Um novo gerador central de padrões de locomoção para geração de trajectórias em tempo real de robôs quadrúpedes

Neste trabalho estuda-se a geração de trajectórias em tempo real de um robô quadrúpede. As trajectórias podem dividir-se em duas componentes: rítmica e discreta. A componente rítmica das trajectórias é modelada por uma rede de oito osciladores acoplados, com simetria 4 2 Z  Z . Cada oscilador é modelado matematicamente por um sistema de Equações Diferenciais Ordinárias. A referida rede foi proposta por Golubitsky, Stewart, Buono e Collins (1999, 2000), para gerar os passos locomotores de animais quadrúpedes. O trabalho constitui a primeira aplicação desta rede à geração de trajectórias de robôs quadrúpedes. A derivação deste modelo baseia-se na biologia, onde se crê que Geradores Centrais de Padrões de locomoção (CPGs), constituídos por redes neuronais, geram os ritmos associados aos passos locomotores dos animais. O modelo proposto gera soluções periódicas identificadas com os padrões locomotores quadrúpedes, como o andar, o saltar, o galopar, entre outros. A componente discreta das trajectórias dos robôs usa-se para ajustar a parte rítmica das trajectórias. Este tipo de abordagem é útil no controlo da locomoção em terrenos irregulares, em locomoção guiada (por exemplo, mover as pernas enquanto desempenha tarefas discretas para colocar as pernas em localizações específicas) e em percussão. Simulou-se numericamente o modelo de CPG usando o oscilador de Hopf para modelar a parte rítmica do movimento e um modelo inspirado no modelo VITE para modelar a parte discreta do movimento. Variou-se o parâmetro g e mediram-se a amplitude e a frequência das soluções periódicas identificadas com o passo locomotor quadrúpede Trot, para variação deste parâmetro. A parte discreta foi inserida na parte rítmica de duas formas distintas: (a) como um offset, (b) somada às equações que geram a parte rítmica. Os resultados obtidos para o caso (a), revelam que a amplitude e a frequência se mantêm constantes em função de g. Os resultados obtidos para o caso (b) revelam que a amplitude e a frequência aumentam até um determinado valor de g e depois diminuem à medida que o g aumenta, numa curva quase sinusoidal. A variação da amplitude das soluções periódicas traduz-se numa variação directamente proporcional na extensão do movimento do robô. A velocidade da locomoção do robô varia com a frequência das soluções periódicas, que são identificadas com passos locomotores quadrúpedes.

Assessment of genotoxic effects in nurses handling cytostatic drugs

Several antineoplastic drugs have been classified as carcinogens by the International Agency for Research on Cancer (IARC) on the basis of epidemiological findings, animal carcinogenicity data, and outcomes of in vitro genotoxicity studies. 5-Fluorouracil (5-FU), which is easily absorbed through the skin, is the most frequently used antineoplastic agent in Portuguese hospitals and therefore may be used as an indicator of surface contamination. The aims of the present investigation were to (1) examine surface contamination by 5-FU and (2) assess the genotoxic risk using cytokinesis-block micronucleus assay in nurses from two Portuguese hospitals. The study consisted of 2 groups: 27 nurses occupationally exposed to cytostatic agents (cases) and 111 unexposed individuals (controls). Peripheral blood lymphocytes (PBL) were collected in order to measure micronuclei (MN) in both groups. Hospital B showed a higher numerical level of contamination but not significantly different from Hospital A. However; Hospital A presented the highest value of contamination and also a higher proportion of contaminated samples. The mean frequency of MN was significantly higher in exposed workers compared with controls. No significant differences were found among MN levels between the two hospitals. The analysis of confounding factors showed that age is a significant variable in MN frequency occurrence. Data suggest that there is a potential genotoxic damage related to occupational exposure to cytostatic drugs in oncology nurses.

Análise do comportamento angular da articulação do joelho no plano frontal em bailarinas amadoras de élite durante o demi-plié: um estudo piloto

Mestrado em Fisioterapia

Frequency of TERT promoter mutations in human cancers

Reactivation of telomerase has been implicated in human tumorigenesis, but the underlying mechanisms remain poorly understood. Here we report the presence of recurrent somatic mutations in the TERT promoter in cancers of the central nervous system (43%), bladder (59%), thyroid (follicular cell-derived, 10%) and skin (melanoma, 29%). In thyroid cancers, the presence of TERT promoter mutations (when occurring together with BRAF mutations) is significantly associated with higher TERT mRNA expression, and in glioblastoma we find a trend for increased telomerase expression in cases harbouring TERT promoter mutations. Both in thyroid cancers and glioblastoma, TERT promoter mutations are significantly associated with older age of the patients. Our results show that TERT promoter mutations are relatively frequent in specific types of human cancers, where they lead to enhanced expression of telomerase.

Fractional order modelling of zero length column desorption response for adsorbents with variable particle sizes

This manuscript analyses the data generated by a Zero Length Column (ZLC) diffusion experimental set-up, for 1,3 Di-isopropyl benzene in a 100% alumina matrix with variable particle size. The time evolution of the phenomena resembles those of fractional order systems, namely those with a fast initial transient followed by long and slow tails. The experimental measurements are best fitted with the Harris model revealing a power law behavior.

The effect of fractional order in variable structure control

This paper studies fractional variable structure controllers. Two cases are considered namely, the sliding reference model and the control action, that are generalized from integer into fractional orders. The test bed consists in a mechanical manipulator and the effect of the fractional approach upon the system performance is evaluated. The results show that fractional dynamics, both in the switching surface and the control law are important design algorithms in variable structure controllers.

Postpartum depression among women with unintended pregnancy

OBJECTIVE To analyze the association between unintended pregnancy and postpartum depression.METHODS This is a prospective cohort study conducted with 1,121 pregnant aged 18 to 49 years, who attended the prenatal program devised by the Brazilian Family Health Strategy, Recife, PE, Northeastern Brazil, between July 2005 and December 2006. We interviewed 1,121 women during pregnancy and 1,057 after childbirth. Unintended pregnancy was evaluated during the first interview and postpartum depression symptoms were assessed using the Edinburgh Postnatal Depression Screening Scale. The crude and adjusted odds ratios for the studied association were estimated using logistic regression analysis.RESULTS The frequency for unintended pregnancy was 60.2%; 25.9% presented postpartum depression symptoms. Those who had unintended pregnancies had a higher likelihood of presenting this symptoms, even after adjusting for confounding variables (OR = 1.48; 95%CI 1.09;2.01). When the Self Reporting Questionnaire (SRQ-20) variable was included, the association decreased, however, remained statistically significant (OR = 1.42; 95%CI 1.03;1.97).CONCLUSIONS Unintended pregnancy showed association with subsequent postpartum depressive symptoms. This suggests that high values in Edinburgh Postnatal Depression Screening Scale may result from unintended pregnancy.

Factors associated with dietary supplement use by people who exercise at gyms

OBJECTIVE To assess the factors associated with the use of dietary supplements by people who exercise at gyms.METHODS A cross-sectional study with a sample defined by convenience, considering the number of gyms registered in the Conselho Regional de Educação Física (Regional Council of Physical Education) of Sao Luis, MA, Northeastern Brazil, from July 2011 to July 2012. The final sample comprised 723 individuals who exercise at gyms. The dependent variable was supplement use, and the explanatory variables were length of time and motivation of the physical exercises, duration, goal and self-perception of training, weekly frequency of gym attendance, sex, age, educational attainment, self-perception of body weight, smoking and self-perception of diet. The association between variables was analysed by hierarchical Poisson regression based on a theoretical model.RESULTS Supplement use was reported by 64.7% of the participants. Most of the sample was male (52.6%). The most frequent age group was 20 to 39 years (74.4%). Most participants (46.1%) had been exercising for over a year. The following variables were associated with supplement use: self-perceiving body weight as below ideal (p < 0.001), smoking (p < 0.001), exercising for 7 to 12 months (p = 0.028) or more than one year (p < 0.001), spending more than two hours at the gym (p = 0.051), and perceiving training as moderate (p = 0.024) or intense (p = 0.001).CONCLUSIONS The use of supplements lacks proper professional guidance, being motivated by individuals unsatisfied with their low body weight and who perceive their workout as intense, which raises the need for monitoring this population.

Política(s) de saúde no trabalho : um inquérito sociológico às empresas portuguesas

Resumo Política(s) de saúde no trabalho: um inquérito sociológico às empresas portuguesas A literatura portuguesa sobre políticas, programas e actividades de Segurança, Higiene e Saúde no Trabalho (abreviadamente, SH&ST) é ainda escassa. Com este projecto de investigação pretende-se (i) colmatar essa lacuna, (ii) melhorar o conhecimento dos sistemas de gestão da saúde e segurança no trabalho e (iii) contribuir para a protecção e a promoção da saúde dos trabalhadores. Foi construída uma tipologia com cinco grupos principais de políticas, programas e actividades: A (Higiene & Segurança no Trabalho / Melhoria do ambiente físico de trabalho); B (Avaliação de saúde / Vigilância médica / Prestação de cuidados de saúde); C (Prevenção de comportamentos de risco/ Promoção de estilos de vida saudáveis); D (Intervenções a nível organizacional / Melhoria do ambiente psicossocial de trabalho); E (Actividades e programas sociais e de bem-estar). Havia uma lista de mais de 60 actividades possíveis, correspondendo a um índice de realização de 100%. Foi concebido e desenhado, para ser auto-administrado, um questionário sobre Política de Saúde no Local de Trabalho. Foram efectuados dois mailings, e um follow-up telefónico. O trabalho de campo decorreu entre a primavera de 1997 e o verão de 1998. A amostra (n=259) é considerada representativa das duas mil maiores empresas do país. Uma em cada quatro é uma multinacional. A taxa de sindicalização rondava os 30% da população trabalhadora, mas apenas 16% dos respondentes assinalou a existência de representantes dos trabalhadores eleitos para a SH&ST. A hipótese de investigação principal era a de que as empresas com um sistema integrado de gestão da SH&ST seriam também as empresas com um (i) maior número de políticas, programas e actividades de saúde; (ii) maior índice de saúde; (iii) maior índice de realização; e (iv) maior percentagem dos encargos com a SH&ST no total da massa salarial. As actividades de tipo A e B, tradicionalmente associadas à SH&ST, representavam, só por si, mais de 57% do total. Os resultados, correspondentes às respostas da Secção C do questionário, apontam, para (i) a hipervalorização dos exames de medicina do trabalho; e por outro para (ii) o subaproveitamento de um vasto conjunto de actividades (nomeadamente as de tipo D e E), que são correntemente levadas a cabo pelas empresas e que nunca ou raramente são pensadas em termos de protecção e promoção da saúde dos trabalhadores. As actividades e os programas de tipo C (Prevenção de comportamentos de risco/Promoção de estilos de vida saudáveis), ainda eram as menos frequentes entre nós, a seguir aos Programas sociais e de bem-estar (E). É a existência de sistemas de gestão integrados de SH&ST, e não o tamanho da empresa ou outra característica sociodemográfica ou técnico-organizacional, que permite predizer a frequência de políticas de saúde mais activas e mais inovadores. Os três principais motivos ou razões que levam as empresas portuguesas a investir na protecção e promoção da saúde dos seus trabalhadores eram, por ordem de frequência, (i) o absentismo em geral; (ii) a produtividade, qualidade e/ou competitividade, e (iii) a filosofia de gestão ou cultura organizacional. Quanto aos três principais benefícios que são reportados, surge em primeiro lugar (i) a melhoria da saúde dos trabalhadores, seguida da (ii) melhoria do ambiente do ambiente de trabalho e, por fim, (iii) a melhoria da produtividade, qualidade e/ou competitividade.Quanto aos três principais obstáculos que se põem, em geral, ao desenvolvimento das iniciativas de saúde, eles seriam os seguintes, na percepção dos respondentes: (i) a falta de empenho dos trabalhadores; (ii) a falta de tempo; e (iii) os problemas de articulação/ comunicação a nível interno. Por fim, (i) o empenho das estruturas hierárquicas; (ii) a cultura organizacional propícia; e (iii) o sentido de responsabilidade social surgem, destacadamente, como os três principais factores facilitadores do desenvolvimento da política de saúde no trabalho. Tantos estes factores como os obstáculos são de natureza endógena, susceptíveis portanto de controlo por parte dos gestores. Na sua generalidade, os resultados deste trabalho põem em evidência a fraqueza teóricometodológica de grande parte das iniciativas de saúde, realizadas na década de 1990. Muitas delas seriam medidas avulsas, que se inserem na gestão corrente das nossas empresas, e que dificilmente poderão ser tomadas como expressão de uma política de saúde no local de trabalho, (i) definida e assumida pela gestão de topo, (ii) socialmente concertada, (iii) coerente, (iv) baseada na avaliação de necessidades e expectativas de saúde dos trabalhadores, (v) divulgada, conhecida e partilhada por todos, (vi) contingencial, flexível e integrada, e, por fim, (vii) orientada por custos e resultados. Segundo a Declaração do Luxemburgo (1997), a promoção da saúde engloba o esforço conjunto dos empregadores, dos trabalhadores, do Estado e da sociedade civil para melhorar a segurança, a saúde e o bem-estar no trabalho, objectivo isso que pode ser conseguido através da (i) melhoria da organização e das demais condições de trabalho, da (ii) participação efectiva e concreta dos trabalhadores bem como do seu (iii) desenvolvimento pessoal. Abstract Health at work policies: a sociological inquiry into Portuguese corporations Portuguese literature on workplace health policies, programs and activities is still scarce. With this research project the author intends (i) to improve knowledge on the Occupational Health and Safety (shortly thereafter, OSH) management systems and (ii) contribute to the development of health promotion initiatives at a corporate level. Five categories of workplace health initiatives have been identified: (i) Occupational Hygiene and Safety / Improvement of Physical Working Environment (type A programs); (ii) Health Screening, Medical Surveillance and Other Occupational Health Care Provision (type B programs); (iii) Preventing Risk Behaviours / Promoting Healthy Life Styles (type C programs); (iv) Organisational Change / Improvement of Psycho-Social Working Environment (type D programs); and (v) Industrial and Social Welfare (type E programs). A mail questionnaire was sent to the Chief Executive Officer of the 1500 largest Portuguese companies, operating in the primary and secondary sectors (≥ 100 employees) or tertiary sector (≥ 75 employees). Response rate has reached about 20% (259 respondents, representing about 300 companies). Carried out between Spring 1997 and Summer 1998, the fieldwork has encompassed two direct mailings and one phone follow-up. Sample is considered to be representative of the two thousand largest companies. One in four is a multinational. Union membership rate is about 30%, but only 16% has reported the existence of a workers’ health and safety representative. The most frequent workplace health initiatives were those under the traditional scope of the OSH field (type A and B programs) (57% of total) (e.g., Periodical Medical Examinations; Individual Protective Equipment; Assessment of Working Ability). In SMEs (< 250) it was less likely to find out some time-consuming and expensive activities (e.g., Training on OSH knowledge and skills, Improvement of environmental parameters as ventilation, lighting, heating).There were significant differences in SMEs, when compared with the larger ones (≥ 250) concerning type B programs such as Periodical medical examinations, GP consultation, Nursing care, Other medical and non-medical specialities (e.g., psychiatrist, psychologist, ergonomist, physiotherapist, occupational social worker). With regard to type C programs, there were a greater percentage of programs centred on Substance abuse (tobacco, alcohol, and drug) than on Other health risk behaviours. SMEs representatives reported very few prevention- oriented programs in the field of Drug abuse, Nutrition, Physical activity, Off- job accidents, Blood pressure or Weight control. Frequency of type D programs included Training on Human Resources Management, Training on Organisational Behaviour, Total Quality Management, Job Design/Ergonomics, and Workplace rehabilitation. In general, implementation of this type of programs (Organisational Change / Improvement of Psychosocial Working Environment) is not largely driven by health considerations. Concerning Industrial and Social Welfare (Type E programs), the larger employers are in a better position than SMEs to offer to their employees a large spectrum of health resources and facilities (e.g., Restaurant, Canteen, Resting room, Transport, Infra-structures for physical activity, Surgery, Complementary social protection, Support to recreational and cultural activities, Magazine or newsletter, Intranet). Other workplace health promotion programs like Training on Stress Management, Employee Assistance Programs, or Self-help groups are uncommon in the Portuguese worksites. The existence of integrated OSH management systems, not the company size, is the main variable explaining the implementation of more active and innovative workplace health policies in Portugal. The three main prompting factors reported by employers for health protection and promotion initiatives are: (i) Employee absenteeism; (ii) Productivity, quality and/or competitiveness; and (iii) Corporate culture/management philosophy. On the other hand, (i) Improved staff’s health, (ii) Improved working environment and (iii) Improved productivity, quality and/or competitiveness were the three main benefits reported by companies’ representatives, as a result of successful implementation of workplace health initiatives. (i) Lack of staff commitment; (ii) Lack of time; and (iii) Problems of co-operation and communication within company or establishment (iii) are perceived to be the main barriers companies must cope with. Asked about the main facilitating factors, these companies have pointed out the following ones: (i) Top management commitment; (ii) Corporate culture; and (iii) Sense of social responsibility. This sociological research report shows the methodological weaknesses of workplace health initiatives, carried out by Portuguese companies during the last ‘90s. In many cases, these programs and actions were not part of a corporate health strategy and policy, (i) based on the assessment of workers’ health needs and expectancies, (ii) advocated by the employer or the chief executive officer, (ii) planned and implemented with the staff consultation and participation or (iv) evaluated according to a cost-benefit analysis. In short, corporate health policy and action were still rather based on more traditional OSH approaches and should be reoriented towards Workplace Health Promotion (WHP) approach. According to the Luxembourg Declaration of Workplace Health Promotion in the European Union (1997), WHP is “a combination of: (i) improving the work organisation and environment; (ii) promoting active participation; (iii) encouraging personal development”.Résumée Politique(s) de santé au travail: une enquête sociologique aux entreprises portugaises Au Portugal on ne sait presque rien des politiques de santé au travail, adoptés par les entreprises. Avec ce projet de recherche, on veut (i) améliorer la connaissance sur les systèmes de gestion de la santé et de la sécurité au travail et, au même temps, (ii) contribuer au développement de la promotion de la santé des travailleurs. Une typologie a été usée pour identifier les politiques, programmes et actions de santé au travail: A. Amélioration des conditions de travail / Sécurité au travail; B. Médecine du travail /Santé au travail; C. Prévention des comportements de risque / Promotion de styles de vie sains; D. Interventions organisationnelles / Amélioration des facteurs psychosociaux au travail; E. Gestion de personnel et bien-être social. Un questionnaire postal a été envoyé au représentant maximum des grandes entreprises portugaises, industrielles (≥ 100 employés) ou des services (≥ 75 employés). Le taux de réponse a été environ 20% (259 répondants, concernant trois centaines d’entreprises et d’établissements). La recherche de champ, conduite du printemps 1997 à l’été 1998, a compris deux enquêtes postales et un follow-up téléphonique. L´échantillon est représentatif de la population des deux miles plus grandes entreprises. Un quart sont des multinationales. Le taux de syndicalisation est d’environ 30%. Toutefois, il y a seulement 16% de lieux de travail avec des représentants du personnel pour la santé et sécurité au travail. Les initiatives de santé au travail les plus communes sont celles concernant le domaine plus traditionnel (types A et B) (57% du total): par exemple, les examens de médecine du travail, l’équipement de protection individuelle, les tests d’aptitude au travail. En ce qui concerne les programmes de type C, les plus fréquents sont le contrôle et la prévention des addictions (tabac, alcool, drogue). Les interventions dans le domaine de du système technique et organisationnelle du travail peuvent comprendre les courses de formation en gestion de ressources humaines ou en psychosociologie des organisations, l’ergonomie, le travail posté ou la gestion de la qualité totale. En général, la protection et la promotion de la santé des travailleurs ne sont pas prises en considération dans l’implémentation des initiatives de type D. Il y a des différences quand on compare les grandes entreprises et les moyennes en matière de politique de gestion du personnel e du bien-être (programmes de type E, y compris l’allocation de ressources humaines ou logistiques comme, par exemple, restaurant, journal d’entreprise, transports, installations et équipements sportifs). D’autres activités de promotion de la santé au travail comme la formation en gestion du stress, les programmes d’ assistance aux employés, ou les groupes de soutien et d’auto-aide sont encore très peu fréquents dans les entreprises portugaises. C’est le système intégré de gestion de la santé et de la sécurité au travail, et non pas la taille de l’entreprise, qui aide à prédire l’existence de politiques actives et innovatrices dans ce domaine. Les trois facteurs principaux qui encouragent les actions de santé (prompting factors, en anglais) sont (i) l’absentéisme (y compris la maladie), (ii) les problèmes liés à la productivité, qualité et/ou la compétitivité, et aussi (iii) la culture de l’entreprise/philosophie de gestion. Du coté des bénéfices, on a obtenu surtout l’amélioration (i) de la santé du personnel, (ii) des conditions de travail, et (iii) de la productivité, qualité et/ou compétitivité.Les facteurs qui facilitent les actions de santé au travail sont (i) l’engagement de la direction, (ii) la culture de l’entreprise, et (iii) le sens de responsabilité sociale. Par contre, les obstacles à surmonter, selon les organisations qui ont répondu au questionnaire, seraient surtout (i) le manque d’engagement des travailleurs et de leur représentants, (ii) le temps insuffisant, et (iii) les problèmes de articulation/communication au niveau interne de l’entreprise/établissement. Ce travail de recherche sociologique montre la faiblesse méthodologique des services et activités de santé et sécurité au travail, mis en place par les entreprises portugaises dans les années de 1990, à la suite des accords de concertation sociale de 1991. Dans beaucoup de cas, (i) ces politiques de santé ne font pas partie encore d’un système intégré de gestion, (ii) il n’a pas d’évaluation des besoins et des expectatives des travailleurs, (iii) c’est très bas ou inexistant le niveau de participation du personnel, (iv) on ne fait pas d’analyse coût-bénéfice. On peut conclure que les politiques de santé au travail sont plus proches de la médecine du travail et de la sécurité au travail que de la promotion de la santé des travailleurs. Selon la Déclaration du Luxembourg sur la Promotion de la Santé au Lieu de Travail dans la Communauté Européenne (1997), celle-ci « comprend toutes les mesures des employeurs, des employés et de la société pour améliorer l'état de santé et le bien être des travailleurs » e « ceci peut être obtenu par la concentration des efforts dans les domaines suivants: (i) amélioration de l'organisation du travail et des conditions de travail ; (ii) promotion d'une participation active des collaborateurs ; (iii) renforcement des compétences personnelles ».

Experimenting through the web a linear variable differential transformer

Aiming for teaching/learning support in sciences and engineering areas, the Remote Experimentation concept (an E-learning subset) has grown in last years with the development of several infrastructures that enable doing practical experiments from anywhere and anytime, using a simple PC connected to the Internet. Nevertheless, given its valuable contribution to the teaching/learning process, the development of more infrastructures should continue, in order to make available more solutions able to improve courseware contents and motivate students for learning. The work presented in this paper contributes for that purpose, in the specific area of industrial automation. After a brief introduction to the Remote Experimentation concept, we describe a remote accessible lab infrastructure that enables users to conduct real experiments with an important and widely used transducer in industrial automation, named Linear Variable Differential Transformer.

Complex variable positive definite functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.

Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

Compact, frequency reconfigurable, printed monopole antenna

This paper proposes a possible implementation of a compact printed monopole antenna, useful to operate in UMTS and WLAN bands. In order to accomplish that, a miniaturization technique based on the application of chip inductors is used in conjunction with frequency reconfiguration capability. The chip inductors change the impedance response of the monopole, allowing to reduce the resonant frequency. In order to be able to operate the antenna in these two different frequencies, an antenna reconfiguration technique based on PIN diodes is applied. This procedure allows the change of the active form of the antenna leading to a shift in the resonant frequency. The prototype measurements show good agreement with the simulation results.

Time-frequency and time-scale characterisation of the beat-by-beat high-resolution electrocardiogram

Proceedings of the Sixth Portuguese Conference on Bioemedical Engineering faro, Portugal

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.