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.

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

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
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.

scholarly journals Generation of Multi-Scroll Chaotic Attractors from a Jerk Circuit with a Special Form of a Sine Function

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 842
Author(s):  
Pengfei Ding ◽  
Xiaoyi Feng
Keyword(s):  
Chaotic System ◽  
Special Form ◽  
Electronic Circuit ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Sine Function ◽  
Sign Function ◽  
System Parameters ◽  
Left And Right

A novel chaotic system for generating multi-scroll attractors based on a Jerk circuit using a special form of a sine function (SFSF) is proposed in this paper, and the SFSF is the product of a sine function and a sign function. Although there are infinite equilibrium points in this system, the scroll number of the generated chaotic attractors is certain under appropriate system parameters. The dynamical properties of the proposed chaotic system are studied through Lyapunov exponents, phase portraits, and bifurcation diagrams. It is found that the scroll number of the chaotic system in the left and right part of the x-y plane can be determined arbitrarily by adjusting the values of the parameters in the SFSF, and the size of attractors is dominated by the frequency of the SFSF. Finally, an electronic circuit of the proposed chaotic system is implemented on Pspice, and the simulation results of electronic circuit are in agreement with the numerical ones.

Fractal Basin Boundaries and Chaotic Motions in Spring-Pendulum System

2003 ◽  
Author(s):  
Aria Alasty ◽  
Rasool Shabani
Keyword(s):  
Numerical Simulations ◽  
Multiple Scales ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Chaotic Motions ◽  
Pendulum System ◽  
Fractal Basin Boundaries ◽  
System Parameters ◽  
Basin Boundaries

This study investigates chaotic response in the spring-pendulum system. In this system beside of strange attractors, multiple regular attractors may coexist for some values of system parameters, where it is important to study the global behavior of the system using the basin boundaries of the attractors. Multiple scales method is used to distinguish the regions of stable and unstable attractors. In unstable regions, bifurcation diagram and poincare´ maps are used to study the existence of quasi-periodic and chaotic attractors. Results show that the jumping phenomena may occur when multiple regular attractors exist and for this case fractal basins of attraction are developed using numerical simulations.

Stability and Hopf bifurcation analysis of a new four-dimensional hyper-chaotic system

2020 ◽  
Vol 34 (29) ◽  
pp. 2050327
Author(s):  
Liangqiang Zhou ◽  
Ziman Zhao ◽  
Fangqi Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Local Dynamic ◽  
Chaotic Motions ◽  
System Parameters ◽  
Lyapunov Coefficient

With both analytical and numerical methods, local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional hyper-chaotic system are studied in this paper. All the equilibrium points and their stability conditions are obtained with the Routh–Hurwitz criterion. It is shown that there may exist one, two, or three equilibrium points for different system parameters. Via Hopf bifurcation theory, parameter conditions leading to Hopf bifurcation is presented. With the aid of center manifold and the first Lyapunov coefficient, it is also presented that the Hopf bifurcation is supercritical for some certain parameters. Finally, numerical simulations are given to confirm the analytical results and demonstrate the chaotic attractors of this system. It is also shown that the system may evolve chaotic motions through periodic bifurcations or intermittence chaos while the system parameters vary.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1477-1483 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Positive Definite ◽  
First Integrals ◽  
Invariant Sets ◽  
Yang Mills ◽  
First Order ◽  
Special Case ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

scholarly journals A Novel Method for Constructing Grid Multi-Wing Butterfly Chaotic Attractors via Nonlinear Coupling Control

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yun Huang
Keyword(s):  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Piecewise Linear Function ◽  
Chaotic Attractors ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
First Order ◽  
Novel Method ◽  
Chaotic Characteristic

A new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. Through the analysis of the equilibrium points, Lyapunov exponent spectrums, bifurcation diagrams, and Poincaré mapping in this system, the chaotic characteristic of the system is verified. Apart from the research above, an electronic circuit is designed to implement the system. The circuit experimental results are in accordance with the results of numerical simulation, which verify the availability and feasibility of this method.

BOUNDS FOR THE DOMAIN CONTAINING ALL COMPACT INVARIANT SETS OF THE SYSTEM MODELING DYNAMICS OF ACOUSTIC GRAVITY WAVES

2009 ◽  
Vol 19 (10) ◽  
pp. 3425-3432 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Gravity Waves ◽  
Lorenz System ◽  
System Modeling ◽  
Invariant Sets ◽  
Localization Problem ◽  
Acoustic Gravity Waves ◽  
First Order ◽  
The Lorenz System ◽  
Quadratic Surfaces ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of the system modeling the dynamics of acoustic gravity waves constructed by Stenflo (the Lorenz–Stenflo system). We discuss relations between compact localization domains and trapping domains with the help of extended invariance principle due to Rodrigues et al. Based on this analysis, we compute the trapping domain for the system modeling the dynamics of acoustic gravity waves. By using the first order extremum conditions and a comparison with localization results obtained earlier for the Lorenz system we find that all compact invariant sets of the Lorenz–Stenflo system are located in the intersection of one-parameter set of ellipsoids with a few domains bounded by some quadratic surfaces. Further, we derive polytopic bounds for the locus of compact invariant sets. Finally, we present the general formulae validating the application of rational localizing functions and use these formulae for the Lorenz–Stenflo system and for the Lorenz system.

CAN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS QUADRATIC CHAOTIC SYSTEM GENERATE A SINGLE FOUR-SCROLL ATTRACTOR?

2004 ◽  
Vol 14 (04) ◽  
pp. 1395-1403 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Phase Space ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
System Parameters ◽  
New Chaotic System

Recently, we have investigated a new chaotic system of three-dimensional autonomous quadratic ordinary differential equations, and found that the system visually displays a four-scroll chaotic attractor confirmed by both numerical simulations and circuit implementation. In this paper, we further study the following question: Is it really true that this system can generate a four-scroll chaotic attractor, or is it only a numerical artifact? By a more careful theoretical analysis along with some further numerical simulations, we conclude that the four-scroll chaotic attractor of this system, which we observed on both computer and oscilloscope, cannot actually exist in theory. The fact is that this system has two co-existing two-scroll chaotic attractors that are arbitrarily close in the phase space for some system parameters, therefore extremely tiny numerical round-off errors or signal fluctuations will nudge the system orbit to switch from one attractor to another, thereby forming the seemingly single four-scroll chaotic attractor on screen display.

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 The Dynamics and Synchronization of a Fractional-Order System with Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoya Yang ◽  
Xiaojun Liu ◽  
Honggang Dang ◽  
Wansheng He
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Complex Variables ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
System Parameters ◽  
The Stability

A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

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