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Discontinuous spirals of stable periodic oscillations.

Sack A, Freire JG, Lindberg E, Pöschel T, Gallas JA - Sci Rep (2013)

Bottom Line: This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators.Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point.They are useful to optimize locking into desired oscillatory modes and to control complex systems.

View Article: PubMed Central - PubMed

Affiliation: Institute for Multiscale Simulations, Friedrich-Alexander Universität, D-91052 Erlangen, Germany.

ABSTRACT
We report the experimental discovery of a remarkable organization of the set of self-generated periodic oscillations in the parameter space of a nonlinear electronic circuit. When control parameters are suitably tuned, the wave pattern complexity of the periodic oscillations is found to increase orderly without bound. Such complex patterns emerge forming self-similar discontinuous phases that combine in an artful way to produce large discontinuous spirals of stability. This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators. Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point. They are useful to optimize locking into desired oscillatory modes and to control complex systems. The organization of oscillations into discontinuous spirals is expected to be generic for a class of nonlinear oscillators.

No MeSH data available.


Comparison of structural details between (a) the new discontinuous spiral composed of the terminations shown in Fig. [6](a), and (b) an example of the known continuous spiral, composed of the “shrimp” in Fig. [6](b), here displaying Lyapunov exponents for a model of an abstract chemical reaction3.Both spirals have a rather distinct structure and cannot be deformed one into the other.
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f5: Comparison of structural details between (a) the new discontinuous spiral composed of the terminations shown in Fig. [6](a), and (b) an example of the known continuous spiral, composed of the “shrimp” in Fig. [6](b), here displaying Lyapunov exponents for a model of an abstract chemical reaction3.Both spirals have a rather distinct structure and cannot be deformed one into the other.

Mentions: Figure [2] illustrates five typical wave patterns x(t) obtained numerically for our circuit, together with their respective return maps, as indicated in the figure caption. The patterns labeled A, B, C display multipeaked periodic oscillations while patterns a and b illustrate chaotic oscillations. Patterns A, B, C show clearly that the Duffing-like proxy displays precisely the same antiperiodic oscillations that were discovered recently in a different experiment4. In contrast with that earlier work, which is governed by four differential equations, the Duffing-like proxy generates antiperiodicity in a considerably simpler setup, involving just three equations. In panels A, B, C of Fig. [2] we also compare numerically computed with experimentally measured return maps (shown in red) for the maxima in the x(t) variable: x(tmax) × x(tmax+1). They illustrate the very good agreement obtained between measurements and simulations. Similar agreement is found for the other variables (not shown). The periodic waveforms A, B, C were obtained for (b, k) = (0.3646, 2.114), (0.3481, 1.997) and (0.3373, 1.914), and have periods TA = 42.696, TB = 50.175, and TC = 57.924 ms, respectively. The corresponding experimental points are (b, k) = (0.4906, 1.976), (0.4766, 1.892), and (0.4669, 1.804). The waveforms labeled a and b illustrate non-periodic (chaotic) oscillations obtained for (b, k) = (0.356, 2.05) and (0.342, 1.95), respectively. The location in parameter space of all points A, a, B, b, C is indicated in Fig. [5](a) and serve to indicate how the oscillations unfold when moving towards the focal point.


Discontinuous spirals of stable periodic oscillations.

Sack A, Freire JG, Lindberg E, Pöschel T, Gallas JA - Sci Rep (2013)

Comparison of structural details between (a) the new discontinuous spiral composed of the terminations shown in Fig. [6](a), and (b) an example of the known continuous spiral, composed of the “shrimp” in Fig. [6](b), here displaying Lyapunov exponents for a model of an abstract chemical reaction3.Both spirals have a rather distinct structure and cannot be deformed one into the other.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3842083&req=5

f5: Comparison of structural details between (a) the new discontinuous spiral composed of the terminations shown in Fig. [6](a), and (b) an example of the known continuous spiral, composed of the “shrimp” in Fig. [6](b), here displaying Lyapunov exponents for a model of an abstract chemical reaction3.Both spirals have a rather distinct structure and cannot be deformed one into the other.
Mentions: Figure [2] illustrates five typical wave patterns x(t) obtained numerically for our circuit, together with their respective return maps, as indicated in the figure caption. The patterns labeled A, B, C display multipeaked periodic oscillations while patterns a and b illustrate chaotic oscillations. Patterns A, B, C show clearly that the Duffing-like proxy displays precisely the same antiperiodic oscillations that were discovered recently in a different experiment4. In contrast with that earlier work, which is governed by four differential equations, the Duffing-like proxy generates antiperiodicity in a considerably simpler setup, involving just three equations. In panels A, B, C of Fig. [2] we also compare numerically computed with experimentally measured return maps (shown in red) for the maxima in the x(t) variable: x(tmax) × x(tmax+1). They illustrate the very good agreement obtained between measurements and simulations. Similar agreement is found for the other variables (not shown). The periodic waveforms A, B, C were obtained for (b, k) = (0.3646, 2.114), (0.3481, 1.997) and (0.3373, 1.914), and have periods TA = 42.696, TB = 50.175, and TC = 57.924 ms, respectively. The corresponding experimental points are (b, k) = (0.4906, 1.976), (0.4766, 1.892), and (0.4669, 1.804). The waveforms labeled a and b illustrate non-periodic (chaotic) oscillations obtained for (b, k) = (0.356, 2.05) and (0.342, 1.95), respectively. The location in parameter space of all points A, a, B, b, C is indicated in Fig. [5](a) and serve to indicate how the oscillations unfold when moving towards the focal point.

Bottom Line: This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators.Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point.They are useful to optimize locking into desired oscillatory modes and to control complex systems.

View Article: PubMed Central - PubMed

Affiliation: Institute for Multiscale Simulations, Friedrich-Alexander Universität, D-91052 Erlangen, Germany.

ABSTRACT
We report the experimental discovery of a remarkable organization of the set of self-generated periodic oscillations in the parameter space of a nonlinear electronic circuit. When control parameters are suitably tuned, the wave pattern complexity of the periodic oscillations is found to increase orderly without bound. Such complex patterns emerge forming self-similar discontinuous phases that combine in an artful way to produce large discontinuous spirals of stability. This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators. Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point. They are useful to optimize locking into desired oscillatory modes and to control complex systems. The organization of oscillations into discontinuous spirals is expected to be generic for a class of nonlinear oscillators.

No MeSH data available.