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Stability of coupled van der pol oscillators and applications to gait control in simple animals /Low, Lesley Ann. January 2002 (has links)
Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 153-160).
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A study of dynamics of coupled nonlinear circuitsHernández, José Luis Sánchez. January 2004 (has links) (PDF)
Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2005. / Feodor Vainstein, Committee Member ; Dieci Luca, Committee Member ; Yi Yingfei, Committee Member ; Wang Yang, Committee Member ; Shui-Nee, Chow, Committee Chair. Vita. Includes bibliographical references.
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General non linear perturbation model of phase noise in LC oscillatorsMukherjee, Jayanta, January 2006 (has links)
Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 113-114).
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Μελέτη της εξίσωσης van der Rol στο επίπεδο και υπό την παρουσία περιοδικών διαταραχώνΠαπανικολάου, Ξενοφών 30 July 2014 (has links)
Η παρούσα διατριβή εκπονήθηκε ως μια διπλώματική εργασία υπό την επίβλεψη του καθηγητή Αναστάσιου Μπούντη (Τμήμα Μαθηματικό Πανεπιστήμιο Πατρών), κατά την διάρκεια του ακαδημαικού έτους 2012-2013. Στόχος μας ήταν να μελετήσουμε τόσο θεωρητικά όσο και αριθμητικά μη τετριμένες λύσεις και να κατανοήσουμε, σε γενικές γραμμές, τη συμπεριφορά της μη γραμμικής διαφορικής εξίσωσης δεύτερης τάξης van der Pol.
Στη μελέτη που ακολουθεί εξετάζονται δύο περιπτώσεις της εξίσωσης van der Pol: η αυτόνομη μορφή και η μή αυτόνομη με περιοδικό εξαναγκασμό. Η εξίσωση που μελετάμε είναι μη γραμμική, οπότε
για την ανάλυσή της χρησιμοποιείται η θεωρία διαταραχών μη γραμμικών
διαφορικών εξισώσεων. Η θεωρία αυτή χρησιμοποιείται για τη
κατασκευή προσεγγιστικών λύσεων, οι οποίες
στη συνέχεια συγκρίνονται με τα αντίστοιχα αποτελέσματα που παράγονται
μέσω αριθμητικής ολοκλήρωσης. Σχολιάζονται οι ομοιότητες και οι
διαφορές μεταξύ των μεθόδων, τα πλεονεκτήματα και οι αδυναμίες τους.
Συζητούνται επίσης ορισμένες από τις πιο χαρακτηριστικές ιδιότητες των λύσεων
τόσο στη αυτόνομη, όσο και τη μη αυτόνομη μορφή της εξίσωσης.
Ειδικότερα στο Kεφάλαιο 2 επικεντρωνόμαστε στην αυτόνομη μορφή και παραθέτουμε βασικούς ορισμούς και θεωρήματα της θεωρίας μη γραμμικών
Σ.Δ.Ε, για την ποιοτική μελέτη της εξίσωσης. Μελετάται το
είδος και η ευστάθεια των σημείων ισορροπίας και αποδεικνύεται η ύπαρξη
οριακού κύκλου μέσω της θεωρίας Poincare-Bendixson. Με χρήση των μεθόδων ασυμπτωτικής επέκτασης,
Poincare-Lindstedt και πολλαπλών χρονικών κλιμάκων της
θεωρίας διαταραχών, προσδιορίζονται διαφορετικές προσεγγίσεις του
οριακού κύκλου της εξίσωσης για 0<ε<<1. Σε κάθε περίπτωση
κατασκευάζονται συγκριτικά διαγράμματα, όπου περιγράφονται οι λύσεις που δίνουν η
αριθμητική ολοκλήρωση και οι αναλυτικές
προσεγγίσεις.
Στο Kεφάλαιο 3 αναλύονται μη αυτόνομες μορφές της εξίσωσης και
διακρίνονται δύο περιπτώσεις: Διέγερση συχνότητας κοντά σε αυτή του
αυτόνομου συστήματος και διέγερση συχνότητας μακριά από αυτή του
αυτόνομου συστήματος.
Στην πρώτη περίπτωση υπολογίζονται προσεγγίσεις των περιοδικών
λύσεων της εξίσωσης με τις μεθόδους Poincare-Lindstedt και
πολλαπλών χρονικών κλιμάκων και παρουσιάζονται σε διαγράμματα οι περιοδικές και οι σχεδόν-περιοδικές λύσεις για ορισμένες τιμές των παραμέτρων.
Στη δεύτερη περίπτωση υπολογίζονται προσεγγιστικές λύσεις με τη
μέθοδο δύο χρονικών κλιμάκων και κατασκευάζονται συγκριτικά
διαγράμματα με τη λύση που
δίνει η αριθμητική ολοκλήρωση, για τιμές παραμέτρων που αντιστοιχούν σε περιοδικές και σχεδόν-περιοδικές καταστάσεις.
Στο τέλος του κεφαλαίου δείχνεται η ύπαρξη χαοτικής συμπεριφοράς στο σύστημα μας.
Το Παράρτημα Α περιλαμβάνει τα κυριότερα στοιχεία, ορισμούς και
θεωρήματα, της θεωρίας μη γραμμικών Σ.Δ.Ε, τα οποία
αναφέρονται και εφαρμόζονται στα Κεφάλαια 2 και 3. Τέλος περέχονται όλα τα προγράμματα σε Mathematica, με τα οποία κατασκευάστηκαν τα διαγράμματα της
εργασίας και πραγματοποιήθηκε η αριθμητική ολοκλήρωση των εξισώσεων. / This thesis elaborated as diploma work under the supervision of Professor Anastasios Buddhi (Department of Mathematics University of Patras), during the academic year 2012-2013. Our aim was to study both theoretically and numerically non- trivial solutions and to understand, in general, the behavior of non- linear differential equation of second order van der Pol.
The following study examined both cases the equation van der Pol: the independent form and the non- autonomous by periodic forcing. The equation is nonlinear study, so
for analysis using the perturbation theory of nonlinear
differential equations. The theory is used to
construction of approximate solutions which
then compared with the corresponding results obtained
through numerical integration. Commented on the similarities and
differences between the methods, strengths and weaknesses.
Also discussed some of the most characteristic properties of the solutions
both autonomous and non- autonomous form of the equation.
In particular in Chapter 2 we focus on autonomous form and quote basic definitions and theorems of the theory of nonlinear
SDE for the qualitative study of the equation. studied the
nature and stability of equilibria and prove the existence of
incremental cycle through theory Poincare-Bendixson. Using the methods asymptotic expansion
Poincare-Lindstedt and multiple time scales of
perturbation theory, identified different approaches
boundary circle of the equation for 0 < e << 1. In each case
made comparative charts describing the solutions that give the
numerical integration and analytical
approaches.
In Chapter 3 details the forms of non- autonomous equation
there are two cases: Excitation frequency close to that of
autonomous system and stimulation frequency from that of
autonomous system.
In the first case calculated approximations of periodic
solutions of the equation by the methods of Poincare-Lindstedt and
multiple time scales and charts presented in periodic and quasi- periodic solutions for certain values of parameters.
In the second case calculated approximate solutions with
method two time scales and made comparatively
charts with the solution
gives the numerical integration for parameter values corresponding to periodic and quasi- periodic statements.
At the end of the chapter shows the existence of chaotic behavior in our system.
Appendix A contains the main elements and definitions
theorems of the theory of nonlinear SDE, which
referred to and applied in Chapters 2 and 3. Finally presentation given all programs in Mathematica, the constructed diagrams of
work and performed the numerical integration of the equations.
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Sur l’existence de solutions pour l’équation de van der Pol et pour certaines équations différentielles du second ordre, en présence d’impulsions ; sur la moyennisation pour les équations différentielles flouesGuen, Rahma January 2016 (has links)
Cette thèse est constituée de deux parties :
Dans la première partie nous étudions l’existence de solutions périodiques,
de periode donnée, et à variations bornées, de l’équation de van der
Pol en présence d’impulsions. Nous étudions, en premier, le cas où les impulsions
ne dépendent pas de l’état. Ensuite, nous considèrons le cas où les
impulsions dépendent de la moyenne de l’état et enfin, nous traitons le cas
général où les impulsions dépendent de l’état. La méthode de résolution est
basée sur le principe de point fixe de type contraction.
Nous nous intéressons ensuite à l’étude d’un problème avec trois points
aux limites, associé à certaines équations différentielles impulsives du second
ordre. Nous obtenons un premier résultat d’existence de solutions en appliquant
le théorème de point fixe de Schaefer. Un deuxième résultat est obtenu
en utilisant le théorème de point fixe de Sadovskii. Pour le résultat d’unicité
des solutions nous appliquons, enfin, un théorème de point fixe de type
contraction.
La deuxième partie est consacrée à la justification de la technique de
moyennisation dans le cadre des équations différentielles floues. Les conditions
sur les données que nous imposons sont moins restrictives que celles de
la littérature.
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Estudo da Interação entre o Sistema Cardiovascular e o Respiratório à Luz da Teoria da Informação de ShannonCristine Brasileiro Valença, Anelle January 2006 (has links)
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Previous issue date: 2006 / Estuda-se a dinâmica do sistema cardiorespiratório à luz da Teoria de Shannon. O
estudo tem dois suportes metodológicos, um é o estudo de dois osciladores de van der
Pol acoplados de diferentes maneiras e o outro é a ferramenta estatística, Informação
Mútua, da Teoria da Informação de Shannon. Faz-se, primeiramente, uma revisão da
fisiologia dos dois sistemas: o cardiovascular e o respiratório. Explora-se a interação
entre a freqüência respiratória e a freqüência cardíaca como um possível marcador de
eventuais disfunções fisiológicas. Analisa-se as possíveis formas de acoplamento entre
dois osciladores de van der Pol. Os resultados reais, obtidos com o estudo de quatro
pacientes do Instituto do Coração da Faculdade de Medicina da Universidade de São
Paulo, confirmaram a grande potencialidade da ferramenta Informação Mútua como
um indicador fisiológico
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Μέθοδοι διαταραχών και εφαρμογές αυτών / Perturbation methods for non-linear differential equationsΤαβουλάρη, Δέσποινα 12 April 2013 (has links)
Σε αυτή τη διπλωματική εργασία παρουσιάζονται μερικές μέθοδοι ομαλών διαταραχών και η εφαρμογή τους στις "διάσημες" μη γραμμικές συνήθεις διαφορικές εξισώσεις Duffing, Castor και van der Pol.Οι μέθοδοι διαταραχών μπορούν να χρησιμοποιηθούν για να βρούμε προσεγγιστικές λύσεις σε διαφορικές εξισώσεις οι οποίες είναι μη γραμμικές και μια ακριβή λύση δεν μπορεί να βρεθεί. Η μέθοδος της θεωρίας διαταραχών γίνεται με σεβασμό ως προς μια μικρή παράμετρο ε, 0<ε<<1. Οι προσεγγιστικές αυτές μέθοδοι προυποθέτουν ότι γνωρίζουμε πλήρως τη λύση του προβλήματος για την τιμή ε=0 μιας παραμέτρου και επιχειρούμε να εκφράσουμε τη γενική λύση, για 0<ε<<1, υπό μορφή σειράς όρων του ε,ε^2,...κ.τ.λ. / In this thesis we present some regular perturbation methods and their applications to the famous non-linear ordinary differential equations of Duffing, Castor and van der Pol. The method of perturbations can be used to develop approximate solutions to differential equations, which have nonlinearities or variable coefficients so that an exact solution cannot be constructed. The method of perturbation expansion is carried out with respect to a small parameter ε,0<ε<<1. These approximate methods require that we know the solution of the problem for ε=0 and try to expess the general solution, for 0<ε<<1, as a series of terms ε,ε^2,...e.t.c.
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Frequency Pulling of the van der Pol OscillatorOutram, Ian Hugh 05 1900 (has links)
<p> The frequency pulling of the van dcr Pol nonlinear oscillator due to an external forcing signal is investigated. The nonlinearity is of the zero-memory symmetric-cut-off type following a cube law.</p> <p> An experimental oscillator was built, and curves of the frequency shift of the oscillator fundamental against the magnitude of the input forcing signal are shown, both for a sinusoidal input and for a narrow band noise input. An empirical result is derived.</p> <p> The case of the sinusoidal input is examined theoretically. The importance of harmonic and intermodulation frequencies in the oscillator output is shown, and relations giving the oscillator frequency shift are given for small forcing amplitudes and for large amplitudes when
the oscillator is nearly synchronized.</p> / Thesis / Master of Engineering (MEngr)
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Remodelagem das equações da membrana da fibra do neurônio: relação com a equação de Van der Pol e elaboração de novo circuito equivalente / Remodeling the equations of the neuron fiber membrane: its relationship with the Van der Pol equation and elaboration of a new equivalent circuitBarboza, Ruy 13 November 1992 (has links)
Neste trabalho as equações fenomenológicas (tetra-dimensionais) de Hodgkin-Huxley [5], para a membrana da fibra do neurônio, são estudadas mediante transformações não-lineares de variáveis. As transformações de variáveis visam estabelecer um processo controlado de redução de variáveis até chegar a um modelo bidimensional com o menor prejuízo quantitativo possível. O objetivo primordial é aprofundar o entendimento da aparente relação das equações de Hodgkin-Huxley com uma versão da equação de 2ª ordem de van der Pol, conhecida na literatura pelos nomes de equação de FitzHugh-Nagumo [83], equação de Nagumo [84] ou equação Bonhoeffer-van der Pol [7]. É proposta também uma nova formulação matemática para o modelo da corrente de potássio. Estas modificações possibilitam a elaboração de uma remodelagem do aspecto e funcionamento interno do circuito equivalente da membrana. Este circuito, além de facilitar as simplificações para comparar as novas equações em relação ao modelo tipo van der pol, apresenta também potencial teórico mais desenvolvido do que o circuito equivalente original de Hodgkin-Huxley, já que ao contrário deste os elementos do novo circuito podem ser mais facilmente reconhecidos e manipulados dentro da teoria usual de circuitos elétricos. Uma primeira conseqüência da concepção do novo circuito, aqui explorada, é a formulação do modelo da membrana na linguagem da mecânica analítica. / The phenomenological four-variable equations of Hodgkin and Huxley [5] for the neuron fiber membrane are studied by means of nonlinear transformations of variables . The purpose is gradually reduce the number of variables to a three and then to a two-dimensional model, with smallest possible deviations from the quantitative properties of the original model. The primary aim is to get better insights into the apparent connect ion between the Hodgkin-Huxley equations and a version of the second order equation of van der Pol, usually called FitzHugh-Nagumo equation [83], or Nagumo equation [84], or Bonhoeffer- van der Pol equation [7]. An alternative formulation for the potassium current is also proposed. The above modifications lead to an alternative circuit model for the nerve membrane. Such circuit helps the comparison with the van der Pol-type model. It exhibits also better theoretical appeal than the original circuit of Hodgkin and Huxley in the sense that the circuit elements are now properly defined in terms of usual electrical circuit theory. An application of the proposed equivalent circuit i s a description of the Hodgkin-Huxley membrane model according to the formalism of analytical mechanics.
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Remodelagem das equações da membrana da fibra do neurônio: relação com a equação de Van der Pol e elaboração de novo circuito equivalente / Remodeling the equations of the neuron fiber membrane: its relationship with the Van der Pol equation and elaboration of a new equivalent circuitRuy Barboza 13 November 1992 (has links)
Neste trabalho as equações fenomenológicas (tetra-dimensionais) de Hodgkin-Huxley [5], para a membrana da fibra do neurônio, são estudadas mediante transformações não-lineares de variáveis. As transformações de variáveis visam estabelecer um processo controlado de redução de variáveis até chegar a um modelo bidimensional com o menor prejuízo quantitativo possível. O objetivo primordial é aprofundar o entendimento da aparente relação das equações de Hodgkin-Huxley com uma versão da equação de 2ª ordem de van der Pol, conhecida na literatura pelos nomes de equação de FitzHugh-Nagumo [83], equação de Nagumo [84] ou equação Bonhoeffer-van der Pol [7]. É proposta também uma nova formulação matemática para o modelo da corrente de potássio. Estas modificações possibilitam a elaboração de uma remodelagem do aspecto e funcionamento interno do circuito equivalente da membrana. Este circuito, além de facilitar as simplificações para comparar as novas equações em relação ao modelo tipo van der pol, apresenta também potencial teórico mais desenvolvido do que o circuito equivalente original de Hodgkin-Huxley, já que ao contrário deste os elementos do novo circuito podem ser mais facilmente reconhecidos e manipulados dentro da teoria usual de circuitos elétricos. Uma primeira conseqüência da concepção do novo circuito, aqui explorada, é a formulação do modelo da membrana na linguagem da mecânica analítica. / The phenomenological four-variable equations of Hodgkin and Huxley [5] for the neuron fiber membrane are studied by means of nonlinear transformations of variables . The purpose is gradually reduce the number of variables to a three and then to a two-dimensional model, with smallest possible deviations from the quantitative properties of the original model. The primary aim is to get better insights into the apparent connect ion between the Hodgkin-Huxley equations and a version of the second order equation of van der Pol, usually called FitzHugh-Nagumo equation [83], or Nagumo equation [84], or Bonhoeffer- van der Pol equation [7]. An alternative formulation for the potassium current is also proposed. The above modifications lead to an alternative circuit model for the nerve membrane. Such circuit helps the comparison with the van der Pol-type model. It exhibits also better theoretical appeal than the original circuit of Hodgkin and Huxley in the sense that the circuit elements are now properly defined in terms of usual electrical circuit theory. An application of the proposed equivalent circuit i s a description of the Hodgkin-Huxley membrane model according to the formalism of analytical mechanics.
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