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).

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...

Chaos via mixed-mode oscillations

1991 ◽  
Vol 337 (1646) ◽  
pp. 291-298 ◽  
Keyword(s):  
Mixed Mode ◽  
Phase Locking ◽  
Period Doubling ◽  
Specific Model ◽  
Dynamical Behaviour ◽  
Object A ◽  
Mixed Mode Oscillations ◽  
Catalysed Oxidation ◽  
Periodic State ◽  
Peroxidase Enzyme

The route by which chaos arises from mixed-mode periodic states in a model of the peroxidase enzyme catalysed oxidation of NADH is described. The specific model studied displays a rich variety of exotic dynamical behaviour including simple oscillations, quasiperiodicity, bistability between periodic states, complex periodic oscillations (including the mixed-mode type) and chaos. The route to chaos in this system involves a torus attractor which becomes destabilized and breaks up into a fractal object, a strange attractor. The mixed-mode states correspond to phase-locking on this fractal attractor and are arranged in staircases according to the complexity of the state. In this paper, we investigate the sequence leading from a mixed-mode periodic state to a chaotic one in the staircase region and find a familiar cascade of period-doubling bifurcations, which finally culminate in chaos.

Symmetric Fold/Super-Hopf Bursting, Chaos and Mixed-Mode Oscillations in Pernarowski Model of Pancreatic Beta-Cells

2016 ◽  
Vol 26 (09) ◽  
pp. 1630022 ◽  
Author(s):  
Haniyeh Fallah
Keyword(s):  
Blood Glucose ◽  
Blood Glucose Level ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Current System ◽  
Period Doubling ◽  
Slow Manifold ◽  
Mixed Mode Oscillations ◽  
Route To Chaos

Pancreatic beta-cells produce insulin to regularize the blood glucose level. Bursting is important in beta cells due to its relation to the release of insulin. Pernarowski model is a simple polynomial model of beta-cell activities indicating bursting oscillations in these cells. This paper presents bursting behaviors of symmetric type in this model. In addition, it is shown that the current system exhibits the phenomenon of period doubling cascades of canards which is a route to chaos. Canards are also observed symmetrically near folds of slow manifold which results in a chaotic transition between [Formula: see text] and [Formula: see text] spikes symmetric bursting. Furthermore, mixed-mode oscillations (MMOs) and combination of symmetric bursting together with MMOs are illustrated during the transition between symmetric bursting and continuous spiking.

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

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 Autocatalytic and oscillatory reaction networks that form guanidines and products of their cyclization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Alexander I. Novichkov ◽  
Anton I. Hanopolskyi ◽  
Xiaoming Miao ◽  
Linda J. W. Shimon ◽  
Yael Diskin-Posner ◽  
...  
Keyword(s):  
Chemical Cues ◽  
Native Chemical Ligation ◽  
Reaction Networks ◽  
Oscillatory Reaction ◽  
Experimental Conditions ◽  
Chemical Ligation ◽  
Chemical Systems ◽  
Life On Earth ◽  
Emergence Of Life ◽  
Equilibrium Chemical

AbstractAutocatalytic and oscillatory networks of organic reactions are important for designing life-inspired materials and for better understanding the emergence of life on Earth; however, the diversity of the chemistries of these reactions is limited. In this work, we present the thiol-assisted formation of guanidines, which has a mechanism analogous to that of native chemical ligation. Using this reaction, we designed autocatalytic and oscillatory reaction networks that form substituted guanidines from thiouronium salts. The thiouronium salt-based oscillator show good stability of oscillations within a broad range of experimental conditions. By using nitrile-containing starting materials, we constructed an oscillator where the concentration of a bicyclic derivative of dihydropyrimidine oscillates. Moreover, the mixed thioester and thiouronium salt-based oscillator show unique responsiveness to chemical cues. The reactions developed in this work expand our toolbox for designing out-of-equilibrium chemical systems and link autocatalytic and oscillatory chemistry to the synthesis of guanidinium derivatives and the products of their transformations including analogs of nucleobases.

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.

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