuss flows on a torus. Characteristic oscillatory features, dependent on the ratio of the two radii of the torus, emerge in the solutions for the expansion, shear and rotation. Singular (within a finite time) and non-singular behavior of the solutions are also discussed. Finally, we conclude with a generalization to three-dimensional spaces of constant curvature, a summary of some of the generic features obtained and a comparison of our results with those for flows in flat space.

A NEW TYPE OF STRANGE ATTRACTOR RELATED TO THE CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 361-374 ◽  
Author(s):  
V. N. BELYKH ◽  
L. O. CHUA
Keyword(s):  
Strange Attractor ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Two Dimensional ◽  
Reverse Time ◽  
Mathematical Properties ◽  
New Type ◽  
Chaotic Nature ◽  
Three Dimensional Vector Field ◽  
Hyperbolic Points

We present a new type of strange attractors generated by an odd-symmetric three-dimensional vector field with a saddle-focus having two homoclinic orbits at the origin. This type of attractor is intimately related to the double-scroll Chua's attractor. We present the mathematical properties which proved rigorously the chaotic nature of this strange attractor to be different from that of a Lorenz-type attractor or a quasi-attractor. In particular, we proved that for certain nonempty intervals of parameters, our two-dimensional map has a strange attractor with no stable orbits. Unlike other known attractors, this strange attractor contains not only a Cantor set structure of hyperbolic points typical of horseshoe maps, but also there exist unstable points (i.e. stable in reverse time) belonging to the attractor as well. This implies that the points from the stable manifolds of the hyperbolic points must necessarily attract the unstable points.

scholarly journals MODEL PENULARAN PENYAKIT DEMAM BERDARAH DENGUE (DBD) DALAM SYSTEM DYNAMIK BERDIMENSI DUA

2021 ◽  
Vol 7 (1) ◽  
pp. 11-18
Author(s):  
Posma Lumbanraja ◽  
Keyword(s):  
Dynamic Behavior ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Hemorrhagic Fever ◽  
Heteroclinic Orbits ◽  
Dimensional System ◽  
Two Dimensional ◽  
Globally Asymptotically Stable ◽  
Trapping Region ◽  
Time Period

Here we examine the dynamic model of the spread of Dengue Hemorrhagic Fever (DHF) assuming a constant number of host and vector populations. In this paper, the model is reduced from a three-dimensional system to a two-dimensional system so that the dynamic behavior can be analyzed in the R2 plane. In the two-dimensional model, if the threshold parameter R > 1, the endemic state becomes globally asymptotically stable. During the analysis of its dynamic behavior, a trapping region is found which contains a heteroclinic orbit connecting the slowing point, namely the origin and the endemic point. By using heteroclinic orbits, it can be estimated the time period required from a state to reach a certain state.

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

EXACT HOMOCLINIC ORBITS AND HETEROCLINIC FAMILIES FOR A THIRD-ORDER SYSTEM IN THE CHAZY CLASS XI (N = 3)

2011 ◽  
Vol 21 (11) ◽  
pp. 3305-3322 ◽  
Author(s):  
JIBIN LI ◽  
FENGJUAN CHEN
Keyword(s):  
Dynamical System ◽  
Level Sets ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Order System ◽  
Dimensional System ◽  
Third Order ◽  
Unbounded Solutions ◽  
Three Dimensional System

For a differential equation in the Chazy class XI (N = 3), the corresponding three-dimensional system is studied by using dynamical system methods and Cosgrove's results. In all level sets, the exact explicit parametric representations of homoclinic orbits, the families of heteroclinic orbits and periodic orbits, as well as the families of unbounded solutions are obtained.

CLASSIFICATION OF CHAOS IN 3-D AUTONOMOUS QUADRATIC SYSTEMS-I: BASIC FRAMEWORK AND METHODS

2006 ◽  
Vol 16 (09) ◽  
pp. 2459-2479 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
GUANGRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Classification Problem ◽  
Heteroclinic Orbits ◽  
Orbit Type ◽  
Quadratic Systems ◽  
Hybrid Type

This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

scholarly journals A novel data hiding method by using a chaotic system without equilibrium points

2019 ◽  
Vol 33 (29) ◽  
pp. 1950357 ◽  
Author(s):  
Akif Akgul ◽  
Irene M. Moroz ◽  
Ali Durdu
Keyword(s):  
Equilibrium Point ◽  
Chaotic System ◽  
Data Hiding ◽  
Measurement Method ◽  
Equilibrium Points ◽  
Random Number Generator ◽  
Three Dimensional ◽  
Image Distortion ◽  
Heteroclinic Orbits ◽  
Secret Data

In this paper, we investigate how special is the choice of parameter values in the three-dimensional nonlinear system, proposed by Akgul and Pehlivan (2016), in producing a system, which exhibits chaos but has no real equilibrium states. Also, a data hiding method with a three-dimensional chaotic system without equilibrium point, developed by Akgul and Pehlivan, is realized. Numerous encryption studies have recently been made based on chaos. Encryption processes that are used with chaos bring about some security deficiencies in some cases. Steganography, unlike encryption studies, helps communicate the secret data by hiding it in an innocent-looking cover in order to avoid detection by third parties at first glance. In this work, a novel chaos-based data hiding method that hides an image with a different color into color images is proposed. Via the proposed method, data are hidden in bit spaces with the help of the chaotic random number generator (RNG). The generated random numbers are found with a chaotic system without equilibrium point, which is new in the literature. Shilnikov method cannot be applied to find whether the system is chaotic or not because they cannot have homoclinic or heteroclinic orbits. Thus, it can be useful in several engineering applications, especially in chaos-based cryptology and coding information. In the study, bits are hidden in pixels indicated by numbers generated by RNG. As the order of the hiding process is made randomly on a chaotic level, it has made data hiding algorithm stronger. The proposed method hides the data in cover image in such a way that it cannot be easily detected. Furthermore, the proposed method has been evaluated with steganalysis methods and image distortion measurement method PSNR. The chaos-based steganography method realized here has produced more best results in image distortion measurement method PSNR than other studies in the literature.

Existence and uniqueness of heteroclinic orbits for the equation λu‴ + u′ = f(u)

1988 ◽  
Vol 109 (1-2) ◽  
pp. 23-36 ◽  
Author(s):  
J. F. Toland
Keyword(s):  
Unstable Manifold ◽  
Stable Manifold ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Heteroclinic Orbits ◽  
Two Dimensional ◽  
Two Equilibria

SynopsisIffis a continuous even function which is decreasing on (0,∞) and such that±α are its only zeros and are simple, then in three-dimensional phase spacethe unstable manifold of the equilibrium u = −α and the stable manifold of u = α are both two dimensional. If λ<0 it is shown that there is a unique bounded orbit of the equation λu‴ + u′ = f(u), and that this is a heteroclinic orbit joining these two equilibria. Other results on the existence and uniqueness of heteroclinic orbits are also established when f is not even and when f is not monotone on (0, ∞).

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