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Relevant bibliographies by topics / Saddle node bifurcation

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Academic literature on the topic 'Saddle node bifurcation'

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Published: 4 June 2021

Last updated: 6 February 2022

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Journal articles on the topic "Saddle node bifurcation"

1

ROCŞOREANU, CARMEN, NICOLAIE GIURGIŢEANU, and ADELINA GEORGESCU. "CONNECTIONS BETWEEN SADDLES FOR THE FITZHUGH–NAGUMO SYSTEM." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 533–40. http://dx.doi.org/10.1142/s0218127401002213.

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By studying the two-dimensional FitzHugh–Nagumo (F–N) dynamical system, points of Bogdanov–Takens bifurcation were detected (Sec. 1). Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle connection bifurcation (Sec. 2). Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allowed us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node–saddle connection bifurcation and saddle-node–saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented in Sec. 3. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is also presented in Sec. 3.

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Ueta, Tetsushi, Daisuke Ito, and Kazuyuki Aihara. "Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?" International Journal of Bifurcation and Chaos 25, no. 13 (December 15, 2015): 1550185. http://dx.doi.org/10.1142/s0218127415501850.

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We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

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Kuznetsov, Yuri. "Saddle-node bifurcation." Scholarpedia 1, no. 10 (2006): 1859. http://dx.doi.org/10.4249/scholarpedia.1859.

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Fang, Ding, Yongxin Zhang, and Wendi Wang. "Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate." Complexity 2020 (May 8, 2020): 1–16. http://dx.doi.org/10.1155/2020/7310347.

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An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

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HIZANIDIS, J., R. AUST, and E. SCHÖLL. "DELAY-INDUCED MULTISTABILITY NEAR A GLOBAL BIFURCATION." International Journal of Bifurcation and Chaos 18, no. 06 (June 2008): 1759–65. http://dx.doi.org/10.1142/s0218127408021348.

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We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.

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Shilnikov, Leonid, and Andrey Shilnikov. "Shilnikov saddle-node bifurcation." Scholarpedia 3, no. 4 (2008): 4789. http://dx.doi.org/10.4249/scholarpedia.4789.

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GLENDINNING, PAUL, and COLIN SPARROW. "SHILNIKOV’S SADDLE-NODE BIFURCATION." International Journal of Bifurcation and Chaos 06, no. 06 (June 1996): 1153–60. http://dx.doi.org/10.1142/s0218127496000643.

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In 1969, Shilnikov described a bifurcation for families of ordinary differential equations involving n≥2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols. We investigate a simple (piecewise-smooth) example with an infinite number of homoclinic loops. We also present a smooth example which shows how Shilnikov’s mechanism is related to the Lorenz bifurcation by considering the unfolding of a previously unstudied codimension two bifurcation point.

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AFRAIMOVICH, V. S., and M. A. SHERESHEVSKY. "THE HAUSDORFF DIMENSION OF ATTRACTORS APPEARING BY SADDLE-NODE BIFURCATIONS." International Journal of Bifurcation and Chaos 01, no. 02 (June 1991): 309–25. http://dx.doi.org/10.1142/s0218127491000233.

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We consider the strange attractors which appear as a result of saddle-node vanishing bifurcations in two-dimensional, smooth dynamical systems. Some estimates and asymptotic formulas for the Hausdorff dimension of such attractors are obtained. The estimates demonstrate a dependence of the dimension growth rate after the bifurcation upon the "pre-bifurcational" picture.

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YET, NGUYEN TIEN, DOAN THAI SON, TOBIAS JÄGER, and STEFAN SIEGMUND. "NONAUTONOMOUS SADDLE-NODE BIFURCATIONS IN THE QUASIPERIODICALLY FORCED LOGISTIC MAP." International Journal of Bifurcation and Chaos 21, no. 05 (May 2011): 1427–38. http://dx.doi.org/10.1142/s0218127411029124.

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We provide a local saddle-node bifurcation result for quasiperiodically forced interval maps. As an application, we give a rigorous description of saddle-node bifurcations of 3-periodic graphs in the quasiperiodically forced logistic map with small forcing amplitude.

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KIRK, VIVIEN, and EDGAR KNOBLOCH. "A REMARK ON HETEROCLINIC BIFURCATIONS NEAR STEADY STATE/PITCHFORK BIFURCATIONS." International Journal of Bifurcation and Chaos 14, no. 11 (November 2004): 3855–69. http://dx.doi.org/10.1142/s0218127404011752.

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We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near the saddle-node/pitchfork instability. By construction and analysis of a return map, we show that the codimension-one bifurcation emerges from a codimension-two point at which there is a heteroclinic bifurcation between two saddle equilibria, one hyperbolic and the other nonhyperbolic. We find the same phenomenon occurs in the normal form equations for the hysteresis/pitchfork bifurcation, in this case arbitrarily close to the instability, and show there are restrictions regarding the way in which such dynamics can occur near pitchfork/pitchfork bifurcations. These conclusions carry over to analogous phenomena in normal forms for steady state/Hopf bifurcations.

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Dissertations / Theses on the topic "Saddle node bifurcation"

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Young, Todd Ray. "Saddle-node bifurcations with homoclinic orbits." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/29855.

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Borquez, Caballero Rodrigo Edgardo. "Calculating the Distance to the Saddle-Node Bifurcation Set." Thesis, KTH, Elektriska energisystem, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-119236.

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A power system will experience voltage collapse when the loads increase up to a certain critical limit, where the system physically cannot support the amount of connected load. This point identified as a Saddle- Node Bifurcation (SNB), corresponds to a generic instability of parameterized differential equation models and represents the intersection point where different branches of equilibria meet. At this point the jacobian matrix of the system is singular and the system loses stability bringing the typical scenario of voltage collapse. To prevent voltage instability and collapse, the computation of the closest distance from a present operating point to the saddle-node bifurcation set can be used as a loadability index useful in power system operation and planning. The power margin is determined by applying the iterative or direct method described in [16]. Numerical examples of both methods applied to IEEE 9-bus system and IEEE 39-bus system shows that the iterative method is more reliable although it requires a longer computation time. The stability of the system is negatively affected in two ways when generators reach their reactive power limits: the voltage stability margin is deteriorated, or immediate voltage instability and collapse is produced.

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Hesse, Janina. "Implications of neuronal excitability and morphology for spike-based information transmission." Doctoral thesis, Humboldt-Universität zu Berlin, 2017. http://dx.doi.org/10.18452/18583.

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Signalverarbeitung im Nervensystem hängt sowohl von der Netzwerkstruktur, als auch den zellulären Eigenschaften der Nervenzellen ab. In dieser Abhandlung werden zwei zelluläre Eigenschaften im Hinblick auf ihre funktionellen Anpassungsmöglichkeiten untersucht: Es wird gezeigt, dass neuronale Morphologie die Signalweiterleitung unter Berücksichtigung energetischer Beschränkungen verstärken kann, und dass selbst kleine Änderungen in biophysikalischen Parametern die Aktivierungsbifurkation in Nervenzellen, und damit deren Informationskodierung, wechseln können. Im ersten Teil dieser Abhandlung wird, unter Verwendung von mathematischen Modellen und Daten, die Hypothese aufgestellt, dass Energie-effiziente Signalweiterleitung als starker Evolutionsdruck für unterschiedliche Zellkörperlagen bei Nervenzellen wirkt. Um Energie zu sparen, kann die Signalweiterleitung vom Dendrit zum Axon verstärkt werden, indem relativ kleine Zellkörper zwischen Dendrit und Axon eingebaut werden, während relativ große Zellkörper besser ausgelagert werden. Im zweiten Teil wird gezeigt, dass biophysikalische Parameter, wie Temperatur, Membranwiderstand oder Kapazität, den Feuermechanismus des Neurons ändern, und damit gleichfalls Aktionspotential-basierte Informationsverarbeitung. Diese Arbeit identifiziert die sogenannte "saddle-node-loop" (Sattel-Knoten-Schlaufe) Bifurkation als den Übergang, der besonders drastische funktionale Auswirkungen hat. Neben der Änderung neuronaler Filtereigenschaften sowie der Ankopplung an Stimuli, führt die "saddle-node-loop" Bifurkation zu einer Erhöhung der Netzwerk-Synchronisation, was möglicherweise für das Auslösen von Anfällen durch Temperatur, wie bei Fieberkrämpfen, interessant sein könnte.
Signal processing in nervous systems is shaped by the connectome as well as the cellular properties of nerve cells. In this thesis, two cellular properties are investigated with respect to the functional adaptations they provide: It is shown that neuronal morphology can improve signal transmission under energetic constraints, and that even small changes in biophysical parameters can switch spike generation, and thus information encoding. In the first project of the thesis, mathematical modeling and data are deployed to suggest energy-efficient signaling as a major evolutionary pressure behind morphological adaptations of cell body location: In order to save energy, the electrical signal transmission from dendrite to axon can be enhanced if a relatively small cell body is located between dendrite and axon, while a relatively large cell body should be externalized. In the second project, it is shown that biophysical parameters, such as temperature, membrane leak or capacitance, can transform neuronal excitability (i.e., the spike onset bifurcation) and, with that, spike-based information processing. This thesis identifies the so-called saddle-node-loop bifurcation as the transition with particularly drastic functional implications. Besides altering neuronal filters and stimulus locking, the saddle-node-loop bifurcation leads to an increase in network synchronization, which may potentially be relevant for the initiation of seizures in response to increased temperature, such as during fever cramps.

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Bronzi, Marcus Augusto. "Intersecções homoclínicas /." São José do Rio Preto : [s.n.], 2006. http://hdl.handle.net/11449/94242.

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Orientador: Vanderlei Minori Horita
Banca: Ali Tahzibi
Banca: Paulo Ricardo Silva
Resumo: Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica.
Abstract: We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency.
Mestre

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5

Fuhrmann, G., M. Gröger, and T. Jäger. "Non-smooth saddle-node bifurcations II: Dimensions of strange attractors." Cambridge University Press, 2018. https://tud.qucosa.de/id/qucosa%3A70708.

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We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

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Fuhrmann, Gabriel. "Non-smooth saddle-node bifurcations I: existence of an SNA." Cambridge University Press, 2016. https://tud.qucosa.de/id/qucosa%3A70707.

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We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are C² -open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

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Bronzi, Marcus Augusto [UNESP]. "Intersecções homoclínicas." Universidade Estadual Paulista (UNESP), 2006. http://hdl.handle.net/11449/94242.

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Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-03Bitstream added on 2014-06-13T20:27:28Z : No. of bitstreams: 1 bronzi_ma_me_sjrp.pdf: 904425 bytes, checksum: 2344eb35a112034c2f1741b2e229f1ec (MD5)
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica.
We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency.

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8

Amaral, Fabíolo Moraes. "Caracterização, estimativas e bifurcações da região de estabilidade de sistemas dinâmicos não lineares." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/18/18154/tde-29102010-145102/.

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Estimar a região de estabilidade de um ponto de equilíbrio assintoticamente estável é importante em aplicações tais como sistemas de potência, economia e ecologia. A compreensão da estrutura qualitativa da fronteira da região de estabilidade é fundamental para estimar com eficiência a região de estabilidade. Caracterizações topológicas e dinâmicas da fronteira da região de estabilidade foram desenvolvidas ao longo das últimas décadas. Estas caracterizações foram desenvolvidas sob hipóteses de hiperbolicidade dos pontos de equilíbrio na fronteira e transversalidade. Para sistemas que dependem de parâmetros, a condição de hiperbolicidade pode ser violada em pontos de bifurcações. Estaremos interessados em estimar a região de estabilidade, para sistemas sujeitos a variações de parâmetros, onde ocorre a violação da condição de hiperbolicidade dos pontos de equilíbrio na fronteira da região de estabilidade devido ao aparecimento de uma bifurcação sela-nó do tipo zero nesta fronteira. Apresentaremos neste trabalho uma caracterização completa da fronteira da região de estabilidade na presença de um ponto de equilíbrio não hiperbólico sela-nó do tipo zero. Motivados também em oferecer um algoritmo conceitual para obter estimativas da região de estabilidade perturbada via conjunto de nível de uma dada função energia na vizinhança de um parâmetro de bifurcação sela-nó do tipo zero, buscaremos exibir resultados que permitam compreender o comportamento da região de estabilidade e de sua fronteira sob a influência das variações do parâmetro, incluindo variações do parâmetro próximo a um parâmetro de bifurcação sela-nó do tipo zero.
Estimating the stability region of an asymptotically stable equilibrium point is fundamental in applications such as power systems, economy and ecology. The knowledge of the qualitative structure of the stability boundary is essential to estimate with efficiency the stability region. Topological and dynamical characterizations of the stability boundary have been developed over the past decades. These characterizations were developed under assumptions of hyperbolicity of equilibrium points on the stability boundary and transversality. For systems that depend on parameters, the condition of hyperbolicity can be violated at points of bifurcations. We will be primarily interested in estimating the stability region, for systems subjected to parameter variations, when the condition of hyperbolicity of equilibrium points on the stability boundary is violated due to the appearance of a type-zero saddle-node bifurcation on the stability boundary. We will develop in this work, a complete characterization of the stability boundary in the presence of a type-zero saddle-node non-hyperbolic equilibrium point. Also, motivated to providing a conceptual algorithm to obtain estimates of the perturbed stability region via level sets of a given energy function in the neighborhood of a type-zero saddle-node bifurcation parameter, we offer results that explain the behavior of the stability region and its boundary under the influence of parameter variations, including variations of the parameter close to a type-zero saddle-node bifurcation parameter.

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Mayol, Serra Catalina. "Dinàmica no lineal de sistemes làsers: potencials de Lyapunov i diagrames de bifurcacions." Doctoral thesis, Universitat de les Illes Balears, 2002. http://hdl.handle.net/10803/9430.

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En aquest treball s'ha estudiat la dinàmica dels làsers de classe A i de classe B en termes del potencial de Lyapunov. En el cas que s'injecti un senyal al làser o es modulin alguns dels paràmetres, apareix un comportament moltmés complex i s'estudia el conjunt de bifurcacions.

1) Als làsers de classe A, la dinàmica determinista s'ha interpretat com el moviment damunt el potencial de Lyapunov. En la dinàmica estocàstica s'obté un flux sostingut per renou per a la fase del camp elèctric.

2) Per als làsers de classe A amb senyal injectat, s'ha descrit el conjunt de bifurcacions complet i s'ha determinat el conjunt d'amplituds i freqüències en el quals el làser respon
ajustant la seva freqüència a la del camp extern.

3) S'ha obtingut un potencial de Lyapunov pels làsers de classe B, només vàlid en el cas determinista, que inclou els termes de saturació de guany i d'emissió espontània.

4) S'ha realitzat un estudi del conjunt de bifurcacions parcial al voltant del règim tipus II de la singularitat Hopf--sella--node en un làser de classe B amb senyal injectat.

5) S'han identificat les respostes òptimes pels làsers de semiconductor sotmesos a modulació periòdica externa. S'han obtingut les corbes que donen la resposta màxima per cada tipus de resonància en el pla definit per l'amplitud relativa de modulació i la freqüència de modulació.
In this work we have studied the dynamics of both class A and class B lasers in terms of Lyapunov potentials. In the case of an injected signal or when some laser parameters are modulated, and more complex behaviour is expected, the bifurcation set is studied. The main results are the following:
1) For class A lasers, the deterministic dynamics has been interpreted as a movement on the potential landscape. In the stochastic dynamics we have found a noise sustained flow for the phase of the electric field.
2) For class A lasers with an injected signal, we have been able to describe the whole bifurcation set of this system and to determine the set of amplitudes frequencies for which the laser responds adjusting its frequency to that of the external field.
3) In the case of class B lasers, we have obtained a Lyapunov potential only valid in the deterministic case, including spontaneous emission and gain saturation terms. The fixed point corresponding to the laser in the on state has been interpreted as a minimum in this potential. Relaxation to this minimum is reached through damped oscillations.
4) We have performed a study of the partial bifurcation set around the type II regime of the Hopf-saddle-node singularity in a class B laser with injected signal.
5) We have identified the optimal responses of a semiconductor laser subjected to an external periodic modulation. The lines that give a maximum response for each type of resonance are obtained in the plane defined by the relative amplitude modulation and frequency modulation.

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Greene, Scott. "Constraint at a saddle node bifurcation." 1993. http://catalog.hathitrust.org/api/volumes/oclc/32710270.html.

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Thesis (M.S.)--University of Wisconsin--Madison, 1993.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 56-58).

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Books on the topic "Saddle node bifurcation"

1

Glendinning, Paul. Shilnikov's saddle-node bifurcation. Bristol [England]: Hewlett Packard, 1996.

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Jäger, Tobias H. The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Providence, R.I: American Mathematical Society, 2009.

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The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Providence, R.I: American Mathematical Society, 2009.

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Jager, Tobias H. The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Providence, R.I: American Mathematical Society, 2009.

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Book chapters on the topic "Saddle node bifurcation"

1

Zhou, Tianshou. "Saddle-Node Bifurcation." In Encyclopedia of Systems Biology, 1889. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_501.

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Zhang, Jia-Zhong, Yan Liu, Pei-Hua Feng, and Jia-Hui Chen. "Formations of Transitional Zones in Shock Wave with Saddle-Node Bifurcations." In Discontinuity and Complexity in Nonlinear Physical Systems, 347–58. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-01411-1_19.

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Yang, Jianke. "Conditions and Stability Analysis for Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schrödinger Equations." In Progress in Optical Science and Photonics, 639–55. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/10091_2012_3.

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"The Saddle-Node Homoclinic Bifurcation. Dynamics of Slow-Fast Systems in the Plane." In Introduction to Nonlinear Oscillations, 123–36. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA, 2015. http://dx.doi.org/10.1002/9783527695942.ch10.

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Chenciner, Alain, and Jacob Palis. "Hamiltonian - Like Phenomena in Saddle - Node Bifurcations of Invariant Curves for Plane Diffeomorphisms." In Singularities & Dynamical Systems, Proceedings of the International Conference on Singularities and Dynamical Systems, 7–14. Elsevier, 1985. http://dx.doi.org/10.1016/s0304-0208(08)72111-x.

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"GLOBAL BIFURCATIONS AT THE DISAPPEARANCE OF SADDLE-NODE EQUILIBRIUM STATES AND PERIODIC ORBITS." In World Scientific Series on Nonlinear Science Series A, 637–86. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812798558_0006.

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Conference papers on the topic "Saddle node bifurcation"

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Kim, Jinki, R. L. Harne, and K. W. Wang. "Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation." In ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/smasis2016-9051.

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Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and adaptive material/structure applications such as structural health monitoring and piezoelectric energy harvesting. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors towards the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept towards a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. Using this archetype example, the stochastic normal form of the saddle-node bifurcation is derived from which expressions of the escape statistics are formulated. Non-stationarity is accounted for using a time dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verify that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

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Hizanidis, Johanne, Roland Aust, and Eckehard Scho¨ll. "Delay-Induced Multistability in a Generic Model for Excitable Dynamics." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34329.

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Motivated by real-world excitable systems such as neuron models and lasers, we consider a paradigmatic model for excitability with a global bifurcation, namely a saddle-node bifurcation on a limit cycle. We study the effect of a time-delayed feedback force in the form of the difference between a system variable at a certain time and at a delayed time. In the absence of delay the only attractor in the system in the excitability regime, below the global bifurcation, is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling as well as saddle-node bifurcations of limit cycles are found in accordance with Shilnikov’s theorems.

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Pak, C. H., and Y. S. Choi. "On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34217.

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It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

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Luo, Albert C. J., and Chuan Guo. "Period-3 Motions in a Parametrically Exited Inverted Pendulum." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22176.

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Abstract In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

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Vitolo, R., H. W. Broer, and C. Simó. "The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms." In Proceedings of the International Conference on SPT 2007. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812776174_0050.

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Gendelman, O. V. "Degenerate Bifurcation Scenarios in the Dynamics of Coupled Oscillators With Symmetric Nonlinearities." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84373.

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We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. Both the potential of the oscillator and of the coupling spring are adopted to be even-power polynomials with nonnegative coefficients. By defining the coupling parameter ε, the dynamics of this system at the limit ε → 0 and for finite ε is investigated. Bifurcation scenario of the nonlinear normal modes is revealed. The degeneracy in the dynamics is manifested by a ‘bifurcation from infinity’ where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by mechanism of successive cusp catastrophes with growth of the coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincare sections). For particular case of pure cubic nonlinearity of the oscillator and the coupling spring good agreement between quantitative analytical predictions and numerical results is observed.

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Kang, Wonmo, Bryan Wilcox, Harry Dankowicz, and Phanikrishna Thota. "Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34441.

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This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

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Yan-Feng Jiang and Hsiao-Dong Chiang. "Saddle-node bifurcation in three-phase unbalanced distribution networks with distributed generators." In 2013 IEEE International Symposium on Circuits and Systems (ISCAS). IEEE, 2013. http://dx.doi.org/10.1109/iscas.2013.6571880.

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Fang, Yong, and Hong-geng Yang. "Saddle-node Bifurcation of Power Systems Analysis in the Simplest Normal Form." In 2012 International Conference on Computer Distributed Control and Intelligent Environmental Monitoring (CDCIEM). IEEE, 2012. http://dx.doi.org/10.1109/cdciem.2012.151.

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Yano, Takeru, Shigeo Fujikawa, and Tao Yu. "Reconsideration of Cavitation Inception Theory." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37177.

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The nonlinear dynamics of a spherical gas bubble in a liquid water is reconsidered on the basis of the Rayleigh-Plesset equation with particularly emphasis on the unstable behavior with respect to infinitesimal perturbations. The evolution of bubble radius after the discontinuous change of ambient pressure is theoretically analyzed, and the classical critical pressure and critical radius are re-derived as a saddle-node bifurcation point, when the center and saddle on the phase plane merge into a degenerate unstable singular point in the phase plane. Before the saddle-node bifurcation, there is a separatrix issuing from and entering into the saddle point in the inviscid limit. We propose a new criterion for cavitation inception: the ambient pressure that makes the separatrix pass through the initial bubble radius. This criterion gives a cavitation inception pressure higher than the classical one. The effects of viscosity and thermal conductivity are also discussed.

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