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Student Number: 25154893 16/03/2014

Measuring Feigenbaums First Constant

using an Electric Circuit

Brittany Barrettbb2g11@soton.ac.uk

Personal Tutor: Graham Reed

Abstract: Any system exhibiting period doubling should produce Feigenbaums constant, theratio 1, at 4.23 from 2

to 8 periods. This ratio should be present in electrical systems, but has been found somewhat inaccurate in

previous experiments. Six results were gathered, only one of which found amplitudes relating

Feigenbaums Constant to an appropriate approximation of the ratio, at 4.24. Although this system was

exhibiting chaotic behaviour, through an exponential relationship between the amount of periods present

in the system and the input amplitude, it was not conclusively exhibiting the ratio 1.It was concluded

that an electrical system is markedly difficult to produce consistent behaviour in, due to its variance withinput, and the inaccuracy when making visual measurements of period doubling.

1. Introduction

Feigenbaum described period doubling towards chaos as

a naturally occurring phenomenon, in which one-

dimensional maps for a system that shows a single

quadratic maximum will bifurcate at the same rate:

+1

+2 +1

(1)

The value is not absolute, and approaches the ratio ,

defined as 4.699 quickly as per each period doublingiteration of the system. This value was described as

being present in all period doubling systems[1].

This theory should therefore apply to electrical systems,

and should be observable through the use of electronic

measurement tools. By analysing a series system

comprising an EMF, inductor and diode, the amplitude

of voltage measured across the diode should exhibit

period doubling behaviour, as described in previous

experiments[2].

The experiment undertaken sought to clarify the value

by relating input voltage to observed bifurcation in

output voltage across the diode. Thereby, period

doubling would be inferred. As the amplitude of a

sinusoidal signal is variant, and all elements of the

system tend towards a higher level of uncertainty, the

amount of periods present in a system becomes difficult

to determine, i.e. chaotic. For this reason, 1 is most

accurately gauged electronically, where 1 is given as

approximately 4.23. Therefore, the three results gathered

must be at 2, 4 and 8 periods present in the system, inorder to gain the ratio 1.

2. Background

When the EMF of the circuit is applied within the

forward-bias region of the diode, the diode will act as a

voltage source. The diode will respond in a capacitive

manner when the EMF is applied within the reverse-bias

region of the diode. The alternation between conduction

and capacitance is not an immediate effect when the

current reaches zero, it continues conduction for a shortperiod of time .

is exponentially dependent on the current applied to the

system. As a result, with a certain applied current, the

system no longer has a single period in which it is

conducting, but the previous period will have an effect

on a succinctly occurring current. Therefore, period

doubling occurs, and determines an exponential

relationship between each succinct period doubling. A

more succinct, mathematical analysis of this behaviour is

given in the paper Chaos In A Diode[3].

Previous experiments have well described this electrical

circuit, but have often failed to draw results within an

accuracy range of 5%[4].

Figure 1: Period doubling exponentially tending towards chaos,

applicable to an electronic system.[5]

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3. Experimental Design

An electric circuit was built comprising an unspecified

diode, 10 20%copper wire coil inductor, wiredusing a breadboard. An EMF was supplied using a

TG550 function generator, with a marked 1% error

margin given on all functional parameters used in thistest, as provided in its data booklet.The function

generator was set to provide a 182 signal, wherevoltage was used as a parameter to investigate period

doubling.Wires used in connecting these components

were either electrically insulated or kept short to avoid

overriding inductance, which could have given

unspecified results.

Figure 2: Schematic of analysed circuit

The measurement equipment comprised of two x10

attenuation oscilloscope probes, and a TDS1000C-EDU

oscilloscope. The probes were calibrated using the

oscilloscopes calibration function to provide a< 1%error in measurement. The oscilloscopes used measured

across the diode for channel 1, and across the functiongenerator for channel 2, with ground as reference, as

given in figure 1.The oscilloscope scale was adjusted as

per each value, to give a precise reading for period

doubling.

The system was analysed by slowly adjusting the

function generators voltage amplitude, and then taking

two measurements, 1 ,2 , per each observed perioddoubling. The first measurement was taken slightly

before period doubling, and the second measurement

slightly after.

2 1

2 (2)

The result using equation (2) would give an average

amplitude, , as to where period doubling had occurred.This average was used due to inconsistency generated by

treating period doubling as a point, as opposed to a

region.

The amplitude of period doubling was measured three

times per system, in order to gain an approximation of

Feigenbaums first constant, using equation 1.The

experiment was repeated multiple times in order to better

approximate .

4. Results

The same system was analysed in six experiments, and

produced a variety of different amplitudes at which

period doubling occurred.

To determine whether or not period doubling hadoccurred, a result would be gathered at a notable change

in the oscilloscopes reading of output amplitude.

Figures 3 and 4 describe the visual changes observed to

gather these amplitudes.

Figure 3: A system exhibiting two periods

Figure 4: A system exhibiting four periods

Experiment 1() 2() 3() 1

First 1.80 2.50 3.22 0.97

Second 1.80 2.70 3.08 2.37

Third 1.80 2.69 2.91 4.24

Fourth 1.96 2.47 2.88 1.24

Fifth 1.66 2.10 2.50 1.10

Sixth 1.76 2.52 2.84 2.375

Table 1: Output amplitude and ratio of period doubling

Due to the inherently chaotic nature of the system, and

the oscillatory function being provided, the results

gathered were found to have been within a 5%

tolerance range. These values varied on the oscilloscope

display, and were estimated using equation (2).

The values gathered attempting to confirmFeigenbaumsConstant varied between 77.10% and +1.03% of

Feigenbaums original value of 4.23.

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Of the six experiments, one result was found to have

been within a tolerable range given the error. Experiment

three produced appropriate results, however was not

carried out in a different manner to the rest of the

experiments. Due to the range of tolerance of each

amplitude, it was found by minor adjustment within the

5%range of each value that each experiment wouldyield a far closer value to 4.23.

Another measurement observed in this system was an

effect of blocking occurring in regards to whether or not

a system was exhibiting local period doubling. To move

from one local period doubling to the next, a certain

amplitude would have to be applied to the system.

However, to observe this change occurring in reverse,

the system would have to cross the same region, meaning

there was a voltage range for each period doubling in

which a system would be in either the higher period

doubling, or the lower, depending on which it hadpreviously been in. This value changed significantly per

each experiment.

Despite inaccurate results being drawn through most

experiments, for the specified component and

measurement tolerance ranges, experiment three yielded

a highly accurate result. Within 1.03% of the

Feigenbaums constant in a system exhibiting up to 8

periods, experiment three shows a working electrical

mode of Feigenbaums hypothesis.

5. Discussion

The system was determined to be difficult to analyse

using the provided hardware. The gathered results in

Table 1 for 1 were, in all experiments other than the

third, more than 40%out of the expected range, which

greatly exceeds the determined error margins of

components and measurement equipment.

The data drawn could have been misrepresented, as

visual observation of peak changes in the output may not

necessarily give an accurate range of when period

doubling is occurring. The first gathered term 1 is far

more consistent than

2and

3. This could show that thesystem is already becoming heavily affected by noise

due to the tendency towards chaos as the system tends to

more periods present. As such, the gathered data would

have no longer represented period doubling at certain

amplitude, but inconsistencies.

Other component conditions were not accounted for in

this experiment. The diodes forward and reverse regions

were not defined before the experiment begun, and may

contribute to an analytical approach which could better

estimate the expected amplitude at period doubling. This

would have in turn lead to more care being taken in

reading experimental results around the points ofexpected period doubling.

Component conditions such as variance with temperature

and inductance are unaccounted for in the experiment,

yet account for changes in resistance, and therefore

current. As a higher peak voltage is applied to the

system, its expected therefore that the diode will raise in

temperature, changing its forward-bias region. The

inductance of the system is also subject to change basedon temperature. As a result, the system may be changing

over time, meaning that the gathering of results is

inherently gathering from different conditional systems.

All of these changes account for minor variations in

gathering appropriate values for amplitude. The blocking

described in the results section also accounted for

variations in gathering results. Due to the variance

involved, and the visual approach to deducing period

doubling, results are heavily influenced by external

factors.

6. Conclusions

Five of the six experiments failed to conclusively prove

Feigenbaums hypothesis. Although period doubling is

observed, and the system tends towards chaos, with an

exponential rise in periods present in the system per

amplitude, the experiments have failed to draw

conclusive linear results.

The circuit conditions involved in this system are

observably difficult to resolve and quantify. These

unreliable factors lead to difficulty in observing specific

points of bifurcation.

As small values of variability produce large changes in

accuracy to approximation of Feigenbaums constant,

these small inaccuracies are most likely to be responsible

for the discrepancy in each experiment, including what

could be wishful thinking in the case of the experiment

that drew conclusively successful results.

References

[1] M. Feigenbaum, Universal Behaviour in NonlinearSystems, 1980, pp. 1-5.

[2] University of Wisconsin, Demonstration of Chaos, 2003,pp. 4.

[3] A. Missert, P. Thompson, Chaos In A Diode, 2008, pp. 1-

2.

[4] B. Prusha, Measuring Feigenbaumsin a Bifurcating

Electric Circuit, 1997, pp. 3.[5] University of Yale, Deterministic Chaos,

http://classes.yale.edu/fractals/chaos/Feigenbaum/Feigenbaum.html

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