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Slow Passage Through a Pitchfork Bifurcation

G. J. M. Marée
1996 SIAM Journal on Applied Mathematics  
Neishtadt (1987 Neishtadt ( , 1988 and Baer, Erneux, and Rinzel (1989) concentrate on the slow passage through a Hopf bifurcation from a stable steady state to a stable time-periodic solution and demonstrate  ...  In literature this phenomenon has commonly been called a pitchfork bifurcation. This pitchfork bifurcation is illustrated in Figures 2 and 3 .  ... 
doi:10.1137/s0036139993257399 fatcat:a5t6olohqza3pcxdq67ogzc2re

Page 6577 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews  
37079 37520 34C23 34C37 70HOS 70K44 Haberman, Richard (1-SMU; Dallas, TX) Slow passage through the nonhyperbolic homoclinic orbit associated with a subcritical pitchfork bifurcation for Hamiltonian systems  ...  Slow passages through the nonhy- perbolic homoclinic orbit, and the resulting change of action, are studied and compared with slow passages through hyperbolic ho- moclinic orbits.  ... 

A Generalized 2D-Dynamical Mean-Field Ising Model with a Rich Set of Bifurcations (Inspired and Applied to Financial Crises)

Damian Smug, Didier Sornette, Peter Ashwin
2018 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
Finally, we examine several transitions through bifurcation curves and study how they could be understood as specific market scenarios.  ...  We show the existence of a rich set of bifurcations as a function of the two parameters quantifying the relative importance of instantaneous versus past social opinions on the formation of the next value  ...  For instance, if a shift occurs from a fixed point Linear passages through bifurcations as described by the system(13).  ... 
doi:10.1142/s0218127418300100 fatcat:ffnvekihjfesbbjbpvn652anfq

A Generalized 2D-Dynamical Mean-Field Ising Model with a Rich Set of Bifurcations (Inspired and Applied to Financial Crises)

Damian Smug, Didier Sornette, Peter Ashwin
2017 Social Science Research Network  
Finally, we examine several transitions through bifurcation curves and study how they could be understood as specific market scenarios.  ...  We show the existence of a rich set of bifurcations as a function of the two parameters quantifying the relative importance of instantaneous versus past social opinions on the formation of the next value  ...  For instance, if a shift occurs from a fixed point Linear passages through bifurcations as described by the system(13).  ... 
doi:10.2139/ssrn.3064673 fatcat:wv2rf26jxjbsfb7fl225oe4oqm

Exploring the Mechanisms of Differentiation, Dedifferentiation, Reprogramming and Transdifferentiation

Li Xu, Kun Zhang, Jin Wang, Yuin-Han Loh
2014 PLoS ONE  
We also classified the mechanisms of cell fate development from our landscape theory: super-critical pitchfork bifurcation, sub-critical pitchfork bifurcation, sub-critical pitchfork with two saddle-node  ...  The dedifferentiation process proceeds through a pluripotent cell state.  ...  A saddle-node bifurcation denotes a collision and disappearance of two equilibria rather than a pitchfork bifurcation [33, 35] .  ... 
doi:10.1371/journal.pone.0105216 pmid:25133589 pmcid:PMC4136825 fatcat:24i3jiqbrbfslck2avrptyvqkq

Multi-agent decision-making dynamics inspired by honeybees [article]

Rebecca Gray and Alessio Franci and Vaibhav Srivastava and Naomi Ehrich Leonard
2018 arXiv   pre-print
To explore and generalize these features to other networks, we design distributed multi-agent network dynamics that exhibit a pitchfork bifurcation, ubiquitous in biological models of decision-making.  ...  We further present a distributed adaptive bifurcation control law and prove how it enhances the network decision-making performance beyond that observed in swarms.  ...  to a decision through a pitchfork bifurcation.  ... 
arXiv:1711.11578v2 fatcat:iw3xl7fcjrfhhah4ebkylsiueu

Synchronization of weakly coupled canard oscillators

Elif Köksal Ersöz, Mathieu Desroches, Martin Krupa
2017 Physica D : Non-linear phenomena  
Phase plane analysis of slow-fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard.  ...  by a small perturbation.  ...  The IP solution undergoes a pitchfork bifurcation through which it loses its stability as a stable OP solution appears (Panel (b) ).  ... 
doi:10.1016/j.physd.2017.02.016 fatcat:abytkylv7nghjf65c3xqvfwwke

Non-equilibrium transitions in multiscale systems with a bifurcating slow manifold

Tobias Grafke, Eric Vanden-Eijnden
2017 Journal of Statistical Mechanics: Theory and Experiment  
It is shown that these non-equilibrium transitions make use of a reaction channel created by the bifurcation structure of the slow manifold, leading to vastly increased transition rates.  ...  Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation  ...  Pitchfork bifurcation A pitchfork bifurcation is another example of a low-dimensional bifurcation structure.  ... 
doi:10.1088/1742-5468/aa85cb fatcat:kcg5yhxprbcrxfhgop26ybu2hi

An Organizing Center in a Planar Model of Neuronal Excitability

Alessio Franci, Guillaume Drion, Rodolphe Sepulchre
2012 SIAM Journal on Applied Dynamical Systems  
Excitability is explored by unfolding a pitchfork bifurcation that is shown to organize five different types of excitability.  ...  The paper studies the excitability properties of a generalized FitzHugh-Nagumo model.  ...  The transition from Region I to Region IV is through a transcritical bifurcation at which a saddle and a node exchange their stability.  ... 
doi:10.1137/120875016 fatcat:sdmugvyh3rfyjpgzwf2x3mqpka

An organizing center in a planar model of neuronal excitability [article]

Alessio Franci, Guillaume Drion, Rodolphe Sepulchre
2012 arXiv   pre-print
Excitability is explored by unfolding a pitchfork bifurcation that is shown to organize five different types of excitability.  ...  The paper studies the excitability properties of a generalized FitzHugh-Nagumo model.  ...  The transition from Region I to Region IV is through a transcritical bifurcation at which a saddle and a node exchange their stability.  ... 
arXiv:1204.5686v1 fatcat:6lke7epwlzh3njtkke3yrimyha

Saddle-Node Bifurcation and Vibrational Resonance in a Fractional System with an Asymmetric Bistable Potential

J. H. Yang, Miguel A. F. Sanjuán, F. Tian, H. F. Yang
2015 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
When the asymmetric parameter vanishes, the saddle-node bifurcation turns into a pitchfork bifurcation. There are three kinds of vibrational resonance existing in the system.  ...  We investigate the saddle-node bifurcation and vibrational resonance in a fractional system that has an asymmetric bistable potential.  ...  In (b), the pitchfork bifurcation appears. fractional-order is a supercritical pitchfork bifurcation.  ... 
doi:10.1142/s0218127415500236 fatcat:sz7nbt3k7rfapa6b75mr6ekmyi

Discrete breathers in dissipative lattices

J. L. Marín, F. Falo, P. J. Martínez, L. M. Floría
2001 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
Using Floquet spectral analysis, we give a complete characterization of the most relevant bifurcations, and we find (spatial) symmetry-breaking bifurcations which are linked to breather mobility, just  ...  For instance, it is possible to form "bound states" of moving breathers, through the interaction of their phonon tails.  ...  J.L.M. acknowledges a Return Grant from the Spanish MEC.  ... 
doi:10.1103/physreve.63.066603 pmid:11415238 fatcat:lcr5d45clzboja3zilb7egyr6m

Page 7532 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
The systems exhibit passage through various bifurcations. Results of the normal form analysis are compared with numerical solutions. James A.  ...  (l-PURD-SME; West Lafayette, IN) On the non-stationary passage through bifurcations in resonantly forced Hamiltonian oscillators. (English summary) Internat. J.  ... 

The effect of noise on pitchfork and Hopf bifurcations

A. Juel, A. G. Darbyshire, T. Mullin
1997 Proceedings of the Royal Society A  
We present the results of an experimental and numerical investigation of the effects of noise on pitchfork and Hopf bifurcations.  ...  In the case of the pitchfork we find that natural imperfections override the effects of the noise. However, novel noise amplification effects have been uncovered in the study of the Hopf bifurcation.  ...  , the disconnection has a profound influence on the dynamics depending on the rate that the control parameter is swept through the bifurcation.  ... 
doi:10.1098/rspa.1997.0140 fatcat:gflt5yzn3rc7bnyk6n3stbjtcy

Page 4212 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
Summary: “We consider the following problem as a model for the slow passage through a steady bifurcation: dy/dt = A(t)y — y>+ 6, where A is a slowly increasing function of ¢ given by 4 =A; + 58 GLOBAL  ...  Bruce Stewart (Upton, N.Y.) 88h:58082 58F14 Mandel, Paul (B-ULB-P); Erneux, Thomas (1-NW-F) The slow passage through a steady bifurcation: delay and memory effects. J. Statist.  ... 
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