Demonstration of Chaos

Circuits

Sajjad

Circuit Diagram

+

-

+

-

30.8

58 mH

330

330

2K 3.3K

20K

20K

5K

0.4 F 0.033 F

1 2 3

TL087TL087

Negative resistance - 1

Negative resistance - 2

LC tankcircuit Critical control

For CHAOS

Negative Resistance

+

-

330

330

2K

3

2 6

7TL087mA

V

V

I

4

Negative resistance

I

Nonlinear negative resistance

+

-

330

330

2K

3

2 6

7TL087

mA

V

4

+

-

20K

20K

3.3K

3

26

7TL087

4

Parallel Negative resistance

Damped and undamped oscillation

+

-

330

330

2K

3

2 6

7TL087

4

5K

0.1 F

Find: L = ? RL = ?

RL

L

One negative resistance is connected to a LC tank circuit with some losses in the internal resistanceOf the inductor L. Connecting the negative resistance, forces the oscillation to continue

Oscillation will stop if this value is bigger than the negative resistance

V-I Characteristics of parallel NR

Negative resistance - 1

Negative resistance - 2

Parallel Negative resistance

Break points Critical adjustment of Resistance “R” stops Oscillation at these “Break points”. Then Oscillation can start from any point close toOrigin and go either negative or positiveDirection, which is chaotic.

The state equations

€

C1dvc1dt=G(vc2 − vc1) − g(vc1)

C2dvc2dt

=G(vc1 − vc2) + iL

L diLdt= −vc2

G = 1/Rg is a piecewise linear function defined by:

€

g(vR) = m0vR +12m1 −m0( ) vR + Bp − vR −Bp[ ]

R

C2 C1 NR

Final circuit diagram with values

+

-

+

-

30.8

58 mH

330

330

2K 3.3K

20K

20K

5K

0.4 F 0.033 F

CH1 CH2

oscilloscope

TL087TL087

Oscilloscope traces

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

-4

-3

-2

-1

0

1

2

3

4

5

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-4

-3

-2

-1

0

1

2

3

4

5

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Y-t X-Y

Photograph of the fabricated circuit