Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

Computation of Normal Forms of Differential Equations Associated with Non-Semisimple Zero Eigenvalues

1998 ◽  
Vol 08 (12) ◽  
pp. 2279-2319 ◽  
Author(s):  
Q. Bi ◽  
P. Yu
Keyword(s):  
Differential Equations ◽  
Center Manifold ◽  
Normal Forms ◽  
Specific Problem ◽  
Dimensional System ◽  
Center Manifold Theory ◽  
Zero Eigenvalue ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.

scholarly journals Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanhui Zhai ◽  
Haiyun Bai ◽  
Ying Xiong ◽  
Xiaona Ma
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Price Model ◽  
Stability Switches ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

STABILITY AND BIFURCATION FOR A CLASS OF TRI-NEURON NETWORKS WITH BIDIRECTIONALLY DELAYED CONNECTIONS AND SELF-FEEDBACK

2011 ◽  
Vol 21 (06) ◽  
pp. 1601-1616
Author(s):  
SHAOLIANG YUAN ◽  
XUEMEI LI
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Center Manifold Theory ◽  
Neuron Network ◽  
Normal Form Theory ◽  
Neuron Networks ◽  
Stability And Instability ◽  
Form Theory ◽  
Manifold Theory

In this paper, a tri-neuron network with bidirectionally delay and self-feedback is considered. We derive some sufficient conditions dependent or independent of delays for the local stability and instability of this model. Regarding the self-connection delay as the parameter, the Hopf bifurcation analysis is carried out. The direction and stability of the Hopf bifurcation are worked out by applying the normal form theory and the center manifold theory. An example is given and numerical simulations are presented to illustrate the obtained results.

scholarly journals Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Chuandong Li ◽  
Wenfeng Hu ◽  
Tingwen Huang
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Normal Form Theory ◽  
Computer Viruses ◽  
Stability And Bifurcation ◽  
Form Theory ◽  
The Stability

We extend the three-dimensional SIR model to four-dimensional case and then analyze its dynamical behavior including stability and bifurcation. It is shown that the new model makes a significant improvement to the epidemic model for computer viruses, which is more reasonable than the most existing SIR models. Furthermore, we investigate the stability of the possible equilibrium point and the existence of the Hopf bifurcation with respect to the delay. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. An analytical condition for determining the direction, stability, and other properties of bifurcating periodic solutions is obtained by using the normal form theory and center manifold argument. The obtained results may provide a theoretical foundation to understand the spread of computer viruses and then to minimize virus risks.

CONTROL OF CODIMENSION ONE STATIONARY BIFURCATIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 575-582 ◽  
Author(s):  
FERNANDO VERDUZCO
Keyword(s):  
Control Systems ◽  
Vector Fields ◽  
Center Manifold ◽  
Control Law ◽  
Center Manifold Theory ◽  
Zero Eigenvalue ◽  
Codimension One ◽  
Pitchfork Bifurcations ◽  
Manifold Theory ◽  
Dimension One

The control of the saddle-node, transcritical and pitchfork bifurcations are analyzed in nonlinear control systems with one zero eigenvalue. It is shown that, provided some conditions on the vector fields are satisfied, it is possible to design a control law such that the bifurcation properties can be modified in some desirable way. To simplify the analysis to dimension one, the center manifold theory is used.

Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity

2021 ◽  
Vol 31 (06) ◽  
pp. 2150089
Author(s):  
Biruk Tafesse Mulugeta ◽  
Liping Yu ◽  
Jingli Ren
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Additional Food ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Generalized Hopf Bifurcation ◽  
Form Theory

In this paper, a three-dimensional one-prey and two-predators model, with additional food and harvesting in the presence of toxicity is proposed. Additional food is being provided to one predator. The dynamics and bifurcations of the system are investigated using center manifold theorem, normal form theory and Sotomayor’s theorem. It is proved that the system undergoes transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, generalized Hopf bifurcation, Bogdanov–Takens bifurcation and cusp bifurcation with respect to different parameters. Bifurcation diagrams of the system with respect to toxic effect and harvesting effect are illustrated. The phase portraits and solution curves are also presented to verify the dynamic behavior. The results show that the combined effect of the factors has the power of transforming simple ecosystems into complex ecosystems.

Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field

2014 ◽  
Vol 24 (06) ◽  
pp. 1450083 ◽  
Author(s):  
Yun Tian ◽  
Pei Yu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Polynomial Systems ◽  
Computationally Efficient ◽  
Planar Vector

In this paper, we show that a simple three-dimensional quadratic vector field can have at least seven small-amplitude limit cycles, bifurcating from a Hopf critical point. This result is surprisingly higher than the Bautin's result for quadratic planar vector fields which can only have three small-amplitude limit cycles bifurcating from an elementary focus or an elementary center. The methods used in this paper include computing focus values, and solving multivariate polynomial systems using modular regular chains. In order to obtain higher-order focus values for nonplanar dynamical systems, computationally efficient approaches combined with center manifold computation must be adopted. A recently developed explicit, recursive formula and Maple program for computing the normal form and center manifold of general n-dimensional systems is applied to compute the focus values of the three-dimensional vector field.

Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model

2016 ◽  
Vol 26 (05) ◽  
pp. 1650079 ◽  
Author(s):  
Wenjing Zhang ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Limit Cycles ◽  
Dynamical Behavior ◽  
Disease Model ◽  
Equilibrium Solutions ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory ◽  
Disease Free

This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon — blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement.

Hopf Bifurcation of KdV–Burgers–Kuramoto System with Delay Feedback

2020 ◽  
Vol 30 (14) ◽  
pp. 2050213
Author(s):  
Junbiao Guan ◽  
Jie Liu ◽  
Zhaosheng Feng
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Center Manifold ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

Chaotic phenomena may exist in nonlinear evolution equations. In many cases, they are undesirable but can be controlled. In this study, we deal with the chaos control of a three-dimensional chaotic system, reduced from a KdV–Burgers–Kuramoto equation. By adding a single delay feedback term into the chaotic system, we investigate the local stability and occurrence of Hopf bifurcation near the equilibrium point. Some dynamical properties including the direction and stability of bifurcated periodic solutions are presented by using the normal form theory and the center manifold theorem. Numerical simulations are illustrated which agree well with the theoretical results.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

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