A Class of Three-Dimensional Quadratic Systems with Ten Limit Cycles

2016 ◽  
Vol 26 (09) ◽  
pp. 1650149 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Singular Value ◽  
Singular Points ◽  
Quadratic Systems ◽  
Symmetric Points ◽  
Fifth Order

Our work is concerned with a class of three-dimensional quadratic systems with two symmetric singular points which can yield ten small limit cycles. The method used is singular value method, we obtain the expressions of the first five focal values of the two singular points that the system has. Both singular symmetric points can be fine foci of fifth order at the same time. Moreover, we obtain that each one bifurcates five small limit cycles under a certain coefficient perturbed condition, consequently, at least ten limit cycles can appear by simultaneous Hopf bifurcation.

scholarly journals Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Peiluan Li ◽  
Yusen Wu ◽  
Xiaoquan Ding
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Singular Points ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles ◽  
Center Conditions

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

New Double Bifurcation of Nilpotent Focus

2021 ◽  
Vol 31 (04) ◽  
pp. 2150053
Author(s):  
Feng Li ◽  
Hongwei Li ◽  
Yuanyuan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Limit Cycles ◽  
Second Order ◽  
Singular Points ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Planar Systems ◽  
Nilpotent Singular Point

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

Bifurcation analysis on a class of three-dimensional quadratic systems with twelve limit cycles

2019 ◽  
Vol 363 ◽  
pp. 124577
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Quadratic Systems

scholarly journals Quadratic systems with a weak focus

1991 ◽  
Vol 44 (3) ◽  
pp. 511-526 ◽  
Author(s):  
Zhang Pingguang ◽  
Cai Suilin
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Relative Position ◽  
Quadratic System ◽  
Singular Points ◽  
Finite Plane ◽  
Quadratic Systems ◽  
Weak Focus

In this paper we study the number and the relative position of the limit cycles of a plane quadratic system with a weak focus. In particular, we prove the limit cycles of such a system can never have (2, 2)-distribution, and that there is at most one limit cycle not surrounding this weak focus under any one of the following conditions:(i) the system has at least 2 saddles in the finite plane,(ii) the system has more than 2 finite singular points and more than 1 singular point at infinity,(iii) the system has exactly 2 finite singular points, more than 1 singular point at infinity, and the weak focus is itself surrounded by at least one limit cycle.

SINGULAR POINTS OF QUADRATIC SYSTEMS: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE ℝ12

2008 ◽  
Vol 18 (02) ◽  
pp. 313-362 ◽  
Author(s):  
JOAN C. ARTES ◽  
JAUME LLIBRE ◽  
NICOLAE VULPE
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Complete Classification ◽  
Quadratic System ◽  
Singular Points ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Algebraic Functions ◽  
Algebraic Invariants

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert,1900], are still open for this class. Even when not dealing with limit cycles, still some problems have remained unsolved like a complete classification of different phase portraits without limit cycles. For some time it was thought (see [Coppel, 1966]) there could exist a set of algebraic functions whose signs would completely determine the phase portrait of a quadratic system. Nowadays we already know that this is not so, and that there are some analytical, nonalgebraic functions that also play a role when dealing with limit cycles and separatrix connections. However, it is possible to find out a set of algebraic functions whose signs determine the characteristics of all finite and infinite singular points. Most of the work up to now has dealt with this problem studying it using different normal forms adapted to some subclasses of quadratic systems. A general work useful for any quadratic system regardless of affine changes has only been done for the study of infinite singular points [Schlomiuk et al., 2005]. In this paper, we give a complete global classification of quadratic differential systems according to their topological behavior in the vicinity of the finite singular points. Our classification Main Theorem gives us a complete dictionary describing the local behavior of finite singular points using algebraic invariants and comitants which are a powerful tool for algebraic computations. Linking the result of this paper with the main one of [Schlomiuk et al., 2005] which uses the same algebraic invariants, it is possible to complete the algebraic classification of singular points (finite and infinite) for quadratic differential systems.

On the Analysis of Hopf Bifurcation Associated with a Two-Fold Zero Eigenvalue, Part 1: Autonomous System

1999 ◽  
Vol 121 (1) ◽  
pp. 101-104 ◽  
Author(s):  
M. Moh’d ◽  
K. Huseyin
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Autonomous System ◽  
Periodic Motions ◽  
Zero Eigenvalue ◽  
The Family ◽  
Dynamic Bifurcations ◽  
Formula Type

The static and dynamic bifurcations of an autonomous system associated with a twofold zero eigenvalue (of index one) are studied. Attention is focused on Hopf bifurcation solutions in the neighborhood of such a singularity. The family of limit cycles are analyzed fully by applying the formula type results of the Intrinsic Harmonic Balancing method. Thus, parameter-amplitude and amplitude-frequency relationships as well as an ordered form of approximations for the periodic motions are obtained explicitly. A verification technique, with the aid of MAPLE, is used to show the consistency of ordered approximations.

Optimization of the Navy’s three-dimensional mine impact burial prediction simulation model, Impact35, using high-order numerical methods

2021 ◽  
pp. 154851292110286
Author(s):  
Athanasios Donas ◽  
Ioannis Famelis ◽  
Peter C Chu ◽  
George Galanis
Keyword(s):  
Numerical Analysis ◽  
Numerical Methods ◽  
Prediction Model ◽  
Simulation Model ◽  
Three Dimensional ◽  
Simulation System ◽  
High Order ◽  
Alternative Methodology ◽  
Runge Kutta ◽  
Fifth Order

The aim of this paper is to present an application of high-order numerical analysis methods to a simulation system that models the movement of a cylindrical-shaped object (mine, projectile, etc.) in a marine environment and in general in fluids with important applications in Naval operations. More specifically, an alternative methodology is proposed for the dynamics of the Navy’s three-dimensional mine impact burial prediction model, Impact35/vortex, based on the Dormand–Prince Runge–Kutta fifth-order and the singly diagonally implicit Runge–Kutta fifth-order methods. The main aim is to improve the time efficiency of the system, while keeping the deviation levels of the final results, derived from the standard and the proposed methodology, low.

Recent Developments In Monochromatic Microprobe X-Ray Fluorescence (MMXRF)

1998 ◽  
Vol 4 (S2) ◽  
pp. 378-379
Author(s):  
Z. W. Chen ◽  
D. B. Wittry
Keyword(s):  
Materials Science ◽  
High Vacuum ◽  
Three Dimensional ◽  
High Sensitivity ◽  
X Rays ◽  
Fluorescence Excitation ◽  
X Ray ◽  
Quantitative Microanalysis ◽  
Recent Developments ◽  
Fifth Order

A monochromatic x-ray microprobe based on a laboratory source has recently been developed in our laboratory and used for fluorescence excitation. This technique provides high sensitivity (ppm to ppb), nondestructive, quantitative microanalysis with minimum sample preparation and does not require a high vacuum specimen chamber. It is expected that this technique (MMXRF) will have important applications in materials science, geological sciences and biological science.Three-dimensional focusing of x-rays can be obtained by using diffraction from doubly curved crystals. In our MMXRF setup, a small x-ray source was produced by the bombardment of a selected target with a focused electron beam and a toroidal mica diffractor with Johann pointfocusing geometry was used to focus characteristic x-rays from the source. In the previous work ∼ 108 photons/s were obtained in a Cu Kα probe of 75 μm × 43 μm in the specimen plane using the fifth order reflection of the (002) planes of mica.

Simultaneous Zip Bifurcation and Limit Cycles in Three Dimensional Competition Models

2006 ◽  
Vol 5 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Eduardo Sáez ◽  
Eduardo Stange ◽  
Iván Szántó
Keyword(s):  
Limit Cycles ◽  
Three Dimensional ◽  
Competition Models

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

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