Mixed‐mode oscillations in chemical systems

1992 ◽  
Vol 97 (9) ◽  
pp. 6191-6198 ◽  
Author(s):  
Valery Petrov ◽  
Stephen K. Scott ◽  
Kenneth Showalter
Keyword(s):  
Mixed Mode ◽  
Chemical Systems ◽  
Mixed Mode Oscillations

Period doubling and other complex bifurcations in non-isothermal chemical systems

1990 ◽  
Vol 332 (1624) ◽  
pp. 51-68 ◽  
Keyword(s):  
Mixed Mode ◽  
Period Doubling ◽  
Heat Rate ◽  
Oscillatory Behaviour ◽  
Chemical Heat ◽  
Experimental Conditions ◽  
Chemical Systems ◽  
Mixed Mode Oscillations ◽  
Self Heating ◽  
Thermal Feedback

Chemical feedback in the form of chain-branching or autocatalysis can give rise to oscillatory behaviour in very simple models involving only two variables. Many chemical reactions are also exothermic. This chemical heat release can give rise to self-heating and hence to thermal feedback, where the temperature varies as well as the concentrations. When chemical and thermal feedback are coupled, the range of responses that can be observed are increased dramatically. These features are demonstrated through the simple non-isothermal autocatalator scheme p-> A rate = A + 2B-> 3B rate = k1 2, A-^ B rate = kza, C + heat rate = At its simplest, the reaction can be steady or can show simple period-1 oscillations. More complex oscillations, with higher periodicity appear as the experimental conditions are varied, with period doubling, mixed-mode oscillations and aperiodicity (chemical chaos).

Complex Oscillations and Chaos

1998 ◽  
Author(s):  
Irving R. Epstein ◽  
John A. Pojman
Keyword(s):  
Mixed Mode ◽  
Dynamical Behavior ◽  
Periodic Oscillation ◽  
Periodic Behavior ◽  
Bz Reaction ◽  
Chemical Systems ◽  
Mixed Mode Oscillations ◽  
Dynamical Complexity ◽  
Complex Modes ◽  
Temporal Oscillation

After studying the first seven chapters of this book, the reader may have come to the conclusion that a chemical reaction that exhibits periodic oscillation with a single maximum and a single minimum must be at or near the apex of the pyramid of dynamical complexity. In the words of the song that is sung at the Jewish Passover celebration, the Seder, “Dayenu” (It would have been enough). But nature always has more to offer, and simple periodic oscillation is only the beginning of the story. In this chapter, we will investigate more complex modes of temporal oscillation, including both periodic behavior (in which each cycle can have several maxima and minima in the concentrations) and aperiodic behavior, or chaos (in which no set of concentrations is ever exactly repeated, but the system nonetheless behaves deterministically). Most people who study periodic behavior deal with linear oscillators and therefore tend to think of oscillations as sinusoidal. Chemical oscillators are, as we have seen, decidedly nonlinear, and their waveforms can depart quite drastically from being sinusoidal. Even after accepting that chemical oscillations can look as nonsinusoidal as the relaxation oscillations shown in Figure 4.4, our intuition may still resist the notion that a single period of oscillation might contain two, three, or perhaps twenty-three, maxima and minima. As an example, consider the behavior shown in Figure 8.1, where the potential of a bromide-selective electrode in the BZ reaction in a CSTR shows one large and two small extrema in each cycle of oscillation. The oscillations shown in Figure 8.1 are of the mixed-mode type, in which each period contains a mixture of large-amplitude and small-amplitude peaks. Mixedmode oscillations are perhaps the most commonly occurring form of complex oscillations in chemical systems. In order to develop some intuitive feel for how such behavior might arise, we employ a picture based on slow manifolds and utilized by a variety of authors (Boissonade, 1976; Rössler, 1976; Rinzel, 1987; Barkley, 1988) to analyze mixed-mode oscillations and other forms of complex dynamical behavior.

scholarly journals Complexity of a Peroxidase-Oxidase Reaction Model

2021 ◽  
Author(s):  
Jason Gallas ◽  
Marcus Hauser ◽  
Lars Folke Olsen
Keyword(s):  
Chemical Reaction ◽  
Mixed Mode ◽  
Period Doubling ◽  
Reaction Model ◽  
Chaotic Behaviour ◽  
Mixed Mode Oscillations ◽  
Oscillating Reaction ◽  
Oxidase Reaction

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study...

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

2018 ◽  
Vol 32 (05) ◽  
pp. 1850043 ◽  
Author(s):  
Feibiao Zhan ◽  
Shenquan Liu ◽  
Jing Wang ◽  
Bo Lu
Keyword(s):  
Purkinje Cell ◽  
Mixed Mode ◽  
Cell Model ◽  
Dynamical Behavior ◽  
Characteristic Index ◽  
Mixed Mode Oscillations ◽  
Codimension One ◽  
Bifurcation Phenomenon ◽  
Lyapunov Coefficient ◽  
Potential Link

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

Intermittency, Secondary Bifurcation and Mixed Mode Oscillations in a Swirl-Stabilized Annular Combustor: Experiments and Modeling

2021 ◽  
Author(s):  
Samarjeet Singh ◽  
Amitesh Roy ◽  
K V Reeja ◽  
Asalatha A. S. Nair ◽  
Swetaprovo Chaudhuri ◽  
...  
Keyword(s):  
Mixed Mode ◽  
Mixed Mode Oscillations ◽  
Annular Combustor ◽  
Secondary Bifurcation

One variable map prediction of Belousov–Zhabotinskii mixed mode oscillations

1984 ◽  
Vol 80 (11) ◽  
pp. 5610-5615 ◽  
Author(s):  
John Rinzel ◽  
Ira B. Schwartz
Keyword(s):  
Mixed Mode ◽  
Mixed Mode Oscillations
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