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Structurally unstable quadratic vector fields of codimension two: families possessing one finite saddle-node and a separatrix connection

, 2023
Artés JC.
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ANALYTIC CENTER OF NILPOTENT CRITICAL POINTS

International Journal of Bifurcation and Chaos, 2012
For third-order nilpotent critical points of a planar dynamical system, the analytic center problem is completely solved in this article by using the integrating factor method. The associated quasi-Lyapunov constants are defined and their computation method is given.
Liu, Tao, Wu, Lianggang, Li, Feng
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Local analytic integrability for nilpotent centers

Ergodic Theory and Dynamical Systems, 2003
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Chavarriga, Javier, Giacomin, Hector, Giné, Jaume, Llibre, Jaume   +3 more
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On the Global Nilpotent Centers of Cubic Polynomial Hamiltonian Systems

Differential Equations and Dynamical Systems, 2022
A global center for a vector field in the plane is a singular point p having R2 filled of periodic orbits with the exception of the singular point p. Polynomial differential systems of degree 2 have no global centers. In this paper we classify the global nilpotent centers of planar cubic polynomial Hamiltonian systems symmetric with respect to the y ...
Luis Barreira, Jaume Llibre, Clàudia Valls   +2 more
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Nilpotent centers of cubic systems

Differential Equations, 2017
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Andreev, A. F., Andreeva, I. A., Detchenya, L. V., Makovetskaya, T. V., Sadovskii, A. P.   +4 more
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Analytic nilpotent centers with analytic first integral

Nonlinear Analysis: Theory, Methods & Applications, 2010
This paper considers an analytic system having an isolated nilpotent singularity at the origin: \[ \dot{x}=y+X_2(x,y), \dot{y}=Y_2(x,y). \] A necessary condition for local analytic integrability is given. The condition can be verified using the normal form theory.
García, Isaac A., Giné, Jaume
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HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

International Journal of Bifurcation and Chaos, 2012
As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a
Su, Jing, Yang, Junmin, Han, Maoan
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On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

International Journal of Bifurcation and Chaos, 2016
In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be ...
Asheghi, R., Bakhshalizadeh, A.
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Nondegenerate and Nilpotent Centers for a Cubic System of Differential Equations

Qualitative Theory of Dynamical Systems, 2018
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Antonio Algaba, Cristóbal García, Jaume Giné   +2 more
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