Scaling relations and critical exponents for two dimensional two parameter maps

In this paper we calculate the critical scaling exponents describing the variation of both the positive Lyapunov exponent, λ+, and the mean residence time, τ , near the second order phase transition critical point for dynamical systems experiencing crisis-induced intermittency. We study in detail 2-dimensional 2-parameter nonlinear quadratic mappings of the form: Xn+1 = f1(Xn, Yn; A,B) and Yn+1 = f2(Xn, Yn; A,B) which contain in their parameter space (A,B) a region where there is crisis induced intermittent behaviour. Specifically, the Henon, the Mira 1, and Mira 2 maps are investigated in the vicinity of the crises.We show that near a critical point the following scaling relations hold: τ ∼ |A−Ac|−γ, (λ+ −λ+c ) ∼| A−Ac |βA and (λ+ −λ+c ) ∼| B −Bc |βB. The subscript c on a quantity denotes its value at the critical point. All these maps exhibit a chaos to chaos second order phase transition across the critical point. We find these scaling exponents satisfy the scaling relation γ = βB( 1 βA − 1), which is analogous to Widom’s scaling law. We find strong agreement between the scaling relationship and numerical results

Predictors of outcome in sciatica patients following an epidural steroid injection:the POiSE prospective observational cohort study protocol

INTRODUCTION: Sciatica can be very painful and, in most cases, is due to pressure on a spinal nerve root from a disc herniation with associated inflammation. For some patients, the pain persists, and one management option is a spinal epidural steroid injection (ESI). The aim of an ESI is to relieve leg pain, improve function and reduce the need for surgery. ESIs work well in some patients but not in others, but we cannot identify these patient subgroups currently. This study aims to identify factors, including patient characteristics, clinical examination and imaging findings, that help in predicting who does well and who does not after an ESI. The overall objective is to develop a prognostic model to support individualised patient and clinical decision-making regarding ESI. METHODS: POiSE is a prospective cohort study of 439 patients with sciatica referred by their clinician for an ESI. Participants will receive weekly text messages until 12 weeks following their ESIand then again at 24 weeks following their ESI to collect data on leg pain severity. Questionnaires will be sent to participants at baseline, 6, 12 and 24 weeks after their ESI to collect data on pain, disability, recovery and additional interventions. The prognosis for the cohort will be described. The primary outcome measure for the prognostic model is leg pain at 6 weeks. Prognostic models will also be developed for secondary outcomes of disability and recovery at 6 weeks and additional interventions at 24 weeks following ESI. Statistical analyses will include multivariable linear and logistic regression with mixed effects model. ETHICS AND DISSEMINATION: The POiSE study has received ethical approval (South Central Berkshire B Research Ethics Committee 21/SC/0257). Dissemination will be guided by our patient and public engagement group and will include scientific publications, conference presentations and social media.</p

Longitudinal Outcomes of Gender Identity in Children (LOGIC): study protocol for a retrospective analysis of the characteristics and outcomes of children referred to specialist gender services in the UK and the Netherlands.

INTRODUCTION: Specialist gender services for children and young people (CYP) worldwide have experienced a significant increase in referrals in recent years. As rates of referrals increase, it is important to understand the characteristics and profile of CYP attending these services in order to inform treatment pathways and to ensure optimal outcomes. METHODS AND ANALYSIS: A retrospective observational study of clinical health records from specialist gender services for CYP in the UK and the Netherlands. The retrospective analysis will examine routinely collected clinical and outcome measures data including demographic, clinical, gender identity-related and healthcare resource use information. Data will be reported for each service and also compared between services. This study forms part of a wider programme of research investigating outcomes of gender identity in children (the Longitudinal Outcomes of Gender Identity in Children study). ETHICS AND DISSEMINATION: The proposed study has been approved by the Health Research Authority and London-Hampstead Research Ethics Committee as application 19/LO/0181. The study findings will be published in peer-reviewed journals and presented at both conferences and stakeholder events

Universality and scaling in chaotic attractor-to-chaotic attractor transitions.

In this paper we discuss chaotic attractor-to-chaotic attractor transitions in two-dimensional multiparameter maps as an external parameter is varied. We show that the transitions are sharply de®ned and may be classed as second-order phase transitions

Universality and scaling in chaotic attractor-to-chaotic attractor transitions.

In this paper we discuss chaotic attractor-to-chaotic attractor transitions in two-dimensional multiparameter maps as an external parameter is varied. We show that the transitions are sharply de®ned and may be classed as second-order phase transitions