Constructive Role of Noise in Hysteretic Systems

1
Constructive Role of Noise in Hysteretic Systems
Mihai Dimian Department of Electrical
Engineering and Computer Science Stefan cel
Mare University, Suceava, Romania
Thanks to Max Planck Institute for Applied
Math, Leipzig (Germany) Howard University,
Washington DC (U.S.A.) University of Maryland,
College Park (U.S.A.) and All InterContinental
Airways
2
Outline

• Noise and Dithering
• Stochastic and Coherence Resonance
• Stochastically driven Hysteretic Systems
• Stochastic processes on graphs (SPG)
• Calculation of Spectral Density
• Coherence Resonance Phenomena in Hysteretic
  Systems
• Stochastic Resonance Phenomena in Hysteretic
  systems
• Conclusions

3
Noise as a Nuisance

• Everybody hates noise
• and the world tends to become
  even noisier ...
• Example
• World Health Organization recommends an average
  noise level of about 35 decibels in patient
  areas.
• A John Hopkins Hospital study found that the
  average noise level in hospitals rose from 57
  decibels in 1965 to 72 decibels in 2005
• Scientists and Engineers have even more reasons
  to do so, especially in electronics and
  communications areas
• And for those working on magnetic systems, noise
  tends to bankrupt one of the main financial
  support and technological justification for our
  work, namely magnetic recording industry

4
Music as noise
Voss and Clarke, Nature, 258, 1975
Jan Beran, Chance, 17 (4), 2004
5
Lena Söderberg
Lenna or Lena is the name given to a standard
test image originally cropped from a Playboy
magazine, a Swedish model who posed for the
November 1972 issue. The image is probably the
most widely used test image for all sorts of
image processing algorithms and related
publications.
For a complete picture and details please consult
the following reference Playboy, 40 (11), 1972
6
Dithering Noise benefit in signal processing
Let us apply a pixel threshold to the gray-scale
Lena image. The input images pixels vary from
zero (black) to one (white). The threshold is
0.05. The result is the faint image from the
left. Let us keep the noise mean constant
(-0.02) and increase the noise variance 0.00167,
0.0234, 0.167.
B. Kosko and S. Mitaim, Proc. of the IEEE
Special Issue on Intelligent Signal Processing,
86 (11), 1998
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Stochastic Resonance
Benzi et al. (1981, 1982) Nicolis (1982)
Recurrent Ice Ages
Periodic Input
Output
Nonlinear System
F. Sagues, J. Sancho, and J. Ojalvo, Rev. Mod.
Phys. 79 (3), 829 (2007)
Noise
Optical, neuronal, electric, and various
biological systems
i
t
Zero or Small Noise
Intermediate Noise
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Coherence Resonance
Output
Noise
Nonlinear System

• without noise no response too large
  fluctuations a noisy output
• for an appropriate intermediate amount of noise,
  the trajectory of the system might become quite
  regular (stochastic limit cycle)

A theoretical focus neuron models (e.g.
FitzHugh-Nagumo) Experiments electronic
circuits, magnetoplasma, atomic clouds
See, for example, B. Linder et al. (2004, Phys.
Rep.) and references therein.
If I understood correctly, this phenomena takes
place in the experiments on Synchronising Lasers
presented by Dr. Guillaume Huyet on Monday
Measures of stochastic and coherence resonance
Correlation time Coefficient of variation
Power spectrum Degree of coherence
9
Preisach model driven by stochastic input
Various hysteretic nonlinearities with stochastic
input can be constructed through Preisach
formalism as a weighted superposition of
rectangular loop operators that are individually
driven by the same diffusion process.
- Preisach distribution function
Mayergoyz, Korman, Friedman (90s)
Much simpler, but not simple !
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Treatment of non-Markovian Output
B. Linder et al. (2004, Phys. Rep. 2000, Phys.
Rev. E) FitzHugh-Nagumo model for neuron (based
on Melnikov (1993, Phys. Rev. E)) Mayergoyz and
Korman (IEEE Trans. Magnetics 1991, 1996, 1998)
M. I. Freidlin si S. J. Sheu (2000, Probab.
Theory Relat. Fields) Stochastic processes
defined on graphs
With each edge En, a diffusion process is
associated
Task to characterize the behavior of the
diffusion process at the interior vertices of the
graph.
These gluing conditions depend on how much time
the process spends in Vk and on the probabilities
that the process will move from vertex Vk along
the edges En
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Rectangular Loop Hysteresis
Noise Ornstein-Uhlenbeck Processes
ß
a
The output is a non-Markovian binary process.
Stationary Case
- Wiener process - given constants
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SPG and Hysteresis
The output is a non-Markovian binary process.
Because it describes hysteresis with local
memory, the two-component process yt(it,xt) is
Markovian.
E3
i1
V1
V2
i-1
i1
E4
E1
i-1
E2
x
a
ß
Dimian and Mayergoyz (2004, Phys. Rev. E 2005, J.
Phys.)
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Computation of Spectral Density
Introducing auxiliary function
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Computational Results
to appear in Nano Brief review and reports
(2008)

Van-Vleck Formula for Ideal Relay
The asymmetry of the system plays an essential
role in the appearance of coherence resonance
phenomena.
15
Symmetric Preisach Systems with Stochastic Input
On each graph edge
d initial condition and vertex type boundary
conditions
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Spectral Density of the Preisach Output
In the case of symmetric input
No coherence resonance phenomenon was
found (see for
example the Figure from the right)
In the case of non-symmetric input
Preliminary results indicate
coherence resonance phenomena for intermediate
noise
17
Stochastic resonance in Preisach systems
Now, we have a periodic signal in addition to the
white noise.
R.N. Mantegna, B. Spagnolo, L. Testa, and M
Trapanese, Journal of Applied Physics, 97, 10E519
(2005) NUMERICAL SIMULATIONS

• In the absence of noise one can see a small
  hysteresis loop in the center of the major loop.
• When one add the optimum value for the noise
  variance a major loop is observed.

Analytical approach using the method of
stochastic processes on graphs is in progress
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Conclusions

• Hysteretic systems are usually affected by
  internal or external noise leading to a
  stochastic behavior of the systems output
• While it is mostly seen as a disruptive effect,
  noise can also have a constructive role,
  activating a resonance response of the system
• It is predicted that various hysteretic systems
  can exhibit such coherent motion induced solely
  by noise
• The regularity of the stochastic output is
  quantified by the power spectrum, which displays
  a maximum at a nonzero frequency, for a certain
  level of the input noise
• The calculation of the spectral density for the
  hysteretic systems is performed in the framework
  of stochastic processes on graphs
• The method used here can be generalized through
  Preisach formalism to analyze various hysteretic
  nonlinearities

19
Acknowledgements

• This work is supported by the Romanian Executive
  Agency for Higher Education and Research Funding,
  PN II Human Resources, contract number
  13/1.10.2007
• The workshop travel is supported by Romanian
  Executive Agency for Higher Education and
  Research Funding, PN II, Researchers Mobility
  Funds, Jan. 2008