Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system

Kawczyński, Andrzej L.; Strizhak, P. E.

doi:10.1063/1.481222

Cited by 33 publications

(15 citation statements)

References 27 publications

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“…Koper [45] added a third variable to a planar system and used numerical continuation techniques [26,21] to study MMOs [45]. It is very important to note that equations similar or equivalent to (2) have been proposed many times independently by several different research groups [25,66,43,10,47,29] as a "canonical", "minimal" or "typical" model for MMOs. The version (2) of Koper's model was proposed by the author and co-workers in [15]; it is obtained by a coordinate transformation of Koper's original model and has the symmetry…”

Section: The Koper Modelmentioning

confidence: 99%

On decomposing mixed-mode oscillations and their return maps

Kuehn¹

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs has mainly focused on local mechanisms to explain the small oscillations. Numerical continuation studies reported different MMO patterns based on parameter variation. This paper aims at improving the link between local analysis and numerical simulation. Our starting point is a numerical study of a singular return map for the Koper model which is a prototypical example for MMOs that also relates to local normal form theory. We demonstrate that many MMO patterns can be understood geometrically by approximating the singular maps with affine and quadratic maps. Motivated by our numerical analysis we use abstract affine and quadratic return map models in combination with two local normal forms that generate small oscillations. Using this decomposition approach we can reproduce many classical MMO patterns and effectively decouple bifurcation parameters for local and global parts of the flow. The overall strategy we employ provides an alternative technique for understanding MMOs.Keywords: Fast-slow system, Koper model, return map, mixed-mode oscillations, local-global decomposition.Complex oscillatory patterns have been observed in a wide variety of applications. Analyzing these patterns from a dynamical perspective has been an active area of research for decades. However, several mathematical breakthroughs in the last 15 years have provided substantial additional insight into phenomena that describe local oscillations. In the present paper, we provide a numerical study of the singular Poincaré map in the Koper model. We demonstrate that many MMO patterns for the Koper model can already be understood just using approximations of singular limit maps. The results for the Koper model suggest that a local-global numerical simulation approach combining normal forms with discrete maps can be effective. We show that this abstract approach reproduces many typical MMO patterns that have been observed in applications. This methodology aims to close a gap between previous numerical studies of MMO patterns and analytical results about local normal forms.

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Section: The Koper Modelmentioning

confidence: 99%

On decomposing mixed-mode oscillations and their return maps

Kuehn¹

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The system attained an equilibrium in the manner of Sniper bifurcation, which was carefully discussed [1]. The chemical reactions, used by us in our previous article [1] to describe Hopf and Sniper bifurcations, were not able to reproduce the mixed mode region, but the simple theoretical model [2,3] used previously [4] has done it quite well and it is shown in the work.…”

Section: Introductionmentioning

confidence: 81%

“…The simple three-variable model [2,3] which qualitatively describes various asymptotic MMO observed in the BZ reaction, has been studied recently. This model was also helpful in the successful searches of new types of MMO with LS n s m [4] patterns:…”

Section: Mixed Mode Oscillationsmentioning

confidence: 99%

Mixed Mode, Sequential and Relaxation Oscillations in the Belousov-Zhabotinsky System

Rachwalska

2007

Zeitschrift Für Naturforschung A

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The behaviour of a system composed of malonic acid (MA), KBrO 3 , H 2 SO 4 , and ferroin was investigated in batch experiments at various concentrations of oxygen above the chemical mixture when changing the concentration of MA. We could observe that at 10% of oxygen or more and for initial concentrations of malonic acid [MA] 0 between 0.15 M and 0.6 M the system attains an equilibrium by some of mixed mode oscillations. Such a behaviour of the system could be described by a model used lately. Additionally for [MA] 0 = 0.1 M or less at 0 -20% of oxygen a region of sequential oscillations was discovered (observed for the first time in the system with ferroin) and a trial of understanding of the event is suggested. For rather small [MA] 0 ca. 0.025 M no mixed mode oscillations, no sequential ones but the so-called relaxation oscillations were observed.

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“…Recently, we have introduced a simple three-variable model that describes LS i mixed-mode oscillations in chemical systems. 13,[18][19][20] We have illustrated that adding a fourth variable, which dynamics is not affected by remaining variables, describes LS i transient patterns observed in the BZ reaction in batch 19 and CSTR. 20 In the present paper we show that the three-variable model is also capable to describe the LS i s j patterns.…”

Section: Introductionmentioning

confidence: 95%

Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system

Kawczyński

Khavrus

Strizhak

2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A detailed study of a generic model exhibiting new type of mixed-mode oscillations is presented. Period doubling and various period adding sequences of bifurcations are observed. New type of a family of 1D (one-dimensional) return maps is found. The maps are discontinuous at three points and consist of four branches. They are not invertible. The model describes in a qualitative way mixed-mode oscillations with two types of small amplitude oscillations at local maxima and local minima of large amplitude oscillations, which have been observed recently in the Belousov-Zhabotinsky system. (c) 2000 American Institute of Physics.

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Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system

Cited by 33 publications

References 27 publications

On decomposing mixed-mode oscillations and their return maps

On decomposing mixed-mode oscillations and their return maps

Mixed Mode, Sequential and Relaxation Oscillations in the Belousov-Zhabotinsky System

Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system

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