Shrimps: Occurrence, Scaling and Relevance

Martignoli, Stefan; Benner, Philipp; Stoop, Ruedi; Uwate, Yoko

doi:10.1142/s0218127412300327

Cited by 19 publications

(19 citation statements)

References 26 publications

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“…2(a)-(f) show large periodic regions (regions in black) and chaotic windows (yellow-redblue regions) with periodic structures (in black) embedded on them. As far as our knowledge, this feature is commonly found in almost every nonlinear dissipative dynamical systems [8,9,10,11,12,13,14,15,16,17,18,19,31]. Fig.…”

Section: Numerical Resultsmentioning

confidence: 96%

“…2) but a general feature was observed in some regions of these diagrams, namely the existence of periodic structures embedded in chaotic regions. These sets of periodic structures are presented in a wide range of nonlinear systems [9,15,16,17]. An exception is in high-dimensional systems with more than three-dimensions, where hyperchaotic behaviors can occur.…”

Section: Discussionmentioning

confidence: 99%

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Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

et al. 2014

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The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also present multistability shown in the planes of the basin of attractions.

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Section: Numerical Resultsmentioning

confidence: 96%

Section: Discussionmentioning

confidence: 99%

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

et al. 2014

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“…Only a few chaotic circuits have periodicity high resolution parameter spaces experimentally obtained. [1][2][3][4][5][6][7] The relevance of studying parameter spaces of nonlinear systems is that it allows us to understand how periodic behavior, chaos, and bifurcations come about in a nonlinear system. In fact, parameters leading to the different behaviors are strongly correlated.…”

Section: Introductionmentioning

confidence: 99%

Parameter space of experimental chaotic circuits with high-precision control parameters

Sousa

Rubinger

Sartorelli

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report high-resolution measurements that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit. Circuits constructed using this component allow for a comprehensive characterization of the circuit behaviors through high resolution parameter spaces. To illustrate the power of our technological development for the creation and the study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter space. The reliability and stability of the designed component allowed us to obtain data for long periods of time (∼21 weeks), a data set from which an accurate estimation of Lyapunov exponents for the circuit characterization was possible. Moreover, this data, rigorously characterized by the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally, we confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.

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“…For simplicity of the argument we will follow the second approach, using the exposition given in Refs. [14], [15]. The Hénon map can be written in its standard form as [16] f h : {x, y} → {c − dy − x 2 , x}.…”

Section: Featurementioning

confidence: 99%

Parameter properties of electronic and biological circuits and systems

Stoop

Gomez

Schonenberger

et al. 2013

2013 European Conference on Circuit Theory and Design (ECCTD)

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The question what behaviors can be exhibited by a given electronic circuit upon variation of the parameters, is fundamental to electrical engineering. To efficiently adapt behavior according to need, say from stability to instability, or among stable or unstable periodicities, it is crucial to know how systems (generically and specifically) depend on parameters. Shrimps or swallow-tails are generic, characteristic parameter space regions that yield fixed stable periodic behavior. By dividing the parameter space into stable and unstable dynamical behavior, they provide such guidelines. In applications, shrimps have first been described at great details in the context of laser systems and electronic circuits, but it is still unknown whether they could also be found in realistic models of biophysics (and if so, whether biology exploits this as an alternative computing paradigm). Here, we provide first explicit examples of their existence.

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Shrimps: Occurrence, Scaling and Relevance

Cited by 19 publications

References 26 publications

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

Parameter space of experimental chaotic circuits with high-precision control parameters

Parameter properties of electronic and biological circuits and systems

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