Local Bifurcation Analysis and Global Dynamics Estimation of a Novel 4-Dimensional Hyperchaotic System

2017 ◽  
Vol 27 (02) ◽  
pp. 1750021 ◽  
Author(s):  
Leilei Zhou ◽  
Zengqiang Chen ◽  
Jiezhi Wang ◽  
Qing Zhang
Keyword(s):  
Complex Dynamics ◽  
Optimization Method ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Hyperchaotic System ◽  
Center Manifold Theorem ◽  
Local Bifurcation Analysis ◽  
Lyapunov Function Method ◽  
The Stability

In this paper, we present a novel 4-dimensional (4D) smooth quadratic autonomous hyperchaotic system with complex dynamics. In order to investigate the dynamics evolution of the system, the Lyapunov exponent spectrum, bifurcation diagram and various phase portraits are provided. The local dynamics of this hyperchaotic system, such as the stability, pitchfork bifurcation, and Hopf bifurcation of equilibrium point, are analyzed by using the center manifold theorem and bifurcation theory. About the global dynamics, the ultimate bound sets of the system are found by combining the Lyapunov function method and appropriate optimization method. Numerical simulations are given to demonstrate the emergence of the two bifurcations and show the ultimate boundary regions.

A NOVEL HYPERCHAOTIC SYSTEM AND ITS COMPLEX DYNAMICS

2008 ◽  
Vol 18 (11) ◽  
pp. 3309-3324 ◽  
Author(s):  
JIEZHI WANG ◽  
ZENGQIANG CHEN ◽  
GUANRONG CHEN ◽  
ZHUZHI YUAN
Keyword(s):  
Center Manifold ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Pitchfork Bifurcation ◽  
Cross Product ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Center Manifold Theorem ◽  
Product Term ◽  
Cross Product Term

This paper is concerned with a novel four-dimensional continuous autonomous hyperchaotic system, which is obtained by adding a simple dynamical state-feedback controller to a Lorenz-like three-dimensional autonomous chaotic system. This new system contains three parameters and each equation of the system has one quadratic cross-product term. Some basic properties of the system are studied first. Its complex dynamic behaviors are then analyzed by means of Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits and Poincaré sections. It is shown that the system has several large hyperchaotic regions. When the system is evolving in a hyperchaotic region, the two positive LEs are both large, which can be larger than 1 if the system parameters are taken appropriately. The pitchfork bifurcation of the system is finally analyzed by using the center manifold theorem.

scholarly journals Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

LOCAL BIFURCATION ANALYSIS IN THE FURUTA PENDULUM VIA NORMAL FORMS

2000 ◽  
Vol 10 (05) ◽  
pp. 981-995 ◽  
Author(s):  
DANIEL PAGANO ◽  
LUIS PIZARRO ◽  
JAVIER ARACIL
Keyword(s):  
Center Manifold ◽  
Inverted Pendulum ◽  
Pitchfork Bifurcation ◽  
Open Loop ◽  
Normal Form Theory ◽  
Controlled System ◽  
Center Manifold Theorem ◽  
Local Bifurcation Analysis ◽  
Form Theory ◽  
Novel Design

Inverted pendulums are very suitable to illustrate many ideas in automatic control of nonlinear systems. The rotational inverted pendulum is a novel design that has some interesting dynamics features that are not present in inverted pendulums with linear motion of the pivot. In this paper the dynamics of a rotational inverted pendulum has been studied applying well-known results of bifurcation theory. Two classes of local bifurcations are analyzed by means of the center manifold theorem and the normal form theory — first, a pitchfork bifurcation that appears for the open-loop controlled system; second, a Hopf bifurcation, and its possible degeneracies, of the equilibrium point at the upright pendulum position, that is present for the controlled closed-loop system. Some numerical results are also presented in order to verify the validity of our analysis.

Global Dynamics of Memristor Oscillator

2016 ◽  
Vol 26 (12) ◽  
pp. 1650198 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Generalized Hopf Bifurcation ◽  
Limit Cycle Bifurcation ◽  
Sector Method ◽  
Double Limit ◽  
The Continuum

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

Analysis of a Mass-Spring-Relay System With Periodic Forcing

2021 ◽  
Author(s):  
János Lelkes ◽  
Tamás KALMÁR-NAGY
Keyword(s):  
Poincaré Map ◽  
Pitchfork Bifurcation ◽  
Global Dynamics ◽  
Poincare Map ◽  
Periodic Excitation ◽  
Periodic Forcing ◽  
Relay System ◽  
Saddle Type ◽  
The Stability ◽  
Mass Spring

Abstract The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. A Poincare map is constructed to simplify the mathematical analysis. The stability of the xed points of the Poincare map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle type xed point. The global dynamics of the system is investigated, showing discontinuity induced bifurcations of the xed points.

scholarly journals Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nadjette Debbouche ◽  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi ◽  
Mohammed K. A. Kaabar ◽  
...  
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Hopfield Neural Network ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Neural Network System ◽  
The Stability ◽  
Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

scholarly journals The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

2021 ◽  
Author(s):  
Zhaoxia Wang ◽  
Hebai Chen ◽  
Yilei Tang
Keyword(s):  
Limit Cycle ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Homoclinic Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Curve ◽  
Limit Cycle Bifurcation ◽  
Theoretical Results ◽  
Double Limit

Abstract In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Complex Dynamics in the Unified Lorenz-Type System

2014 ◽  
Vol 24 (04) ◽  
pp. 1450055 ◽  
Author(s):  
Qigui Yang ◽  
Yuming Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Type System ◽  
Global Dynamics ◽  
Two Systems ◽  
The Lorenz System ◽  
The Stability

This paper is devoted to the analysis of complex dynamics of the unified Lorenz-type system (ULTS) with six parameters, which contain common chaotic systems as its particular cases. First, some important local dynamics such as pitchfork bifurcation, Hopf bifurcation, and the stability of nondegenerate and double-zero equilibria are systematically investigated using the parameter-dependent center manifold theory combined with some bifurcation theories. Some adequate conditions for guaranteeing the occurrence of degenerate Hopf bifurcation (DHB) and the stability of the equilibria are given. Second, it is found that if DHB does not generate at the trivial equilibrium but generates at two symmetric nontrivial equilibria, then a small perturbation can lead that ULTS to exhibit a chaotic attractor. Interestingly, such a case can take place in the Chen and Lü systems (two common chaotic systems) but cannot take place in the Lorenz and Yang systems (the other two common chaotic systems), essentially distinguishing the Lorenz system from the Chen system. In addition, it is numerically verified that both of the latter two systems can exhibit the coexistence of both a chaotic attractor and multiple limit cycles but the former two systems seem not to have this property. If DHB takes place simultaneously at three equilibria of ULTS, then this system has an invariant algebraic surface, and rigorously prove the existence of some global dynamics such as periodic orbit, center, homoclinic/heteroclinic orbits. Third, it is shown that a singularly degenerate heteroclinic cycle can exist in the case of b = 0 (where b is a parameter of ULTS, like that in the Lorenz system), and a chaotic attractor can be generated by perturbing this cycle for small b > 0. These results altogether indicate that the ULTS can exhibit complex dynamics, and provide a more reasonable classification for chaos in the 3D autonomous chaotic ODE systems that were developed based on the Lorenz system, in contrast to the previous studies.

scholarly journals Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Huayong Zhang ◽  
Ju Kang ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Shengnan Ma ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Food Web ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Food Web Structure ◽  
Center Manifold Theorem ◽  
Dynamical Complexity ◽  
Period 2 ◽  
The Stability

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

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