Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q^H and Q^R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q^H∩Q^R. One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.     

Polynomial differential systems having a given Darbouxian first integral

The Darbouxian theory of integrability allows to determine when a polynomial differential system in has a first integral of the kind f₁^λ₁?f_p^λ_pexp(g/h) where f_i, g and h are polynomials in , and for i=1,…,p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.     

Half-linear ODE and modified Riccati equation: Comparison theorems, integral characterization of principal solution

In this paper, we study the half-linear differential equation with one-dimensional p-Laplacian

(r(t)Φ_p(x^′))^′+c(t)Φ_p(x)=0,     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.     

Existence of periodic solutions for the Lotka-Volterra type systems

In this paper we prove the existence of nonstationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the S¹-degree due to Dylawerski et al. [G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz, An S¹-equivariant degree and the Fuller index, Ann. Polon. Math. 63 (1991) 243-280].     

Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coefficient of nonlinear term

As p-Laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Lu’s continuation theorem, which is an extension of Manásevich-Mawhin, we study the existence of periodic solutions for a Rayleigh type p-Laplacian equation

(φ_p(x^′(t)))^′+f(x^′(t))+g₁(x(t-τ₁(t,|x|_∞)))+β(t)g₂(x(t-τ₂(t,|x|_∞)))=e(t).     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Travelling waves of auto-catalytic chemical reaction of general order—An elliptic approach

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at=axx−af(b), bt=Dbxx+af(b), with f a degenerate nonlinearity. In the context of an auto-catalytic chemical reaction, a is the density of a chemical species called reactant A, b that of another chemical species B called auto-catalyst, and D=DB/DA>0 is the ratio of diffusion coefficients, DB of B and DA of A, respectively. Such a system also arises from isothermal combustion. The nonlinearity is called degenerate, since f(0)=f^′(0)=0. One case of interest in this article is the propagating wave fronts in an isothermal auto-catalytic chemical reaction of order with 1<n<2, and D≠1 due to different molecular weights and/or sizes of A and B. The resulting nonlinearity is f(b)=bn. Explicit bounds v_∗ and v^∗ that depend on D are derived such that there is a unique travelling wave of every speed v?v^∗ and there does not exist any travelling wave of speed v<v_∗. New to the literature, it is shown that v_∗∝v^∗∝D when D<1. Furthermore, when D>1, it is shown rigorously that there exists a vmin such that there is a travelling wave of speed v if and only if v?vmin. Estimates on vmin improve significantly that of early works. Another case in which two different orders of isothermal auto-catalytic chemical reactions are involved is also studied with interesting new results proved.     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

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 orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

Non-isochronicity of the center in polynomial Hamiltonian systems

In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by M_2m−2, can be computed such that M_2m−2≤0. As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of M_2m−2>0 and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included.     

Multi-layer canard cycles and translated power functions

The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with n?2. The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions.