Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function

The local period function for Hamiltonian systems with applications

In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations.Comment: 23 page

On 3-parameter families of piecewise smooth vector fields in the plane

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.Comment: 33 Figures. arXiv admin note: text overlap with arXiv:1104.086

Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane

This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three parameter families of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams are exhibited. Our main result describes the unfolding of the so called Fold-Saddle singularity

Psychological treatments and psychotherapies in the neurorehabilitation of pain. Evidences and recommendations from the italian consensus conference on pain in neurorehabilitation

BACKGROUND: It is increasingly recognized that treating pain is crucial for effective care within neurological rehabilitation in the setting of the neurological rehabilitation. The Italian Consensus Conference on Pain in Neurorehabilitation was constituted with the purpose identifying best practices for us in this context. Along with drug therapies and physical interventions, psychological treatments have been proven to be some of the most valuable tools that can be used within a multidisciplinary approach for fostering a reduction in pain intensity. However, there is a need to elucidate what forms of psychotherapy could be effectively matched with the specific pathologies that are typically addressed by neurorehabilitation teams. OBJECTIVES: To extensively assess the available evidence which supports the use of psychological therapies for pain reduction in neurological diseases. METHODS: A systematic review of the studies evaluating the effect of psychotherapies on pain intensity in neurological disorders was performed through an electronic search using PUBMED, EMBASE, and the Cochrane Database of Systematic Reviews. Based on the level of evidence of the included studies, recommendations were outlined separately for the different conditions. RESULTS: The literature search yielded 2352 results and the final database included 400 articles. The overall strength of the recommendations was medium/low. The different forms of psychological interventions, including Cognitive-Behavioral Therapy, cognitive or behavioral techniques, Mindfulness, hypnosis, Acceptance and Commitment Therapy (ACT), Brief Interpersonal Therapy, virtual reality interventions, various forms of biofeedback and mirror therapy were found to be effective for pain reduction in pathologies such as musculoskeletal pain, fibromyalgia, Complex Regional Pain Syndrome, Central Post-Stroke pain, Phantom Limb Pain, pain secondary to Spinal Cord Injury, multiple sclerosis and other debilitating syndromes, diabetic neuropathy, Medically Unexplained Symptoms, migraine and headache. CONCLUSIONS: Psychological interventions and psychotherapies are safe and effective treatments that can be used within an integrated approach for patients undergoing neurological rehabilitation for pain. The different interventions can be specifically selected depending on the disease being treated. A table of evidence and recommendations from the Italian Consensus Conference on Pain in Neurorehabilitation is also provided in the final part of the pape