Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form , , with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis.     

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles.     

Intersection with the vertical isocline in the generalized Liénard equations

We consider the generalized Liénard system

     

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems , with fm and gn real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681-1701] the author showed that if m?3 and m+1<n<2m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011-2022] proved that the Liénard systems with m=3 and n=5 have no hyperelliptic limit cycles and that there exist Liénard systems with m=4 and 5<n<8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m>4 and m+1<n<2m. In this paper we will prove that there exist Liénard systems with m=5 and m+1<n<2m which have hyperelliptic limit cycles.     

Periodic solutions for a kind of Liénard equation with a deviating argument

By means of continuation theorem of coincidence degree theory, we study a kind of Liénard equation with a deviating argument as follows

     

Almost periodic solutions of forced vectorial Liénard equations

We describe the set of bounded or almost periodic solutions of the following Liénard system: , where is almost periodic, is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R^N. Then, we state a result of existence and uniqueness of almost periodic solution.     

Compactification and desingularization of spaces of polynomial Liénard equations

The paper deals with polynomial Liénard equations of type (m,n), i.e. planar vector fields associated to a scalar second order differential equation , with f and g polynomials of respective degree m and n. It is shown that, besides compactifying the phase plane, or the Liénard plane, one can also compactify and desingularize the space of Liénard equations of type (m,n) for each (m,n) separately, by adding both singular perturbation problems and Hamiltonian perturbation problems.     

Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.     

Small amplitude limit cycles for the polynomial Liénard system

We estimate for the maximal number of limit cycles bifurcating from a focus for the Liénard equation , where f and g are polynomials of degree m and n respectively. These estimates are quadratic in m and n and improve the existing bounds. In the proof we use methods of complex algebraic geometry to bound the number of double points of a rational affine curve.     

Boundedness of solutions for a class of retarded Liénard equation

Consider the retarded Liénard equation

     

On the existence and uniqueness of limit cycles for generalized Liénard systems

In this paper, we consider a generalized Liénard system

     

Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems

In this paper, we consider a three-parameter class of Liénard type nonlinear dissipative systems of the form . Since such dissipative systems admit an eight-parameter Lie group of point transformations, it follows that there exists a (complex) point transformation mapping such a system into the free particle system . Normally, such an explicit point transformation cannot be found. Here we find such an explicit point transformation through exploiting the group properties of the determining equations that lead to it. Consequently, we obtain the explicit general solution of such dissipative systems. Moreover, we completely characterize the asymptotic and/or finite time blow-up behaviour of such systems in terms of their three parameters and initial data.     

Nomadic decompositions of bidirected complete graphs

We use to denote the bidirected complete graph on n vertices. A nomadic Hamiltonian decomposition of is a Hamiltonian decomposition, with the additional property that “nomads” walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length n-1, rather than length n. Bondy asked whether these decompositions of exist for all n. We show that admits a nomadic near-Hamiltonian decomposition when .     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Simultaneous bifurcation of limit cycles from two nests of periodic orbits

Let be an holomorphic differential equation having a center at p, and consider the following perturbation . We give an integral expression, similar to an Abelian integral, whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p. This expression is given in terms of the linearizing map of at p. The result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of , after a polynomial perturbation.     

Homoclinic orbits in generalized Liénard systems

This paper is devoted to the investigation on the existence of homoclinic orbits of the planar system of Liénard type , . Here h(y) is strictly increasing, but is not imposed h(±∞)=±∞. Sufficient conditions are given for a positive orbit of the system starting at a point on the curve h(y)=F(x) to approach the origin without intersecting the x-axis. The obtained theorems include previous results as special cases. Our results are applied to a concrete system and their sharpness are improved.     

Existence results for a class of evolution equations of mixed type

We give an existence result for the evolution equation in the space where is a Banach space and is a non-invertible operator (the equation may be partially elliptic and partially parabolic, both forward and backward) and we study the “Cauchy-Dirichlet” problem associated to this equation (indeed also for the inclusion . We also investigate continuous and compact embeddings of and regularity in time of the solution. At the end we give some examples of different .