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### Entrainment of noise-induced and limit cycle oscillators under weak noise \$(document).ready(function() { // The supplied crossmark code loads this inline before jqplot has finished unitialising, they then unregister the // jQuery causing much hilarity - doing it after page load is safer, we chain all of our requests to hopefully avoid // any kind of race condition var cachedScript = jQuery.cachedScript; cachedScript("https://ajax.googleapis.com/ajax/libs/jquery/1.4.4/jquery.min.js", { success: function () { cachedScript("https://ajax.googleapis.com/ajax/libs/jqueryui/1.8.7/jquery-ui.min.js", { success: function () { var s = document.createElement('script'); s.type = 'text/javascript'; s.src = 'http://crossmark.crossref.org/javascripts/v1.3/crossmark.min.js'; document.body.appendChild(s); } }); } }); });

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Affiliations:
1 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 2100-DK, Denmark
2 Weizmann Institute of Science, 234 Herzl St., Rehovot 76100, Israel
Chaos 23, 023125 (2013)
/content/aip/journal/chaos/23/2/10.1063/1.4808253

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• Namiko Mitarai, Uri Alon and Mogens H. Jensen
• Source: Chaos 23, 023125 ( 2013 );
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http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4808253

## Figures

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FIG. 1.

The time evolution of (solid line) and (dashed line) when there is no external forcing. (a) Limit cycle oscillator with  = 2 and  = 10 without noise ( ). (b) Linear system with  = −0.1 and  = 0 without noise ( ). (c) Nonlinear system with  = −0.1 and  = 1 without noise ( ). For (b) and (c), the initial condition is perturbed from the fixed point to demonstrate the dumped oscillation. (d) Limit cycle oscillator with  = 2 and  = 10 with noise ( ). (e) Linear system with  = −0.1 and  = 0 with noise ( ). (f) Nonlinear system with  = −0.1 and  = 1 with noise ( ).

Click to view

FIG. 2.

The time evolution of (solid line) and (dashed line) when there is additive external forcing (dotted line,  = 1). The external forcing has angular frequency for (a)–(c), and for (d)–(f). (a), (d) Limit cycle oscillator with  = 2 and  = 10 without noise ( ). (b), (e) Linear system with a stable fixed point with  = −0.1 and  = 0 without noise ( ). (c), (f) Nonlinear system with a stable fixed point with  = −0.1 and  = 1 without noise ( ). For the case with a limit cycle oscillator (a), (d), the system's angular frequency can entrain to the external angular Ω with various ratios, while in the linear and nonlinear systems with a stable fixed point case (b), (c), (e), (f) the system can only entrain to one to one ratio.

Click to view

FIG. 3.

The time evolution of (solid line) and (dashed line) when there is additive external forcing (dotted line,  = 1). The external forcing has angular frequency for (a)–(c), and for (d)–(f). (a), (d) Limit cycle oscillator with  = 2 and  = 10 with noise ( ). (b), (e) Linear noise-induced oscillator with  = −0.1 and  = 0 with noise ( ). (c), (f) Nonlinear noise-induced oscillator with  = −0.1 and  = 1 with noise ( ). For the limit cycle oscillator (a), (d), the noise makes the orbit irregular, and the phase sometime slips. In the linear noise-induced oscillator for small external angular frequency, we can clearly see that the noise put the oscillation with angular frequency close to on top of one-to-one entrainment behavior (b). When Ω is larger than (e), the external angular frequency is more visible, due to the smaller noise compared to the amplitude. The nonlinear noise-induced oscillator behaves again very similar to the linear case in entrainment behavior (c), (f), except for the suppression of large amplitude.

Click to view

FIG. 4.

“Devil’s staircase” for limit cycle oscillator (a) and linear and nonlinear systems with a stable fixed point (b) under additive forcing with  = 1. (a) The limit cycle oscillator with  = 2 and  = 10 with (dotted line), (dashed line), and (solid line). (b) The systems with a stable fixed point ( = −0.1). For the case without noise (solid line), both linear ( = 0) and nonlinear ( = 1) systems show only one-to-one entrainment. With noise, the noisy oscillations around the one-to-one entrained orbit is induced, as shown with (linear case with  = 0 is shown by dashed line, and nonlinear case with  = 1 is shown by dotted line).

Click to view

FIG. 5.

“Arnold’s tongue” with additive forcing for limit cycle oscillator without (a) and with (b) noise and for noise-induced oscillator with noise for linear (c) and nonlinear (d) case. The horizontal axis is the external frequency Ω, and the vertical axis is the forcing amplitude ′. Entrainment is defined as is within 1% of the given value. (a) The limit cycle oscillator with  = 2 and  = 10 with , which shows standard “Arnold’s tongue.” Noise ( ) make phases to slip, resulting in smaller region of entrainment (b). For noise induced oscillator with noise (c:  = −0.1,  = 0, , d:  = −0.1,  = 1, ), the tongue-like triangle structure is observed only for 1/1 entrainment.

Click to view

FIG. 6.

Maximum real part of the Floquet exponent for various and Ω, for the linear system with stable fixed point ( = −0.1 and  = 0) without noise ( ).

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FIG. 7.

Without noise: The time evolution of (solid line) and (dashed line) when there is multiplicative external forcing (dotted line,  = 1). The external forcing has angular frequency for (a)–(c), and for (d)–(f). (a), (d) Limit cycle oscillator with  = 2 and  = 10 without noise ( ). (b), (e) Linear system with a stable fixed point with  = −0.1 and  = 0 without noise ( ). The transient behavior is shown. Note that the y-range in (e) is different from other plots. (c), (f) Nonlinear system with a stable fixed point with  = −0.1 and  = 1 without noise ( ). The limit cycle oscillator shows entrainments (a), (d), but the linear system either decays to zero (b) or diverges (e). The nonlinear system either decays (c) or entrains (f).

Click to view

FIG. 8.

With noise: The time evolution of (solid line) and (dashed line) when there is multiplicative external forcing (dotted line,  = 1). The external forcing has angular frequency for (a)–(c), and for (d)–(f). (a), (d) Limit cycle oscillator with  = 2 and  = 10 with noise ( ). (b), (e) Linear noise-induced oscillator with  = −0.1 and  = 0 with noise ( ). Note that the y-range in (e) is different from other plots. (c), (f) Nonlinear noise-induced oscillator with  = −0.1 and  = 1 with noise ( ). The limit cycle oscillator shows entrainments with some phase slips (a), (d). For the linear and nonlinear system, the noise induces the oscillatory behavior, for the parameters where the system would decay without noise (b), (c).

Click to view

FIG. 9.

“Devil’s staircase” for limit cycle oscillator (a) and nonlinear noise-induced oscillator (b) under multiplicative forcing with  = 1. (a) The limit cycle oscillator with  = 2 and  = 10 with (dotted line), (dashed line), and (solid line). (b) The nonlinear system with a stable fixed point with  = −0.1 and  = 1. For the case without noise (solid line), the decaying region where goes to the fixed point is not shown, resulting in three discrete entrainment region. With noise, oscillation is induced in the decaying regime also, resulting in continuous line as shown for (dashed line).

Click to view

FIG. 10.

“Arnold’s tongue” with multiplicative forcing for limit cycle oscillator without (a) and with (b) noise and for nonlinear system with a stable fixed point without (c) and with noise (d). The horizontal axis is the external frequency Ω, and the vertical axis is the forcing amplitude . Entrainment is defined as is within 1% of the given value. (a) The limit cycle oscillator with  = 2 and  = 10 with shows standard “Arnold’s tongue,” while the noise ( ) makes the region of entrainment smaller (b). For nonlinear noise induced oscillator ( = −0.1,  = 1 (c), there are a few entrainment regions for no noise case ( ), but not all the ratios are observed. For (c), the exponentially decaying case were excluded numerically by the following way: The equations are integrated with initial condition (1) = 1 and (2) = 0, and if the average amplitude for is less than 90% of the average amplitude for , then the solution is excluded.

Click to view

FIG. 11.

The oscillations and entrainment for TNF-driven NF-κB system. The left panel shows spontaneous oscillations with [] = 0.5 for (a) no noise (σ = 0) case and (b) σ = 0.001, (c) σ = 0.002. The right panel shows entrainments with  = 0.05 and Ω = 0.0297 with (d) no noise (σ = 0) case and (e) σ = 0.001, (f) σ = 0.002. Solid lines show 3 , and dashed lines shows [].

Click to view

FIG. 12.

“Devil’s staircase” for TNF-driven NF-κB system without and with noise, with (a) , (b) , (c) , and (d) . The entrainment regions are calculated from the frequency of the peaks in the deterministic case. In the finite noise case, we define the nuclear NF-κB peak as follows: We first determined the maximum value and the minimum value of of the steady state in the deterministic simulation for the given parameters. We then calculate two thresholds, and . Next we perform the corresponding simulation with finite σ. We define a switching event from the “low” state to “high” state when exceeds , while the reverse switching happens when becomes smaller than . The number of peaks are calculated from how often the “high” states are reached. This way we can filter out the wiggly motion due to the noise and thus define the overall peak.

## Tables

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Table I.

Variables and the parameters in the TNF-driven NF-kB oscillation, from Ref. .

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Table II.

Summary of entrainment behavior of oscillators under additive and multiplicative forcing. A and M in the “force” column represent additive and multiplicative forcing, respectively.

/content/aip/journal/chaos/23/2/10.1063/1.4808253
2013-06-07
2013-12-09

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