scholarly journals HOPF BIFURCATION AT A DEGENERATE SINGULAR POINT IN 3-DIMENSIONAL VECTOR FIELD

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Chaoxiong Du ◽  
◽  
Wentao Huang ◽  
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Singular Point ◽  
Dimensional Vector ◽  
3 Dimensional ◽  
Degenerate Singular Point

Limit Cycle Bifurcation of Infinity and Degenerate Singular Point in Three-Dimensional Vector Field

2016 ◽  
Vol 26 (09) ◽  
pp. 1650152
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Vector Field ◽  
Singular Point ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Nonlinear Phenomenon ◽  
Dimensional Vector ◽  
Degenerate Singular Point ◽  
Limit Cycle Bifurcation ◽  
Three Dimensional Vector Field

Our work focuses on investigating limit cycle bifurcation for infinity and a degenerate singular point of a fifth degree system in three-dimensional vector field. By using singular value method to compute focal values carefully, we give the expressions of the focal values (Lyapunov constants) at the origin and at infinity. Moreover, we obtain that four limit cycles at most can bifurcate from the origin and three limit cycles can bifurcate from infinity. At the same time, we show the structure of limit cycles from the origin and the infinity. It is interesting for this kind of nonlinear phenomenon that a string of large limit cycles encircle a string of small limit cycles by simultaneous Hopf bifurcation, which is hardly seen for similar published results in three-dimensional vector field, our result is new.

scholarly journals Measuring the criticality of a Hopf bifurcation

2020 ◽  
Vol 101 (4) ◽  
pp. 2541-2549
Author(s):  
Alexei Uteshev ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Quadratic Function ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Numerical Examples ◽  
Lyapunov Constant ◽  
Parameter Point

Abstract This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

scholarly journals New Formulas and Results for 3-Dimensional Vector Fields

2021 ◽  
Vol 12 (11) ◽  
pp. 1058-1096
Author(s):  
Sadanand D. Agashe
Keyword(s):  
Vector Fields ◽  
Dimensional Vector ◽  
3 Dimensional

scholarly journals 3-dimensional Hopf bifurcation via averaging theory

2007 ◽  
Vol 17 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Claudio A. Buzzi ◽  
Paulo R. da Silva ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
3 Dimensional

On the center-focus problem for systems with a degenerate singular point

2008 ◽  
Vol 44 (10) ◽  
pp. 1433-1439
Author(s):  
A. P. Sadovskii ◽  
D. N. Cherginets
Keyword(s):  
Singular Point ◽  
Degenerate Singular Point
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