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.

ŠIL'NIKOV HOMOCLINIC ORBITS IN TWO CLASSES OF 3D AUTONOMOUS NONLINEAR SYSTEMS

2011 ◽  
Vol 25 (20) ◽  
pp. 2697-2712 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU
Keyword(s):  
Computer Simulation ◽  
Nonlinear Systems ◽  
Geometric Structure ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Normal Forms ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Undetermined Coefficient

The Šil'nikov homoclinic theorem provides one analytic criterion for proving the existence of chaos in three-dimensional autonomous nonlinear systems. In applications of the theorem, however, the existence of a homoclinic orbit that usually determines the geometric structure of the chaotic attractor is not easily verified mainly because there are no available efficient methods. In this paper, based on the undetermined coefficient approach we present a framework of how to find homoclinic orbits in two classes of three-dimensional autonomous nonlinear systems of normal forms, including how to set a reasonable form of expanding series of the homoclinic orbit, how to determine all coefficients in the expansion, and how to find a numerical homoclinic orbit. Numerical examples show that the proposed framework in combination with computer simulation is very efficient.

CHEN'S ATTRACTOR EXISTS

2004 ◽  
Vol 14 (09) ◽  
pp. 3167-3177 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
YUN TANG ◽  
GUANRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Series Expansions ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši'lnikov type that connects two nontrivial singular points. The Ši'lnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen's attractor.

scholarly journals Existence of Chaos in the Chen System with Linear Time-Delay Feedback

2019 ◽  
Vol 29 (09) ◽  
pp. 1950114 ◽  
Author(s):  
Kun Tian ◽  
Hai-Peng Ren ◽  
Celso Grebogi
Keyword(s):  
Time Delay ◽  
Projection Method ◽  
Homoclinic Orbit ◽  
Linear Time ◽  
Autonomous Systems ◽  
Technical Problem ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Real Eigenvalue ◽  
Chen System

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

GLOBALLY INDETERMINATE GROWTH PATHS IN THE LUCAS MODEL OF ENDOGENOUS GROWTH

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Giovanni Bella ◽  
Paolo Mattana ◽  
Beatrice Venturi
Keyword(s):  
Endogenous Growth ◽  
Limit Cycle ◽  
Homoclinic Orbit ◽  
Three Dimensional ◽  
Stable Limit Cycle ◽  
Homoclinic Orbits ◽  
Tubular Neighborhood ◽  
Stable Limit ◽  
Stable And Unstable Manifolds ◽  
Endogenous Growth Model

This paper shows that global indeterminacy may characterize the three-dimensional vector field implied by the Lucas [(1988) Journal of Monetary Economics 22, 3–42] endogenous growth model. To achieve this result, we demonstrate the emergence of a family of homoclinic orbits connecting the steady state to itself in backward and forward time, when the stable and unstable manifolds are locally governed by real eigenvalues. In this situation, we prove that if the saddle quantity is negative, and other genericity conditions are fulfilled, a stable limit cycle bifurcates from the homoclinic orbit. Orbits originating in a tubular neighborhood of the homoclinic orbit are then bound to converge to this limit cycle, creating the conditions for the onset of global indeterminacy. Some economic intuitions related to this phenomenon are finally explored.

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.

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.

AN EXTENDED ŠIL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS

2009 ◽  
Vol 19 (05) ◽  
pp. 1679-1693 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU ◽  
GUANRONG CHEN
Keyword(s):  
Theoretical Analysis ◽  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
The Other ◽  
Complex Numbers ◽  
New Chaotic System ◽  
Focus Type

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

New Heteroclinic Orbits Coined

2016 ◽  
Vol 26 (12) ◽  
pp. 1650194 ◽  
Author(s):  
Haijun Wang ◽  
Chang Li ◽  
Xianyi Li
Keyword(s):  
Homoclinic Orbit ◽  
Algebraic Surface ◽  
Type System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Invariant Algebraic Surface

We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and two heteroclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] on its invariant algebraic surface [Formula: see text], formulated in the literature by Yang and Chen [2014], we seize two new heteroclinic orbits of this Unified Lorenz-Type System. Namely, we rigorously prove that this system has another two heteroclinic orbits to [Formula: see text] and [Formula: see text] while no homoclinic orbit when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text].

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.

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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