On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

Cui, Jifeng; Zhang, Wenyu; Liu, Zeng; Sun, Jichao

doi:10.1007/s11075-017-0420-z

Cited by 17 publications

(14 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To be specific, when solving nonlinear oscillator problems with the HAM, it is straightforward to choose time-related trigonometric functions as base functions to express the steady-state solution [23,24] or exponential and trigonometric functions for transient state [25]. But various single-frequency or multifrequency auxiliary linear operators with or without damping terms can all meet the basic requirements for constructing a homotopy [23][24][25][26][27][28], and there are no universally applicable rules on how to choose the effective ones. Currently, a proper choice on L requires not only sufficient mathematical knowledge, but also extensive experience in applying the HAM, especially for problems with strong and hybrid nonlinearities.…”

Section:   Aq and  mentioning

confidence: 99%

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

Song

Zhang

Shao

2022

Preprint

View full text Add to dashboard Cite

It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

show abstract

Section:   Aq and  mentioning

confidence: 99%

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

Song

Zhang

Shao

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The importance of high precision computation in applied mathematics and science need not be overemphasized [5]. The higher precision quantitatively accurate computation of periodic orbits is facilitated in the framework of a novel asymptotic analysis [6][7][8][9], so as to allow significant numerical improvements in the computations of periodic orbits by the conventional asymptotic techniques such as renormalization group method(RGM) [10], multiple scale method (MSM) [1], homotopy analysis method [11,12] etc. As a prototype of strongly nonlinear oscillator, we consider here singularly perturbed Rayleigh Equation (SRLE) [1,4] equation with an external periodic excitation…”

Section: Introductionmentioning

confidence: 99%

“…Existence of slow and fast motions makes traditional analytical techniques ineffective in an efficient estimation of such relaxation oscillations [15,16]. Dynamical systems experiencing the fast-slow motions appear widely in engineering and other applied sciences [12]. The slow-fast periodic motions in such dynamical systems cannot be easily tackled, because such slow-fast periodic motions need many more harmonic terms to get appropriate approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

“…As one or more control (nonlinearity) parameter(s) ε (say) in the nonlinear system is slowly changed from smaller values to larger values progressively, the system experiences a sequence of bifurcations in which a period n orbit is changed suddenly to a period 2n orbit at the bifurcation point ε n leading finally to chaos [2,3,17] . In the absence of exact computability (except for very special cases, for instance, the Duffing oscillator) of amplitudes, phases and solutions of isolated periodic orbits (Limit Cycles), along with precise enumeration of period doubling routes of nonlinear systems, higher precision asymptotic determinations of the periodic 2n cycles are interesting, not only on theoretical ground but also have significant applications [12].…”

Section: Introductionmentioning

confidence: 99%

“…In [22], Luo presented higher precision computations of periodic orbits in the context of so called generalized averaging method. An efficient computation of periodic orbits of forced Van der Pol-Duffing equation was also presented recently [12] in the context of homotopy analysis method. In all these cited works a large number of harmonic terms, however, are necessary for the said higher precision computations of periodic orbits.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient Computation of Periodic Orbits of Forced Rayleigh Equation in the Framework of Novel Asymptotic Structures

Palit,

Datta,

Raut

2021

Preprint

View full text Add to dashboard Cite

Higher precision efficient computation of period 1 relaxation oscillations of strongly nonlinear and singularly perturbed Rayleigh equations with external periodic forcing is presented. The computations are performed in the context of conventional renormalization group method (RGM). We demonstrate that although a slight homotopically modified RGM could generate approximate periodic orbits that agree qualitatively with the exact orbits, the method, nevertheless, fails miserably to reduce the large quantitative disagreement between the theoretically computed results with that of exact numerical orbits. In the second part of the work we present a novel asymptotic analysis incorporating SL(2,R) invariant nonlinear deformation of slower time scales, tn = ε n t, n → ∞, ε < 1, for asymptotic late time t, to a nonlinear time Tn = tnσ(tn), where the deformation factor σ(tn) > 0 respects some well defined SL(2,R) constraints. Motivations and detailed applications of such nonlinear asymptotic structures are explained in performing very high accuracy (> 98%) computations of relaxation orbits. Existence of an interesting condensation and rarefaction phenomenon in connection with dynamically adjustable scales in the context of a slow-fast dynamical system is explained and verified numerically.

show abstract

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Zhang

Han

et al. 2022

Nonlinear Dyn

View full text Add to dashboard Cite

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slowvarying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously dif-

show abstract

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

Cited by 17 publications

References 13 publications

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

Efficient Computation of Periodic Orbits of Forced Rayleigh Equation in the Framework of Novel Asymptotic Structures

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Contact Info

Product

Resources

About