The simplest normal forms associated with a triple zero eigenvalue of indices one and two

Yu, Peifeng; Yuan, Yuejin

doi:10.1016/s0362-546x(01)00250-4

Cited by 21 publications

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“…If i (2) = 0 and i (3) = 0, then A ijk = 0 for i + j + k = 2. Using a result from [13], we transform system (13) into the following normal form:…”

Section: Definementioning

confidence: 99%

Triple-zero singularity of a Kaldor–Kalecki model of business cycles with delay

2013

NAMC

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In this manuscript, we study triple zero singularity of a Kaldor–Kalecki model of business cycles with delay in both the gross product and the capital stock. By using the frameworks of Campbell–Yuan [1] and Faria–Magalhães [2, 3], the normal form on the center manifold is derived for this singularity and hence the corresponding bifurcation diagrams such as Hopf, BT, zero-Hopf, and homoclinic bifurcations are obtained. An example is given to verify some theoretical results.

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“…If i (2) = 0 and i (3) = 0, then A ijk = 0 for i + j + k = 2. Using a result from [13], we transform system (13) into the following normal form:…”

Section: Definementioning

confidence: 99%

Triple-zero singularity of a Kaldor–Kalecki model of business cycles with delay

2013

NAMC

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“…To find the relationship between the coefficients of (38) and (39) we need to know the form of the near-identity transformation that relates them. This transformation is given in [38].…”

Section: Triple Zero Singularitymentioning

confidence: 99%

Zero singularities of codimension two and three in delay differential equations

Campbell

Yuan

2008

Nonlinearity

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We give conditions under which a general class of delay differential equations has a point of Bogdanov-Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to a two or three dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyze the appropriate unfoldings for this case.Mathematics Subject Classification: 92B20, 34K20, 34K15

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“…The stability of system around the equilibrium point can be studied by the Routh-Hurwitz method. This method is frequently used for nonlinear systems (Berger and Sinou 2011) and (Yu and Yuan 2001). It can be concluded on the stability of systems without calculating the eigenvalues of the linearized Jacobin matrix.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of One Degree of Freedom System Equipped with Friction Vibration Absorber

Nasr

Mrad

Nasri

2020

Lecture Notes in Mechanical Engineering

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In this paper, the free vibration response of a linear one degree of freedom system equipped with a friction vibration absorber is analyzed. The friction between the two systems is modeled using Coulomb friction law. The effect of the friction parameters on the stability of the linear one degree of freedom system is studied. First, around the equilibrium point, the local stability is analyzed using the Routh-Hurwitz criterion: the discontinuous non-smooth description of the friction force is approximated by a tangent hyperbolic continuous smooth function. Second, the global stability is analyzed using the Lyapunov direct method. The effect of negative viscous damping and the limit cycle behavior are considered

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The simplest normal forms associated with a triple zero eigenvalue of indices one and two

Cited by 21 publications

References 0 publications

Triple-zero singularity of a Kaldor–Kalecki model of business cycles with delay

Triple-zero singularity of a Kaldor–Kalecki model of business cycles with delay

Zero singularities of codimension two and three in delay differential equations

Stability Analysis of One Degree of Freedom System Equipped with Friction Vibration Absorber

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