Complex Normal Form for Strongly Non-Linear Vibration Systems Exemplified by Duffing–van Der Pol Equation

Leung, A.Y.T.; Zhang, Q.C.

doi:10.1006/jsvi.1998.1561

Cited by 25 publications

(27 citation statements)

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“…Figure 3 shows the solutions given by Eq. (17) are in agreement with the IHB results retaining 20 harmonics (or 10 dominant harmonics because the limit cycle solutions do not contain even-order harmonics). Figure 4 shows the amplitudes of LCOs of Eq.…”

Section: Results Validationsupporting

confidence: 80%

“…(23) AsFig. 2shows, the pitch amplitudes given by Eq (17). are in good agreement with the results given by Eq.…”

supporting

confidence: 80%

“…In this paper, we present such a bifurcation arising in a nonlinear airfoil flutter system. To reveal why this interesting phenomenon happens, the center manifold theory [16] and complex normal form method [17] were employed to obtain the bifurcation equation.…”

Section: Introductionmentioning

confidence: 99%

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Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

陈衍茂

刘济科

2008

Appl. Math. Mech.-Engl. Ed.

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The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.

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Section: Results Validationsupporting

confidence: 80%

“…(23) AsFig. 2shows, the pitch amplitudes given by Eq (17). are in good agreement with the results given by Eq.…”

supporting

confidence: 80%

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Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

陈衍茂

刘济科

2008

Appl. Math. Mech.-Engl. Ed.

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“…Cveticanin [24] extended the HB method, KB method and the elliptic perturbation method to the case of complex strongly non-linear differential equations. Leung and Zhang [25] extended the normal form method to study the asymptotic solutions of cubic non-linear terms. More recently, Belhaq and Lakrad [26] formulated the multiple scales method with elliptic functions for the case of f ðxÞ ¼ x 3 : The main goal of this paper is to present a generalization and an extension of the elliptic multiple scales method proposed in reference [26].…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions of Strongly Non-Linear Oscillators by the Multiple Scales Method

Lakrad

Belhaq

2002

Journal of Sound and Vibration

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“…Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently in analyzing the dynamical behavior of the original system near the dynamic equilibrium points [1][2][3][4][5] . However, it is not a simple task to calculate the normal form for some given ordinary differential equations.…”

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confidence: 99%

Simplest normal forms of generalized Neimark-Sacker bifurcation

Zhang

2009

Trans. Tianjin Univ.

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Abstract：The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given. Keywords：generalized Neimark-Sacker bifurcation; simplest normal form; near identity nonlinear transformationsThe normal form method has been widely used in the fields of dynamical system, ordinary differential equations and nonlinear vibration. Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently in analyzing the dynamical behavior of the original system near the dynamic equilibrium points [1][2][3][4][5] . However, it is not a simple task to calculate the normal form for some given ordinary differential equations. The normal forms of Neimark-Sacker bifurcation, also known as the discrete cases of generalized Hopf bifurcation, as .well as their .applications. have .been .extensively .studied [6][7][8] . This phenomenon appears in some known biological models like the delayed logistic map [9] . It is also of interest because the bifurcation of a limit cycle of a vector field can be transformed into an invariant torus via Poincaré map [10] . Recently this bifurcation has also been detected in some economic models [11] .This paper concentrates on generalized NeimarkSacker bifurcation. This case is subtler due to the possible presence of resonant terms, which are different from the odd order terms in the conventional normal form.In this paper, five theorems are presented to show that the generalized Neimark-Sacker bifurcation can be further simplified to the simplest normal forms in which at most two nonlinear terms remain in the amplitude equations if appropriate nonlinear transformations are chosen. There are two kinds of simplest normal forms which are simpler than the expressions in Refs.[6] and [12]. Their algebraic expression formulas are given below.

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Complex Normal Form for Strongly Non-Linear Vibration Systems Exemplified by Duffing–van Der Pol Equation

Cited by 25 publications

References 0 publications

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

Periodic Solutions of Strongly Non-Linear Oscillators by the Multiple Scales Method

Simplest normal forms of generalized Neimark-Sacker bifurcation

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