Non-algebraic invariant curves for polynomial planar vector fields
Isaac A. García Jaume Giné
Discrete & Continuous Dynamical Systems - A 2004, 10(3): 755-768 doi: 10.3934/dcds.2004.10.755
In this paper we give, as far as we know, the first method to detect non-algebraic invariant curves for polynomial planar vector fields. This approach is based on the existence of a generalized cofactor for such curves. As an application of this algorithmic method we give some Lotka-Volterra systems with non-algebraic invariant curves.
keywords: Nonlinear differential equations Liouvillian and non-Liouvillian first integral. non-algebraic invariant curves
Cyclicity of a class of polynomial nilpotent center singularities
Isaac A. García Douglas S. Shafer
Discrete & Continuous Dynamical Systems - A 2016, 36(5): 2497-2520 doi: 10.3934/dcds.2016.36.2497
In this work we extend techniques based on computational algebra for bounding the cyclicity of nondegenerate centers to nilpotent centers in a natural class of polynomial systems, those of the form $\dot x = y + P_{2m + 1}(x,y)$, $\dot y = Q_{2m + 1}(x,y)$, where $P_{2m+1}$ and $Q_{2m+1}$ are homogeneous polynomials of degree $2m + 1$ in $x$ and $y$. We use the method to obtain an upper bound (which is sharp in this case) on the cyclicity of all centers in the cubic family and all centers in a broad subclass in the quintic family.
keywords: nilpotent center. limit cycle Cyclicity
The three-dimensional center problem for the zero-Hopf singularity
Isaac A. García Claudia Valls
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 2027-2046 doi: 10.3934/dcds.2016.36.2027
In this work we extend well-known techniques for solving the Poincaré-Lyapunov nondegenerate analytic center problem in the plane to the 3-dimensional center problem at the zero-Hopf singularity. Thus we characterize the existence of a neighborhood of the singularity completely foliated by periodic orbits (including continua of equilibria) via an analytic Poincaré return map. The vanishing of the first terms in a Taylor expansion of the associated displacement map provides us with the necessary 3-dimensional center conditions in the parameter space of the family whereas the sufficiency is obtained through symmetry-integrability methods. Finally we use the proposed method to classify the 3-dimensional centers of some quadratic polynomial differential families possessing a zero-Hopf singularity.
keywords: continua of periodic orbits Zero-Hopf singularity Poincaré map. three-dimensional vector fields
Liénard and Riccati differential equations related via Lie Algebras
Isaac A. García Jaume Giné Jaume Llibre
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 485-494 doi: 10.3934/dcdsb.2008.10.485
In this work we study the Liénard systems that can be transformed into Riccati differential equations, using changes of variables more general than the ones used by the classical Lie theory.
keywords: Lie algebra integrability. Liénard system Riccati differential equation
Linearization of smooth planar vector fields around singular points via commuting flows
Isaac A. García Jaume Giné Susanna Maza
Communications on Pure & Applied Analysis 2008, 7(6): 1415-1428 doi: 10.3934/cpaa.2008.7.1415
In this paper we propose a constructive procedure to get the change of variables that linearizes a smooth planar vector field on $\mathbb C^2$ around an elementary singular point (i.e., a singular point with associated eigenvalues $\lambda, \mu \in \mathbb C$ satisfying $\mu$≠$0$) or a nilpotent singular point from a given commutator. Moreover, it is proved that the near--identity change of variables that linearizes the vector field $\mathcal X = (x+\cdots) \partial_x + (y+\cdots) \partial_y$ is unique and linearizes simultaneously all the centralizers of $\mathcal X$. The method is used in order to obtain the linearization of some extracted examples of the existent literature.
keywords: Lie symmetries linearization problem. Planar differential equations

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