On the periodic solutions of a class of Duffing differential equations
Jaume Llibre Luci Any Roberto
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 277-282 doi: 10.3934/dcds.2013.33.277
In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$, $\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions and $\epsilon$ is sufficiently small. Our results are proved using the averaging method of first order.
keywords: bifurcation stability. Duffing differential equation averaging method Periodic solution
On the limit cycles bifurcating from an ellipse of a quadratic center
Jaume Llibre Dana Schlomiuk
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1091-1102 doi: 10.3934/dcds.2015.35.1091
It is well known that invariant algebraic curves of polynomial differential systems play an important role in questions regarding integrability of these systems. But do they also have a role in relation to limit cycles? In this article we show that not only they do have a role in the production of limit cycles in polynomial perturbations of such systems but that algebraic invariant curves can even generate algebraic limit cycles in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.
keywords: quadratic vector fields quadratic center periodic orbit Quadratic systems bifurcation from center inverse integrating factor. limit cycle cyclicity of the period annulus
On the limit cycles of the Floquet differential equation
Jaume Llibre Ana Rodrigues
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 1129-1136 doi: 10.3934/dcdsb.2014.19.1129
We provide sufficient conditions for the existence of limit cycles for the Floquet differential equations $\dot {\bf x}(t) = A{\bf x}(t)+ε(B(t){\bf x}(t)+b(t))$, where ${\bf x}(t)$ and $b(t)$ are column vectors of length $n$, $A$ and $B(t)$ are $n\times n$ matrices, the components of $b(t)$ and $B(t)$ are $T$--periodic functions, the differential equation $\dot {\bf x}(t)= A{\bf x}(t)$ has a plane filled with $T$--periodic orbits, and $ε$ is a small parameter. The proof of this result is based on averaging theory but only uses linear algebra.
keywords: Floquet differential equation averaging theory. periodic solution limit cycle
Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory
Jaume Llibre Clàudia Valls
Discrete & Continuous Dynamical Systems - A 2011, 30(3): 779-790 doi: 10.3934/dcds.2011.30.779
We study the Hopf bifurcation from the singular point with eigenvalues $a$ε$ \ \pm\ bi$ and $c $ε located at the origen of an analytic differential system of the form $ \dot x= f( x)$, where $x \in \R^3$. Under convenient assumptions we prove that the Hopf bifurcation can produce $1$, $2$ or $3$ limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.
keywords: Hopf bifurcation averaging theory differential systems in dimension 3.
Centers for polynomial vector fields of arbitrary degree
Jaume Llibre Claudia Valls
Communications on Pure & Applied Analysis 2009, 8(2): 725-742 doi: 10.3934/cpaa.2009.8.725
We present two new families of polynomial differential systems of arbitrary degree with centers, a two--parameter family and a four--parameter family.
keywords: polynomial vector fields arbitrary degree Centers
The cyclicity of period annuli of some classes of reversible quadratic systems
G. Chen C. Li C. Liu Jaume Llibre
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 157-177 doi: 10.3934/dcds.2006.16.157
The cyclicity of period annuli of some classes of reversible and non-Hamiltonian quadratic systems under quadratic perturbations are studied. The argument principle method and the centroid curve method are combined to prove that the related Abelian integral has at most two zeros.
keywords: reversible systems Limit cycle Abelian integral.
Zero-Hopf bifurcation for a class of Lorenz-type systems
Jaume Llibre Ernesto Pérez-Chavela
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1731-1736 doi: 10.3934/dcdsb.2014.19.1731
In this paper we apply the averaging theory to a class of three-dimensional autonomous quadratic polynomial differential systems of Lorenz-type, to show the existence of limit cycles bifurcating from a degenerate zero-Hopf equilibrium.
keywords: Three-dimensional differential systems Lorenz-type system. limit cycles zero-Hopf equilibrium
Regularization of discontinuous vector fields in dimension three
Jaume Llibre Marco Antonio Teixeira
Discrete & Continuous Dynamical Systems - A 1997, 3(2): 235-241 doi: 10.3934/dcds.1997.3.235
In this paper vector fields around the origin in dimension three which are approximations of discontinuous ones are studied. In a former work of Sotomayor and Teixeira [6] it is shown, via regularization, that Filippov's conditions are the natural ones to extend the orbit solutions through the discontinuity set for vector fields in dimension two. In this paper we show that this is also the case for discontinuous vector fields in dimension three. Moreover, we analyse the qualitative dynamics of the local flow in a neighborhood of the codimension zero regular and singular points of the discontinuity surface.
keywords: Regularization and discontinuous vector fields.
Periodic solutions of El Niño model through the Vallis differential system
Rodrigo Donizete Euzébio Jaume Llibre
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3455-3469 doi: 10.3934/dcds.2014.34.3455
By rescaling the variables, the parameters and the periodic function of the Vallis differential system we provide sufficient conditions for the existence of periodic solutions and we also characterize their kind of stability. The results are obtained using averaging theory.
keywords: El Niño model. Vallis system periodic solutions
Center conditions for a class of planar rigid polynomial differential systems
Jaume Llibre Roland Rabanal
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1075-1090 doi: 10.3934/dcds.2015.35.1075
In general the center--focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial differential systems of the form \[ \dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y), \] where $f(x)$ and $g(y)$ are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.
keywords: limit cycles the center problem Isochronous centers Lyapunov quantities. focal basis

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