Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

Chen, ShuHui; Chen, Yangyang; Sze, K. Y.

doi:10.1007/s11431-010-0069-5

Cited by 14 publications

(22 citation statements)

References 21 publications

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“…27and 29, by which the heteroclinic bifurcation value, µ =µ c can be determined, agree with the Melnikov criterion (Nayfeh and Balachandran, 1995;Guckenheimer and Holmes, 2002). Similar equivalent formulas can also be derived in some works (Chan et al, 1997;Zhang et al, 2008;Cao et al, 2011;Chen et al, 2010).…”

Section: Generalized Hyperbolic Perturbation Methods For Heteroclinic supporting

confidence: 73%

“…(32), (33) and (35).4.2 ExamplesThree examples are presented in this section for assessment of the present method. Cao's perturbation method(Cao et al, 2011) and Chen's hyperbolic Lindstedt-Poincaré method(Chen et al, 2010) are also applied for the examples. As Chen's method is only available for Duffing-type oscillator, it is fail to be performed in Examples 2 and 3.…”

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confidence: 99%

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Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

Chen

Lewei

et al. 2015

Journal of Vibration and Control

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A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of () (, ,) x g x f x x ε µ + =  . The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions (Chen and Chen, 2009) for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method.

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Section: Generalized Hyperbolic Perturbation Methods For Heteroclinic supporting

confidence: 73%

mentioning

confidence: 99%

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

Chen

Lewei

et al. 2015

Journal of Vibration and Control

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“…System (22) has five equilibrium points, so the corresponding stable solution and its phase diagram have a variety of possible cases and complex dynamic behavior. Therefore, system (22) has a good generality, which encompasses the systems discussed in the literature [9,16,17,30].…”

Section: General Non-z 2 Symmetric Nonlinear Quintic Systemmentioning

confidence: 99%

“…For example, analysis of weak nonlinear systems employs techniques such as the elliptic perturbation method [11], the hyperbolic perturbation method [12], and the method developed by Izydorek and Janczewska [13]. Similarly, in a strongly nonlinear system, the undetermined fundamental frequency method [14], the modification of the perturbationincremental method [15], the hyperbolic LP method [16], and the generalized harmonic function perturbation method [17] are all applicable analysis techniques.…”

Section: Introductionmentioning

confidence: 99%

A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

Feng

Zhang

Wang

et al. 2016

Mathematical Problems in Engineering

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In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

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“…Chen et al [10,14,13,11,15] used a generalization of the Lindstedt-Poincaré (L-P) method to study the homoclinic solution to a family of nonlinear oscillators. They applied a nonlinear transformation of time instead of the linear one used in the original L-P method for periodic solutions (see, e.g., [33] for the original L-P method).…”

Section: Introductionmentioning

confidence: 99%

Initialization of Homoclinic Solutions near Bogdanov--Takens Points: Lindstedt--Poincaré Compared with Regular Perturbation Method

Al-Hdaibat¹,

Govaerts²,

Kuznetsov³

et al. 2016

SIAM J. Appl. Dyn. Syst.

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Abstract. To continue a branch of homoclinic solutions starting from a Bogdanov-Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt-Poincaré (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an n-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a "parasitic turn" near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the MATLAB continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Two examples with multidimensional state spaces are included.

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Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

Cited by 14 publications

References 21 publications

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

Initialization of Homoclinic Solutions near Bogdanov--Takens Points: Lindstedt--Poincaré Compared with Regular Perturbation Method

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