Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction

Broens, Morten; Bar‐Eli, K.

doi:10.1021/j100175a053

Cited by 125 publications

(101 citation statements)

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“…However, in certain circumstances the periodic oscillations may follow the unstable part of Σ after the fold. This is the hallmark of the so-called canard phenomenon, and related behavior has been seen, for example, in model equations describing chemical systems [8,35]. Like the present system these systems also exhibit abrupt transitions from small amplitude oscillations to large amplitude relaxation-type oscillations, a behavior that has been called a canard explosion [8,18,35].…”

Section: Case Ii: Relaxation Oscillations and Canardsmentioning

confidence: 59%

Dynamics of nearly inviscid Faraday waves in almost circular containers

Higuera

Knobloch

Vega

2005

Physica D: Nonlinear Phenomena

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Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting 'canards'.

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Section: Case Ii: Relaxation Oscillations and Canardsmentioning

confidence: 59%

Dynamics of nearly inviscid Faraday waves in almost circular containers

Higuera

Knobloch

Vega

2005

Physica D: Nonlinear Phenomena

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“…The square-rootshaped increase of the amplitude can be observed only for a values extremely close to the bifurcation point; for a only slightly larger the amplitude increases very rapidly. This phenomenon is known as canard explosion or canard transition (Benoit et al 1981;Broens & Bar-Eli 1991;Peng et al 1991). The effect is influenced by the time-scale separation ε of the two variables: the smaller ε, the smaller the bifurcation parameter range for which the square-root-shaped increase is observable.…”

Section: (B) the Oscillatory Regimementioning

confidence: 99%

Interplay of time-delayed feedback control and temporally correlated noise in excitable systems

Brandstetter

Dahlem

Schöll

2010

Phil. Trans. R. Soc. A.

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The interplay of time-delayed feedback and temporally correlated coloured noise in a single and two coupled excitable systems is studied in the framework of the FitzHughNagumo (FHN) model. By using coloured noise instead of white noise, the noise correlation time is introduced as an additional time scale. We show that in a single FHN system the major time scale of oscillations is strongly influenced by the noise correlation time, which in turn affects the maxima of coherence with respect to the delay time. In two coupled FHN systems, coloured noise input to one subsystem influences coherence resonance and stochastic synchronization of both subsystems. Application of delayed feedback control to the coloured noise-driven subsystem is shown to change coherence and time scales of noise-induced oscillations in both systems, and to enhance or suppress stochastic synchronization under certain conditions.

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“…Then, increasing again the bromate concentration, the system will stay in a steady state until the dashed line is reached, when the oscillations will start abruptly (subcritical Hopf bifurcation). The dotted line in Figure 1 indicates a canard-like behavior line 59,60 in which the system changes drastically the oscillation amplitude as can be seen in Figure 2. Classification of this phenomena as a canard explosion requires investigation of the presence of hysteresis and excitation.…”

Section: Resultsmentioning

confidence: 86%

Phase diagram and complex patterns in the modeling of the bromate-oxalic acid-Ce-acetone oscillating reaction

Pereira¹,

Faria²

2007

Quím. Nova

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Recebido em 23/1/06; aceito em 3/7/06; publicado na web em 7/2/07Simulations have been carried out on the bromate -oxalic acid -Ce(IV) -acetone oscillating reaction, under flow conditions, using Field and Boyd's model (J. Phys. Chem. 1985, 89, 3707). Many different complex dynamic behaviors were found, including simple periodic oscillations, complex periodic oscillations, quasiperiodicity and chaos. Some of these complex oscillations can be understood as belonging to a Farey sequence. The many different behaviors were systematized in a phase diagram which shows that some regions of complex patterns were nested with one inside the other. The existence of almost all known dynamic behavior for this system allows the suggestion that it can be used as a model for some very complex phenomena that occur in biological systems.

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Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction

Cited by 125 publications

References 1 publication

Dynamics of nearly inviscid Faraday waves in almost circular containers

Dynamics of nearly inviscid Faraday waves in almost circular containers

Interplay of time-delayed feedback control and temporally correlated noise in excitable systems

Phase diagram and complex patterns in the modeling of the bromate-oxalic acid-Ce-acetone oscillating reaction

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