Limits...
Antiperiodic oscillations.

Freire JG, Cabeza C, Marti A, Pöschel T, Gallas JA - Sci Rep (2013)

Bottom Line: The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences.In general, regularity is understood as tantamount to periodicity.However, there is now a flurry of works proving the existence of "antiperiodicity", an unfamiliar type of regularity.

View Article: PubMed Central - PubMed

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

ABSTRACT
The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences. In general, regularity is understood as tantamount to periodicity. However, there is now a flurry of works proving the existence of "antiperiodicity", an unfamiliar type of regularity. Here we report the experimental observation and numerical corroboration of antiperiodic oscillations. In contrast to the isolated solutions presently known, we report infinite hierarchies of antiperiodic waveforms that can be tuned continuously and that form wide spiral-shaped stability phases in the control parameter plane. The waveform complexity increases towards the focal point common to all spirals, a key hub interconnecting them all.

No MeSH data available.


Related in: MedlinePlus

(a) The spiral phase of self-sustained antiperiodic oscillations. Colors denote the number of peaks within one period of v2(t). Black denotes chaos, i.e. lack of numerically detectable repetitions. (b) Magnification of the box in (a) illustrating turning points with high odd-number of spikes (given by the numbers). Note the strong compression of the spiral phase embedded in the wide black background of chaos. (c) Magnification of the box in (b) showing the monotonous convergence towards the focal hub, the accumulation point approached when cycling the spiral anti-clockwisely, where periodic oscillations should have an infinite number of peaks within one period. Each individual panel displays the analysis of 2400 × 2400 = 5.76 × 106 parameter points. The resistances R1 and R2 are measured in Ω.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3675356&req=5

f4: (a) The spiral phase of self-sustained antiperiodic oscillations. Colors denote the number of peaks within one period of v2(t). Black denotes chaos, i.e. lack of numerically detectable repetitions. (b) Magnification of the box in (a) illustrating turning points with high odd-number of spikes (given by the numbers). Note the strong compression of the spiral phase embedded in the wide black background of chaos. (c) Magnification of the box in (b) showing the monotonous convergence towards the focal hub, the accumulation point approached when cycling the spiral anti-clockwisely, where periodic oscillations should have an infinite number of peaks within one period. Each individual panel displays the analysis of 2400 × 2400 = 5.76 × 106 parameter points. The resistances R1 and R2 are measured in Ω.

Mentions: To understand how antiperiodic patterns depend on R1 and R2 we performed an additional numerical experiment, studying the variation of the number of peaks systematically on a 2400 × 2400 = 5.76 × 106 rectangular grid of equally spaced parameter points. The circuit equations were integrated with a standard fourth-order Runge-Kutta algorithm with fixed time-step h = 10−6 s, starting computations always from a fixed initial condition v1 = 8 V, v2 = −5 V, i1 = −1 mA, i2 = 3 mA. The first 80 × 105 integration steps were discarded as transient. The chaotic/periodic/antiperiodic nature of solutions was determined and recorded in so-called isospike diagrams16: after the transient we integrated for an additional 80 × 105 time-steps and recorded extrema (maxima and minima) of a given variable of interest, up to 800 extrema, counting the number of peaks and checking whether for repetitions. Such high-resolution computations are numerically very demanding and, therefore, were performed on a SGI Altix cluster of 1536 high-performance processors running during a period of several weeks to compute many stability diagrams, three of them presented in Fig. 4.


Antiperiodic oscillations.

Freire JG, Cabeza C, Marti A, Pöschel T, Gallas JA - Sci Rep (2013)

(a) The spiral phase of self-sustained antiperiodic oscillations. Colors denote the number of peaks within one period of v2(t). Black denotes chaos, i.e. lack of numerically detectable repetitions. (b) Magnification of the box in (a) illustrating turning points with high odd-number of spikes (given by the numbers). Note the strong compression of the spiral phase embedded in the wide black background of chaos. (c) Magnification of the box in (b) showing the monotonous convergence towards the focal hub, the accumulation point approached when cycling the spiral anti-clockwisely, where periodic oscillations should have an infinite number of peaks within one period. Each individual panel displays the analysis of 2400 × 2400 = 5.76 × 106 parameter points. The resistances R1 and R2 are measured in Ω.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3675356&req=5

f4: (a) The spiral phase of self-sustained antiperiodic oscillations. Colors denote the number of peaks within one period of v2(t). Black denotes chaos, i.e. lack of numerically detectable repetitions. (b) Magnification of the box in (a) illustrating turning points with high odd-number of spikes (given by the numbers). Note the strong compression of the spiral phase embedded in the wide black background of chaos. (c) Magnification of the box in (b) showing the monotonous convergence towards the focal hub, the accumulation point approached when cycling the spiral anti-clockwisely, where periodic oscillations should have an infinite number of peaks within one period. Each individual panel displays the analysis of 2400 × 2400 = 5.76 × 106 parameter points. The resistances R1 and R2 are measured in Ω.
Mentions: To understand how antiperiodic patterns depend on R1 and R2 we performed an additional numerical experiment, studying the variation of the number of peaks systematically on a 2400 × 2400 = 5.76 × 106 rectangular grid of equally spaced parameter points. The circuit equations were integrated with a standard fourth-order Runge-Kutta algorithm with fixed time-step h = 10−6 s, starting computations always from a fixed initial condition v1 = 8 V, v2 = −5 V, i1 = −1 mA, i2 = 3 mA. The first 80 × 105 integration steps were discarded as transient. The chaotic/periodic/antiperiodic nature of solutions was determined and recorded in so-called isospike diagrams16: after the transient we integrated for an additional 80 × 105 time-steps and recorded extrema (maxima and minima) of a given variable of interest, up to 800 extrema, counting the number of peaks and checking whether for repetitions. Such high-resolution computations are numerically very demanding and, therefore, were performed on a SGI Altix cluster of 1536 high-performance processors running during a period of several weeks to compute many stability diagrams, three of them presented in Fig. 4.

Bottom Line: The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences.In general, regularity is understood as tantamount to periodicity.However, there is now a flurry of works proving the existence of "antiperiodicity", an unfamiliar type of regularity.

View Article: PubMed Central - PubMed

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

ABSTRACT
The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences. In general, regularity is understood as tantamount to periodicity. However, there is now a flurry of works proving the existence of "antiperiodicity", an unfamiliar type of regularity. Here we report the experimental observation and numerical corroboration of antiperiodic oscillations. In contrast to the isolated solutions presently known, we report infinite hierarchies of antiperiodic waveforms that can be tuned continuously and that form wide spiral-shaped stability phases in the control parameter plane. The waveform complexity increases towards the focal point common to all spirals, a key hub interconnecting them all.

No MeSH data available.


Related in: MedlinePlus