Period two implies chaos for a class of ODEs: a dynamical system approach

The aim of this note is to set in the field of dynamical systems a recent theorem by Obersnel and Omari about the presence of periodic solutions of all periods for a class of scalar time-periodic first order differential equations without uniqueness, provided a subharmonic solution (and thus, for instance, a solution of period two) does exist. Indeed, making use of the Bebutov flow, we try to clarify in what sense the term "chaos" has to be understood and which dynamical features can be inferred for the system under analysis.Comment: 10 page

Periodic solutions for quasilinear complex-valued differential systems involving singular ϕ\phi-Laplacians

Topological degree is used to obtain sufficient conditions for the existence of periodic solutions of systems of second order complex- valued ordinary differential equations involving a singular $\phi$-Laplacian. Corresponding results for first order equations are also obtained

The variational structure and time-periodic solutions for mean-field games systems

Here, we observe that mean-field game (MFG) systems admit a two-player infinite-dimensional general-sum differential game formulation. We show that particular regimes of this game reduce to previously known variational principles. Furthermore, based on the game-perspective we derive new variational formulations for first-order MFG systems with congestion. Finally, we use these findings to prove the existence of time-periodic solutions for viscous MFG systems with a coupling that is not a non-decreasing function of density.Comment: 31 page

Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a functionu, and prove that the set of bifurcation points for the solutions of the system is notσ-compact. Then, we deal with a linear system depending on a real parameterλ>0and on a functionu, and prove that there existsλ∗such that the set of the functionsu, such that the system admits nontrivial solutions, contains an accumulation point

Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Agraïments: FEDER-UNAB10-4E-378. The second author is supported by a Ciência sem Fronteiras-CNPq grant number 201002/2012-4.Let p be a uniform isochronous cubic polynomial center. We study the maximum number of small or medium limit cycles that bifurcate from p or from the periodic solutions surrounding p respectively, when they are perturbed, either inside the class of all continuous cubic polynomial differential systems, or inside the class of all discontinuous differential systems formed by two cubic differential systems separated by the straight line y = 0. In the case of continuous perturbations using the averaging theory of order 6 we show that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 3, and this number can be reached. For a subfamily of these systems using the averaging theory of first order we prove that at most 3 medium limit cycles can bifurcate from the periodic solutions surrounding p, and this number can be reached. In the case of discontinuous perturbations using the averaging theory of order 6 we prove that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 5, and this number can be reached. For a subfamily of these systems using the averaging method of first order we show that the maximum number of medium limit cycles that can bifurcate from the periodic solutions surrounding p is 7, and this number can be reached. We also provide all the first integrals and the phase portraits in the Poincar'e disc for the uniform isochronous cubic centers

Lie symmetries and exact solutions for a fourth-order nonlinear diffusion equation

In this paper, we consider a fourth-order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one-dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one-dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves

Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system. In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a first-order linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated. In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples