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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: 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.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.


Related in: MedlinePlus

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: 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.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.


Related in: MedlinePlus