scholarly journals Limit cycles appearing from the perturbation of a system with a multiple line of critical points

2012 ◽  
Vol 75 (1) ◽  
pp. 278-285 ◽  
Author(s):  
Armengol Gasull ◽  
Chengzhi Li ◽  
Joan Torregrosa
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Multiple Line ◽  
Line Of Critical Points

Determination of a line of critical points via a transport property

1971 ◽  
Vol 34 (7) ◽  
pp. 352-353 ◽  
Author(s):  
A. Fote ◽  
E. Domb ◽  
S. Bakanowski ◽  
T. Mihalisin ◽  
J. Crow
Keyword(s):  
Transport Property ◽  
Critical Points ◽  
Line Of Critical Points

Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Points

2021 ◽  
Vol 31 (06) ◽  
pp. 2150090
Author(s):  
Liping Sun ◽  
Zhengdong Du
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Parabolic Type ◽  
Piecewise Smooth ◽  
Polar Coordinates ◽  
Quadratic Systems ◽  
Focus Type ◽  
Almost All ◽  
Lyapunov Constants

It is very important to determine the maximum number of limit cycles of planar piecewise smooth quadratic systems and it has become a focal subject in recent years. Almost all of the previous studies on this problem focused on systems with focus–focus type critical points. In this paper, we consider planar piecewise smooth quadratic systems with focus-parabolic type critical points. By using the generalized polar coordinates to compute the corresponding Lyapunov constants, we construct a class of planar piecewise smooth quadratic systems with focus-parabolic type critical points having six limit cycles. Our results improve the results obtained by Coll, Gasull and Prohens in 2001, who constructed a class of such systems with four limit cycles.

scholarly journals Limit Cycles and Integrability in a Class of Systems with High-Order Nilpotent Critical Points

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Feng Li ◽  
Jianlong Qiu
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Limit Cycles ◽  
High Order ◽  
Bifurcation Of Limit Cycles ◽  
Polynomial Differential Systems ◽  
Sufficient And Necessary Conditions ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

A class of polynomial differential systems with high-order nilpotent critical points are investigated in this paper. Those systems could be changed into systems with an element critical point. The center conditions and bifurcation of limit cycles could be obtained by classical methods. Finally, an example was given; with the help of computer algebra system MATHEMATICA, the first 5 Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 5 small amplitude limit cycles created from the high-order nilpotent critical point is also proved.

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

2005 ◽  
Vol 15 (04) ◽  
pp. 1253-1265 ◽  
Author(s):  
M. J. ÁLVAREZ ◽  
A. GASULL
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Stability Problem ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Planar Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Lyapunov Constants

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

On the Parametric Excitation of a Nonlinear Aeroelastic Oscillator

2002 ◽  
Author(s):  
H. Lumbantobing ◽  
T. I. Haaker
Keyword(s):  
Wind Velocity ◽  
Critical Points ◽  
Limit Cycles ◽  
Parametric Excitation ◽  
Original System ◽  
Finite Amplitude ◽  
Stability Diagrams ◽  
Averaged Equations ◽  
Critical Wind Velocity ◽  
Simple Picture

In this paper the following equation for the parametric excitation of a nonlinear aeroelastic oscillator of seesaw type is considered: θ¨+1−εa0cos(ωt)θ=εF(θ,θ˙,μ). In this equation εF represents the aeroelastic force, μ the wind velocity and ε denotes a small parameter. To study the dynamics of the oscillator we use the method of averaging. In absence of parametric excitation one typically finds that above a critical wind velocity the oscillators rest position becomes unstable and stable oscillations with finite amplitude result. Addition of the parametric excitation changes this simple picture. On changing the wind velocity local bifurcations like pitchfork, saddle-node and Hopf bifurcations lead to new nontrivial critical points and limit cycles in the averaged equations. In addition, a global saddle-connection bifurcation is found which either creates or destroys a limit cycle. Note that critical points and limit cycles in the averaged system correspond to periodic solutions and periodically modulated solutions of the original system. An analysis for the possible stability diagrams of the trivial solution and the location of bifurcations in the parameter space is presented. Finally, the numerical calculations performed match with the obtained analytical results and provide phaseportraits for some especial cases.

Existence of a line of critical points in a two-dimensional Lebwohl Lasher model

2016 ◽  
Vol 380 (5-6) ◽  
pp. 667-671 ◽  
Author(s):  
Sabana Shabnam ◽  
Sudeshna DasGupta ◽  
Soumen Kumar Roy
Keyword(s):  
Critical Points ◽  
Two Dimensional ◽  
Line Of Critical Points
Sign in / Sign up

Export Citation Format

Share Document