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

2015 ◽  
Vol 20 (2) ◽  
pp. 148-167 ◽  
Author(s):  
Fengjie Geng ◽  
Xianyi Li
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Basic Properties ◽  
Homoclinic And Heteroclinic Orbits ◽  
Quadratic Nonlinearities ◽  
Singular Orbits

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.

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.

Dynamics of an ecological system

2019 ◽  
Vol 10 (4) ◽  
pp. 355-376
Author(s):  
Shashi Kant
Keyword(s):  
Saddle Point ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Local Stability ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Trivial Equilibrium ◽  
Stability Results ◽  
Predator System

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

scholarly journals A Stochastic Model for Three Species

2018 ◽  
Vol 7 (4.10) ◽  
pp. 497
Author(s):  
Y. Suresh Kumar ◽  
N. Seshagiri Rao ◽  
B. V AppaRao
Keyword(s):  
Stochastic Process ◽  
Stochastic Model ◽  
Global Stability ◽  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Limited Resources

The present work is related to a three species ecosystem including a mutualism interaction between two species and a predator, where the predator is depending on both the mutual species. All three species in this model are considered in limited resources. The sustainability of the system (local stability) is discussed through the perturbed technique at the possible existing each equilibrium points. Using Lyapunov's technique the global stability of the system is also described. Further the nature of the system is observed by introducing the stochastic process to the species and the numerical simulations are studied to know the interaction among the species. 

scholarly journals A Mathematical Model to Study the Effectiveness of Some of the Strategies Adopted in Curtailing the Spread of COVID-19

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Isa Abdullahi Baba ◽  
Bashir Abdullahi Baba ◽  
Parvaneh Esmaili
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Government Policy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Compartmental Models ◽  
Effective Strategy ◽  
Local Stability Analysis ◽  
The Government

In this paper, we developed a model that suggests the use of robots in identifying COVID-19-positive patients and which studied the effectiveness of the government policy of prohibiting migration of individuals into their countries especially from those countries that were known to have COVID-19 epidemic. Two compartmental models consisting of two equations each were constructed. The models studied the use of robots for the identification of COVID-19-positive patients. The effect of migration ban strategy was also studied. Four biologically meaningful equilibrium points were found. Their local stability analysis was also carried out. Numerical simulations were carried out, and the most effective strategy to curtail the spread of the disease was shown.

CHAOTIC ATTRACTORS OF THE CONJUGATE LORENZ-TYPE SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 3929-3949 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN ◽  
KUIFEI HUANG
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Chen System ◽  
Numerical Examples ◽  
Special Cases ◽  
Dynamical Properties

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions and the compound structure of the system are demonstrated with various numerical examples.

scholarly journals A comparison of delayed SIR and SEIR epidemic models

2011 ◽  
Vol 16 (2) ◽  
pp. 181-190 ◽  
Author(s):  
Abdelilah Kaddar ◽  
Abdelhadi Abta ◽  
Hamad Talibi Alaoui
Keyword(s):  
Numerical Simulations ◽  
Incubation Period ◽  
Local Stability ◽  
Sir Model ◽  
Epidemic Models ◽  
Delay Effect ◽  
Epidemiological Research ◽  
Seir Model ◽  
Theoretical Results

In epidemiological research literatures, a latent or incubation period can be medelled by incorporating it as a delay effect (delayed SIR models), or by introducing an exposed class (SEIR models). In this paper we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of local stability. Also some numerical simulations are given to illustrate the theoretical results.

scholarly journals Homoclinic and heteroclinic motions in hybrid systems with impacts

2017 ◽  
Vol 67 (5) ◽  
Author(s):  
Mehmet Onur Fen ◽  
Fatma Tokmak Fen
Keyword(s):  
Hybrid Systems ◽  
Duffing Equation ◽  
Heteroclinic Orbits ◽  
Impulsive Systems ◽  
Homoclinic And Heteroclinic Orbits ◽  
Theoretical Results ◽  
Discrete Map

AbstractIn this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive systems. We rigorously prove the presence of such motions in the case that the systems are under the influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support the theoretical results are represented by means of a Duffing equation with impacts.

3D Grid Multi-Wing Chaotic Attractors

2018 ◽  
Vol 28 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Nan Yu ◽  
Yan-Wu Wang ◽  
Xiao-Kang Liu ◽  
Jiang-Wen Xiao
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Mirror Symmetry ◽  
Electrical Circuit ◽  
Circuit Diagram ◽  
Chaotic Attractors ◽  
Switching Functions ◽  
Rotation Transformation ◽  
Index 2 ◽  
Dynamical Properties

As reported in the existing literature, wing attractors are confined to 1D [Formula: see text]-wing attractors, 2D [Formula: see text]-grid wing attractors. In this paper, we break this limitation and generate 3D [Formula: see text]-grid multi-wing chaotic attractors (GMWCAs). The 3D GMWCAs are produced via the following three steps: (1) applying rotation transformation to a double-wing Lorenz-like system to ensure that its saddle-focus equilibria with index 2 are located on the plane [Formula: see text]; (2) extending the wing attractors of the transformed Lorenz-like system along the [Formula: see text]-axis to have mirror symmetry; (3) introducing stair switching functions to increase the number of saddle-focus equilibria with index 2 along the [Formula: see text]-axis and [Formula: see text]-axis. Furthermore, some basic dynamical properties of the 3D chaotic system, including equilibria, symmetry, dissipativity, Lyapunov exponents and bifurcation diagram, are investigated and a module-based unified circuit diagram is designed. The effectiveness of this approach is confirmed by both numerical simulations and electrical circuit experiment.

scholarly journals Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Agus Suryanto ◽  
Isnani Darti ◽  
Syaiful Anam
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Local Stability ◽  
Allee Effect ◽  
Dynamical Behavior ◽  
Continuous Model ◽  
Positivity Of Solutions ◽  
Existence Conditions ◽  
Dynamical Properties

We analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grünwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grünwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.

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