Slow divergence integral and its application to classical Liénard equations of degree 5

The slow divergence integral is a crucial tool to study the cyclicity of a slow–fast cycle for singularly perturbed planar vector fields. In this paper, we deduce a useful form for this integral in order to apply it to various problems. As an example, we use it to prove that the slow divergence integral along any non-degenerate slow–fast cycle for singular perturbations of classical Liénard equations of degree 5 has at most one zero, and the zero is simple if it exists; hence the cyclicity of any non-degenerate slow–fast cycle in this class of equations is at most 2. Up to now there were many interesting results about Liénard equations of degree 3, 4 and ≥6, but almost nothing is known about degree 5. The result in this paper can be seen as a first stage to study the uniform property for classical Liénard equations of degree 5.     

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.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

The period function of classical Liénard equations

In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree 2?−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ?, can be reduced to the study of slow-fast Liénard equations close to their limiting layer equations. We show that near the central system of degree 2?−1 the number of critical periods is at most 2?−2. We show the occurrence of slow-fast Liénard systems exhibiting 2?−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2?−2 is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even.     

Linearly implicit Liénard systems

The presence of nonlinearities in the capacitance and the inductance in van der Pol type electrical circuits defines a linearly implicit (or quasilinear) counterpart of the classical Liénard systems. When the reactances remain positive, the existence of a unique attracting periodic solution follows, with minor modifications, as in the classical setting. Novel results are obtained when the values of reactances may vanish at certain points of the state space; these points yield singularities of the model, and the existence of an attracting periodic solution can be characterized in terms of the behavior of certain smooth solutions crossing the singular manifold through so-called I-singularities.     

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.     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

Small-amplitude limit cycles of some Liénard-type systems

As we know, the Liénard system and its generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered by most people is the number of limit cycles. In this paper, we investigate two kinds of Liénard systems and obtain the maximal number (i.e. the least upper bound) of limit cycles appearing in Hopf bifurcations by applying some known bifurcation theorems with technical analysis.     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

Periodic solutions for Liénard type p-Laplacian equation with two deviating arguments

In this paper, the Liénard type p$p$-Laplacian equation with two deviating arguments

(_φp(x^′(t)))^′+f(x(t))x^′(t)+_g1(t,x(t−_τ1(t)))+_g2(t,x(t−_τ2(t)))=e(t)${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(x\left(t\right)\right)x^{\prime }\left(t\right)+g_1(t,x(t-{\tau }_1\left(t\right)))+g_2(t,x(t-{\tau }_2\left(t\right)))=e\left(t\right)$

Periodic solutions for a Liénard type p-Laplacian differential equation

By using topological degree theory and some analysis skill, we obtain sufficient conditions for the existence and uniqueness of periodic solutions for Liénard type p$p$-Laplacian differential equation.     

Nonautonomous bifurcation patterns for one-dimensional differential equations

Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained.     

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.     

Borg’s criterion for almost periodic differential equations

Borg’s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction. In this article, this method is generalized to almost periodic differential equations. Both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution. The condition uses a Riemannian metric, and an example for the explicit construction of such a metric is presented.     

Perturbations of symmetric elliptic Hamiltonians of degree four

In this paper four-parameter unfoldings Xλ of symmetric elliptic Hamiltonians of degree four are studied. We prove that in a compact region of the period annulus of X₀ the displacement function of Xλ is sign equivalent to its principal part, which is given by a family induced by a Chebychev system; and we describe the bifurcation diagram of Xλ in a full neighborhood of the origin in the parameter space, where at most two limit cycles can exist for the corresponding systems.     

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation be periodic in time, and let the equilibrium x_∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C² equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v^′=A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v^′=A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.