Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x_0 of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x_0 as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.Comment: To appear in J. Math. Anal. App

Synchronization problems for unidirectional feedback coupled nonlinear systems

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.Comment: To appear in Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Ana

Periodic solutions of periodically perturbed planar autonomous systems: A topological approach

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory

A continuation principle for a class of periodically perturbed autonomous systems

In this paper we evaluate the topological index of periodic solutions otained via the Malkin-Loud bifurcation result. Incidentally, we do not assume that the perturbation is differentiale.Comment: Accepted to Math. Nachr., Vol. 281, (2008), No.

Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbations in Banach spaces

Let A:D(A)\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-periodic solutions for the equation x'=Ax+f(t,x)+e g(t,x,e) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to e=0. We show that by means of a suitable modification of the classical Mel'nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov-Schmidt reduction to the present nonsmooth case.Comment: Submitted to Adv. Nonlinear Stu

On the behavior of periodic solutions of planar autonomous Hamiltonian systems with multivalued periodic perturbations

Aim of the paper is to provide a method to analyze the behavior of $T$-periodic solutions x_\eps, \eps>0, of a perturbed planar Hamiltonian system near a cycle $x_0$, of smallest period $T$, of the unperturbed system. The perturbation is represented by a $T$-periodic multivalued map which vanishes as \eps\to0. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous $T$-periodic term. \noindent Through the paper, assuming the existence of a $T$-periodic solution x_\eps for \eps>0 small, under the condition that $x_0$ is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point $x_0(t)$ and the trajectories x_\eps([0,T]) along a transversal direction to $x_0(t).

Non-trivial, non-negative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.Comment: 39 page

Desensitization properties of P2X3 receptors shaping pain signaling

ATP-gated P2X3 receptors are mostly expressed by nociceptive sensory neurons and participate in transduction of pain signals. P2X3 receptors show a combination of fast desensitization onset and slow recovery. Moreover, even low nanomolar agonist concentrations unable to evoke a response, can induce desensitization via a phenomenon called "high affinity desensitization." We have also observed that recovery from desensitization is agonist-specific and can range from seconds to minutes. The recovery process displays unusually high temperature dependence. Likewise, recycling of P2X3 receptors in peri-membrane regions shows unexpectedly large temperature sensitivity. By applying kinetic modeling, we have previously shown that desensitization characteristics of P2X3 receptor are best explained with a cyclic model of receptor operation involving three agonist molecules binding a single receptor and that desensitization is primarily developing from the open receptor state. Mutagenesis experiments suggested that desensitization depends on a certain conformation of the ATP binding pocket and on the structure of the transmembrane domains forming the ion pore. Further molecular determinants of desensitization have been identified by mutating the intracellular N- and C-termini of P2X3 receptor. Unlike other P2X receptors, the P2X3 subtype is facilitated by extracellular calcium that acts via specific sites in the ectodomain neighboring the ATP binding pocket. Thus, substitution of serine275 in this region (called "left flipper") converts the natural facilitation induced by extracellular calcium to receptor inhibition. Given their strategic location in nociceptive neurons and unique desensitization properties, P2X3 receptors represent an attractive target for development of new analgesic drugs via promotion of desensitization aimed at suppressing chronic pain. \ua9 2013 Giniatullin and Nistri