Difference between revisions of "Group 6"

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[[File: Eqn1.png]]
 
 
[[File: Eqn2.png]]
 
 
[[File: Eqn3.png]]
 
  
 
=== Linear stability analysis ===
 
=== Linear stability analysis ===
  
The function f(x) describes the electrical response of the nonlinear resistor, and its shape depends on the particular configuration of its components. The parameters α and β are determined by the particular values of the circuit components.
 
 
The fixed points can be solved and give the following three equations.
 
 
[[File: fp.png]]
 
 
Linearizing about the fixed points gives the following stability matrix.
 
 
[[File: Linmat.png]]
 
  
 
=== Chaotic dynamics ===
 
=== Chaotic dynamics ===
  
A chaotic attractor, known as "The Double Scroll" because of its shape in the (x,y,z) space, was first observed in a circuit containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function.<ref>{{cite journal | last = Chua | first = Leon O. | authorlink = Leon O. Chua | coauthors = Matsumoto, T., and Komuro, M. | title = The Double Scroll | journal = IEEE Transactions on Circuits and Systems | volume = CAS-32 | issue = 8 | pages = 798–818 | publisher = [[IEEE]] | date = August 1985 | url = http://ieeexplore.ieee.org/iel5/31/23571/01085791.pdf | accessdate = 2008-05-01}}</ref>
 
The easy experimental implementation of the circuit, combined with the existence of a simple and accurate theoretical model, makes Chua's circuit a useful system to study many fundamental and applied issues of [http://en.wikipedia.org/wiki/Chaos_theory chaos theory]. Because of this, it has been object of much study, and appears widely referenced in the literature.
 
 
To date, а huge number of various types of chaotic attractors in Chua's system has been discovered, which can be obtained numerically, with relative ease, by standard computational procedure (after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable zero equilibrium, reaches an attractor and computes it)<ref>{{cite book |author= Bilotta, E., Pantano, P. |title=Gallery of Chua Attractors |publisher=World Scientific |year=2008 |isbn=978-981-279-062-0}}</ref>. Also, recently, a [http://en.wikipedia.org/wiki/Hidden_oscillation hidden Chua's attractor] was discovered<ref name=2011-PLA-Hidden-Chua-attractor>{{cite journal |
 
author = Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. |
 
year = 2011 |
 
title = Localization of hidden Chua's attractors |
 
journal = Physics Letters, Section A |
 
volume = 375 |
 
issue = 23 |
 
pages = 2230&ndash;2233 |
 
doi = 10.1016/j.physleta.2011.04.037}}
 
</ref>.
 
  
 
== Experiment ==
 
== Experiment ==
  
  
[[File: Breadboard.png | thumb | 200px | Breadboard]]
 
 
[[File: ScreenHunter 45 Oct. 11 09.18.png | thumb | 200px | Double scroll]]
 
 
The circuit can be built from easily accessible components on a breadboard and powered with a DC voltage supply. An oscilloscope and digital acquisition card will be used in conjunction to measure the output voltages modeled in the above equations. Below is a list of required components.
 
 
[[File: ScreenHunter 46 Oct. 11 09.22.png | thumb | 200px | Double scroll]]
 
  
 
=== Parts list ===
 
=== Parts list ===

Revision as of 14:36, 11 October 2011

A schematic of Chua's circuit

Chua's circuit is a simple electronic circuit containing standard electrical components (resistors, capacitors, inductors, and an op-amp) developed by Leon Chua <ref>http://en.wikipedia.org/wiki/Leon_O._Chua</ref> that exhibits chaotic behavior.

Background

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Linear stability analysis

Chaotic dynamics

Experiment

Parts list

  1. Breadboard and jumper wires
  2. 5 100 ohm resistors (1/4 Watt)
  3. 3 20 picofarad capacitors
  4. 5 10 microhenry inductors
  5. 2 diodes
  6. 1 20V DC power supply
  7. 1 oscilloscope / DAQ
  8. 3 wire leads with alligator clips

Using the Chua Circuit shown in Figure 4, the voltage waveforms v_{C_1} (t) and v_{C_2} (t) across capacitors C1 and C2, and the current waveform i_L (t) through the inductor L in Figure 1, were observed using an oscilloscope and displayed in Figure 5 (a), (b), and (c) (left column), respectively.

The Lissajous figures associated with 3 permutated pairs of waveforms are displayed on the right column Figure 5; namely, in the v_{C_1}-i_L plane in Figure 5(d), the v_{C_1}-v_{C_2} plane in Figure 5(e), and the v_{C_2}-i_L plane Figure 5(f). They are 2-dimensional projections of the chaotic attractor, called the double scroll, traced out by the 3 waveforms from the left column in the 3-dimensional v_{C_1} - v_{C_2} - i_L space.

It is important to point out that the Chua Circuit is not an analog computer. Rather it is a physical system where the voltage, current, and power associated with each of the 5 circuit elements in Figure 1 can be measured and observed on an oscilloscope, and where the power flow among the elements makes physical sense. In an analog computer (usually using Op Amps interconnected with other electronic components to mimic some prescribed set of differential equations), the measured voltages have no physical meanings because the corresponding currents and powers can not be identified, let alone measured, from the analog computer.


Simulation of Chua's circuit

Matlab code

The following code integrates the system of equations which model the circuit

<source lang="matlab"> % Models Initial Variables %-------------------------

TimeSeries = [x0, y0, z0]'; % models initial conditions

% Optimized Runge-Kutta Variables %--------------------------------

h = step_size*G/C2; h2 = (h)*(.5); h6 = (h)/(6);

anor = Ga/G; bnor = Gb/G; bnorplus1 = bnor + 1; alpha = C2/C1; beta = C2/(L*G*G); gammaloc = (R0*C2)/(L*G);

bh = beta*h; bh2 = beta*h2; ch = gammaloc*h; ch2 = gammaloc*h2; omch2 = 1 - ch2;

k1 = [0 0 0]'; k2 = [0 0 0]'; k3 = [0 0 0]'; k4 = [0 0 0]'; M = [0 0 0]';

% Calculate Time Series %----------------------

M(1) = TimeSeries(3)/E; M(2) = TimeSeries(2)/E; M(3) = TimeSeries(1)/(E*G);

for i=1:dataset_size

   % Runge Kutta
   % Round One
   k1(1) = alpha*(M(2) - bnorplus1*M(1) - (.5)*(anor - bnor)*(abs(M(1) + 1) - abs(M(1) - 1)));
   k1(2) = M(1) - M(2) + M(3);
   k1(3) = -beta*M(2) - gammaloc*M(3);
   % Round Two
   temp = M(1) + h2*k1(1);
   k2(1) = alpha*(M(2) + h2*k1(2) - bnorplus1*temp - (.5)*(anor - bnor)*(abs(temp + 1) - abs(temp - 1)));
   k2(2) = k1(2) + h2*(k1(1) - k1(2) + k1(3));
   k2(3) = omch2*k1(3) - bh2*k1(2);
   % Round Three
   temp = M(1) + h2*k2(1);
   k3(1) = alpha*(M(2) + h2*k2(2) - bnorplus1*temp - (.5)*(anor - bnor)*(abs(temp + 1) - abs(temp - 1)));
   k3(2) = k1(2) + h2*(k2(1) - k2(2) + k2(3));
   k3(3) = k1(3) - bh2*k2(2) - ch2*k2(3);
   % Round Four
   temp = M(1) + h*k3(1);
   k4(1) = alpha*(M(2) + h*k3(2) - bnorplus1*temp - (.5)*(anor - bnor)*(abs(temp + 1) - abs(temp - 1)));
   k4(2) = k1(2) + h*(k3(1) - k3(2) + k3(3));
   k4(3) = k1(3) - bh*k3(2) - ch*k3(3);
   
   M = M + (k1 + 2*k2 + 2*k3 + k4)*(h6);
   
   TimeSeries(3,i+1) = E*M(1);
   TimeSeries(2,i+1) = E*M(2); 
   TimeSeries(1,i+1) = (E*G)*M(3);
   
   i=i+1;

end </source>


References

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