Chaotic Dynamics Lab Report

Mathematica models of non-linear dynamics By Sameer Saini

The aim of this experiment is to analyse the motion of a simple pendulum system and its transition into chaos. The experiment as a whole displayed the differences between the linear pendulum and non-linear pendulum, in terms of what their corresponding motions are capable of and it attempted to show that transition into chaos would follow Feigenbaum’s constant.

Introduction Background Theory Simple Pendulum The equation for a simple linear pendulum is: g M t L  ' t L M  ' ' t = f cos  t The equation for a simple non-linear pendulum is: g M sin tL  ' t L M  ' ' t = f sin t  Where g is acceleration due to gravity, L is the length of the pendulum, M is the Mass, γ is the damping, f is the magnitude of the driving force and Ω is the frequency of the driving force. Feigenbaum constant Feigenbaum’s constant is the limiting ratio of each bifurcation interval to the next, or between the diameters of successive circles on the real axis of the Mandelbrot set. This number can be related to the period-doubling bifurcations in a polynomial mapping of degree 2. It was shown that it held true for all one dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. Fiegenbuam's constant is 4.6692... Aim Investigate chaotic dynamics using mathematica

Method Mathematica was used to simulate the motion of a pendulum, using both linear and non-linear equations. Plots of several wave-functions superimposed on one graph where created, with the driving being varied at small and large angles. This was done to both the linear and non-linear pendulum equations so as to grasp the differences between the two equations relative to the driving forces. Another graph was then created in a similar fashion to the previous graphs, but this time instead of varying the driving force, the initial velocity was varied. From these results it was possible to deduce the effect the initial conditions would have on the pendulum. The second section of this experiment was to gather an understanding of the transition to chaos, for the non-linear pendulum. We knew that as the system would transit into chaos, there would be period doubling. Using poincare plots we attempted to find these regions, by looking for the points at which the periods could be found to start doubling.

Results The following four graphs show several wave-functions superimposed on each other, with the driving force varying according to the initial conditions. Linear Pendulum, Small Angles

Non-Linear Pendulum, Small Angles

Linear Pendulum, Large Angles

Non-Linear Pendulum, Large Angles

Varying initial velocities for non-linear pendulum

Locations of Period Doubling Doubling Type

Location

Absolute error

Relative Error

2

5.67500

±0.000005

±0.000088

4

5.65595

±0.000005

±0.000088

8

5.65150

±0.000005

±0.000088

16

5.65051

±0.000005

±0.000088

Feigenbaum’s constant (γ) is =

r  2−r 1 r 3−r 2

Calculating Feigenbaum’s constant 1=

5.65595−5.67500 5.65150−5.65595

1=4.2809±0.01187 2 =

5.65150−5.65595 5.65051−5.65150

2 =4.4949±0.55503

=

1 2 2

=4.3879±0.034

Discussion Comparison between linear and non-linear pendulums For the linear and non-linear pendulum at small angles for the driving force, the results in the graphs appear to be identical for all of the wave-functions, par a slight phase shift. But for the case of the large angles, the wave-functions for the non-linear pendulum seem to start off similar to those of the linear pendulum when dealing with smaller angled driving forces, but as they increase the amplitude of the wave function seems to lower and the wave-function seems to shift to the left. There also seems to be a point when the wave-function begins to transit into chaotic motion and no longer follow the pattern similar to the linear pendulum, but instead creates seemingly unpredictable patterns. Effects of varying the initial conditions Taking a look at just the non-linear pendulum graphs, it can be seen that as we vary the initial conditions for the wave-function the final settings of the graphs correspondingly change, particularly the cases for the amplitude and angle of the final motion. Feigenbaum’s constant Feigenbaum's constant is approximately equal to 4.6692, in our calculations here we found it to be 4.3879 ± 0.034, which does not contain the exact value within the range of error, but nonetheless is still quite close. There several possibilities as to why this may have turned out this way. Firstly the accuracy of the measurements taken could have possibly played an effect and could be fixed by taking more decimal points into account. Secondly despite the fact that we could increase the accuracy , it would have little to no effect, as attempts to find the regions of period doubling were measured by sight according to the poincare graphs. Which leads to the last point, because we were taking the regions of period doubling by sight of a poincare graph, it was difficult to interpret the exact points at which the period doubling would start as there was a range of values with which the graphs seemed to result in period doubling, with little to no change in graphs.

Conclusion The experiment as a whole went well and helped to display the major differences between the linear pendulum and the non-linear pendulum equations. For the transition into chaos section, the results were a bit a out off what was expected, but that is likely due to human error in measuring the graphs. Otherwise it came out just a feigenbaum's constant predicted.

References: Feigenbaum’s constant: http://mathworld.wolfram.com/FeigenbaumConstant.html Pendulum Physics Simulation: http://www.myphysicslab.com/pendulum1.html#navsite