Melnikov method for discontinuous planar systems

This paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right-hand side. More precisely, we deal with a T$T$-periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit. By means of the so-called “non-smooth” Melnikov function there is shown the existence of a homoclinic solution for a perturbed system. The non-smooth Melnikov function is derived, and the method of how to find it in concrete problems is also introduced.     

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.     

Lagrangian flows and the one-dimensional Peano phenomenon for ODEs

We consider the one-dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative, and satisfies a condition on the amount of zeros. Although it is classically known that this problem lacks uniqueness of classical trajectories, we show that there is uniqueness for the so-called regular Lagrangian flow (by now usual notion of flow in nonsmooth situations), as well as uniqueness of distributional solutions for the associated continuity equation. The proof relies on a space reparametrization argument around the zeros of the vector field.     

Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs

We obtain global and local theorems on the existence of invariant manifolds for perturbations of nonautonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new situations that are far from the hyperbolic setting, our results include, and sometimes improve, some known stable manifold theorems.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities

In this paper an existence result of homoclinic solutions to a nonlinear second-order ODE is presented. To this end, a method based on differential inequalities is used.     

Global qualitative analysis of a quartic ecological model

In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f₀(x,y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f₀(x,y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f₀(x,y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.     

Slow-fast Bogdanov-Takens bifurcations

In this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of Bogdanov-Takens (BT) bifurcations. We introduce, among other precise definitions, the notion of slow-fast BT-bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits. Based on geometric singular perturbation theory, including blow-up, we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane.     

Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C¹-parameterized perturbations originate a family of exponential dichotomies of class C¹ in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form eλρ(t) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ(t)=t.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

On implicit systems of differential equations

This article considers implicit systems of differential equations. The implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit systems of differential equations, we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation. Our approach is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach of this article is different, we have no linearity requirements at all, instead we rely on algebraic geometry. In particular, we use birational mappings to produce an explicit system. The methods developed in this article are easily implemented using various computer algebra systems.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation

u^″=a(x)u−g(u),     

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.