scholarly journals On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

2021 ◽  
Vol 31 (1) ◽  
pp. 35-49
Author(s):  
O.S. Kostromina
Keyword(s):  
Limit Cycle ◽  
Nonlinear Oscillations ◽  
Autonomous Systems ◽  
Type Equation ◽  
Unperturbed System ◽  
Two Dimensional ◽  
Periodic Perturbations ◽  
Averaged Systems ◽  
Theoretical Results ◽  
Double Limit

The problem of the effect of two-frequency quasi-periodic perturbations on systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is studied in the case when the corresponding perturbed autonomous systems have a double limit cycle. Its solution is important both for the theory of synchronization of nonlinear oscillations and for the theory of bifurcations of dynamical systems. In the case of commensurability of the natural frequency of the unperturbed system with frequencies of quasi-periodic perturbation, resonance occurs. Averaged systems are derived that make it possible to ascertain the structure of the resonance zone, that is, to describe the behavior of solutions in the neighborhood of individual resonance levels. The study of these systems allows determining possible bifurcations arising when the resonance level deviates from the level of the unperturbed system, which generates a double limit cycle in a perturbed autonomous system. The theoretical results obtained are applied in the study of a two-frequency quasi-periodic perturbed pendulum-type equation and are illustrated by numerical computations.

scholarly journals On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

2021 ◽  
Vol 23 (1) ◽  
pp. 11-27
Author(s):  
Olga S. Kostromina
Keyword(s):  
Numerical Calculation ◽  
Limit Cycle ◽  
Autonomous System ◽  
Theoretical Study ◽  
Type Equation ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Resonant Structures ◽  
Periodic Perturbations ◽  
Double Limit

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

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.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

The Effect of Froude Number on Entrainment in Two-Dimensional Line Plumes

1981 ◽  
Vol 103 (3) ◽  
pp. 471-477 ◽  
Author(s):  
W. F. Phillips
Keyword(s):  
Froude Number ◽  
Large Distance ◽  
Universal Constant ◽  
Temperature Profiles ◽  
Two Dimensional ◽  
Development Length ◽  
High Discharge ◽  
Entrainment Coefficient ◽  
Theoretical Results

Theoretical results are presented which predict the entrainment coefficient in a forced plume as a function of the local Froude number. The model does not require any external specification of the velocity and temperature profiles. The Froude number for any plume, in a motionless isothermal ambient, approaches a universal constant, at a large distance above the source. However, it is shown here that the development length for the Froude number, in plumes with high discharge Froude number, is of the order of a few hundred times the discharge width.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

Part II: Steady aerofoil-spoiler characteristics

1987 ◽  
Vol 91 (908) ◽  
pp. 359-366
Keyword(s):  
Separated Flow ◽  
Experimental Results ◽  
Surface Singularity ◽  
Separation Region ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Singularity Method ◽  
Total Head ◽  
Low Speeds ◽  
Theoretical Results

Summary A surface singularity method has been formulated to predict two-dimensional spoiler characteristics at low speeds. Vorticity singularities are placed on the aerofoil surface, on the spoiler surface, on the upper separation streamline from the spoiler tip and on the lower separation streamline from the aerofoil trailing edge. The separation region is closed downstream by two discrete vortices. The flow inside the separation region is assumed to have uniform total head. The downstream extent of the separated wake is an empirical input. The flows both external and internal to the separated regions are solved. Theoretical results have been obtained for a range of spoiler-aerofoil configurations which compare reasonably with experimental results. The model is deficient in that it predicts a higher compression ahead of the spoiler than obtained in practice. Furthermore, there is a minimum spoiler angle below which a solution is not possible; it is thought that this feature is related to the physical observation that at small spoiler angles, the separated flow from the spoiler reattaches on the aerofoil upper surface ahead of the trailing edge.

Sign in / Sign up

Export Citation Format

Share Document