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 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.

Global Dynamics of Memristor Oscillator

2016 ◽  
Vol 26 (12) ◽  
pp. 1650198 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Generalized Hopf Bifurcation ◽  
Limit Cycle Bifurcation ◽  
Sector Method ◽  
Double Limit ◽  
The Continuum

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

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.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

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