DOING PHYSICS WITH MATLAB

 

A NUMERICAL APPROACH TO THE PHYSICS OF THE

 ENVIRONMENT AND CLIMATE

 

Ian Cooper

matlabvisualphysics@gmail.com

 

CHAOS IN THE ATMOSPHERE

 

MATLAB

 

Download directory

                   https://drive.google.com/drive/u/3/folders/1j09aAhfrVYpiMavajrgSvUMc89ksF9Jb

 

                   https://github.com/D-Arora/Doing-Physics-With-Matlab/tree/master/mpScripts

 

 

SCRIPTS

 

atmDDP.m

Uses ode45 to solve the equation of motion for a damped driven pendulum (DDP). The main input variable is the drive strength .  The graphical output plots are:  drive stimulus vs time; angular displacement vs time; angular frequency vs time; phase space; Fourier transform of the angular displacement. You need to download the Script simpson1d.m to calculate the Fourier transform by the direct integration of the Fourier integral.

 

The nonlinear ODE for the DDP system is extremely sensitive to the model’s parameters and initial conditions. All input parameters much be chosen with care. Values mostly used in the simulations are those used by J.R. Taylor in his excellent book Classical Physics.

 

Time space: from ~ 10 s to ~ 10 000 s

Drive strength :  ~ 0.2 to ~ 2.0

Drive period TD (cycle time):  TD = 1.00 s    

Natural frequency:    

Damping parameter:         or  

Initial condition for angular displacement:         or      rad

Initial condition for angular frequency:            rad.s-1  

        

            ode45 options:    opts = odeset('RelTol',1e-10); 

 

 

atmPendulumPS.m

A Poincare section for a given drive strength  is plotted by solving the ODE with ode45 function. To construct the Poincare section for chaotic motion, many thousands of time steps have to be used.

 

atmPendulumBD.m

A bifurcation diagram is constructed by solving the equation of motion for a range of drive strengths . After the transient oscillation have decayed, peak values in the angular displacement are used to plot the bifurcation diagram. This method is only suitable for a limited range of which is less than some critical value.

 

atmPendulumBDS.m

A bifurcation diagram is constructed by solving the equation of motion for a range of drive strengths . For each  , the angular displacements in successive drive cycles are plotted.

 

atmPendulum2A.m

Lyapunov Exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.

    

 

 

INTRODUCTION

 

It is without question that accurate weather predictions are of the utmost importance. To forecast the weather, it is necessary to use complex numerical models. In the 1960s Edward Lorenz while developing his models on weather prediction, made a startling discovery leading to the discovery of chaos. Even though a system may be completely deterministic, chaotic systems are extremely sensitive the initial conditions.

 

Although we will not model chaotic weather systems, we can gain much insight to chaotic systems by studying the behaviour of a very simple system, the damped driven pendulum (DDP).

 

For a system to exhibit chaos its equations of motion must be nonlinear. However, nonlinearity does not guarantee chaos. To illustrate many of the principles of chaotic motion, we will consider the nonlinear system of a damped driven pendulum as described by Taylor in his book Classical Mechanics.

 

The equation of motion for the pendulum is

(1)      

 

where

           angular displacement [rad]

     t        time [s]

        angular frequency [rad.s-1]

       damping coefficient [s-1]

      driving strength  [rad-1]

    angular frequency of the driving signal [rad.s-1]

g       acceleration due to gravity  [g = 9.80 m.s-2]

L      length of pendulum  [m]

 

The angular frequency   is the time rate of change of the angular displacement  and T is the period and the f of oscillation

     (2)    

 

The natural angular frequency  and period  of the system for a simple pendulum are

     (3)     

 

The ODE equation 1 is solved using the Matlab function ode45 with the default parameters (S.I. units)

          

           

           

          

For some values of the parameters, equation 1 does lead to chaotic motion. Note, the superposition principle does not apply to nonlinear systems.

 

 

1    Free motion of the pendulum with a small initial displacement

     atmDDP.m

 

         

 

The results of the simulation are shown in figures 1.1 and 1.2.

Fig. 1.1   Simple harmonic motion for small amplitude free motion of the pendulum.

 

Fig. 1.2   Power density plot: Fourier transform of the angular displacement.

                     fN = 1.50 Hz magenta         fD = 1.00 Hz  red       

 

 

The pendulum oscillates at its natural frequency (1.5 Hz) and its motion is described as simple harmonic motion (SHM). Figure 2 displays the Fourier Transform (calculated by direction integration of the Fourier Transform integral).  For small amplitude oscillation the pendulum corresponds to a linear system.

 

 

2  Free motion of the pendulum with a large initial displacement

    atmDDP.m

 

         

 

The results are shown in figures 2.1 and 2.2. The motion is no longer SHM. The motion is periodic with frequency 0.68 Hz and period 1.47 s. The oscillation frequency is lower and its period longer compared with the small amplitude SHM.  The peaks in the angular displacement plot are flatter than for SHM as the restoring force is less than for SHM .

Fig. 2.1   Large amplitude free motion of the pendulum. The motion is periodic but not simple harmonic motion.

                     

 

Fig. 2.2  Fourier Transform spectrum:  peak oscillation frequency f = 0.68 Hz.

                    fN = 1.50 Hz magenta          fD = 1.00 Hz  red        

 

 

3    Free motion of the pendulum and small damping

     atmDDP.m

 

Small initial angular displacement

         

 Fig. 3.1   Small amplitude damped oscillations.

 

Fig. 3.2   Fourier Transform spectrum:  peak oscillation frequency f = 1.00 Hz.

                     fN = 1.50 Hz magenta         fD = 1.00 Hz 

 

 

Large initial angular displacement

        

Fig. 3.3   Large amplitude damped oscillations.

 

Fig.3.4   Fourier Transform spectrum:  peak oscillation frequency f = 1.20 Hz.

                     fN = 1.50 Hz magenta       fD = 1.00 Hz 

  

When the system is damped, the frequency band width is wider than the for the undamped system.

 

The [2D] plot of is called the phase space or state space plot as the two variables  and  completely defines the state of the pendulum. A phase space orbit is simply the trajectory of the two variables  and  as time evolves.  A closed orbit which is an elliptical attractor always evolves in a clockwise sense as shown in the above phase space plots (the green dot shows the initial conditions). The time to complete one closed orbit is the period of the oscillation.

 

 

4    Damped Driven Pendulum   ( DDP )

      atmDDP.m

 

The DDP system is excited by some external sinusoidal driving stimulus as shown in figure 4.1.

Fig. 4.1   A sinusoidal stimulus drives the oscillations of the DDP system with frequency fD = 1.00 Hz and period TD = 1.00 s (the drive cycle).

 

The response of the DDP system can be studied by changing the input model parameters. Plots can be made for the time evolution of the system, phase space, Poincare sections and bifurcation diagrams.

 

PERIODIC OSCILLATIONS

The default parameters mainly are used for each simulation. Only the driving force strength  is successively increased.

 

Relatively weak driving strength                     

Fig. 4.2   The motion of the DDP for relatively weak driving strength . After the initial transient lasting about 3 cycles (3 s), the motion is SHM with a period equal to the driving period, . After the initial transient motion, the orbit in the phase space is an ellipse (the attractor).

 

 

Fig. 4.3   Weak driving strength   . Fourier Transform spectrum:  peak oscillation frequency f = 1.00 Hz.

                     fN = 1.50 Hz magenta      fD = 1.00 Hz   red

    

 

We see that the driver and the response have the same period. Something which intuition from linear problems would say is obvious. The response has two regimes: (1) the decay of an initial transient motion and (2) the steady oscillations at the frequency of the driving signal. The amplitude of the response depends upon the energy balance between the energy supplied by the external driving force and the energy dissipated by the system due to the damping. The phase space plot exhibits a regular orbit which is independent of the initial conditions except for the initial transients which does depend upon the initial conditions. The motion approaches a unique attractor in which the pendulum oscillates sinusoidally with exactly the same frequency as the driving force.

 

In conclusion for the motion of the linear DDP with a sinusoidal driving force:

(1)       There is a unique attractor which the motion approaches, irrespective of the initial conditions applied.

(2)       The motion of the attractor is itself sinusoidal with frequency exactly matching the drive frequency.

 

We can now increase the driver strength to so the approximation  is no longer valid.

 

Weak driving input            

Fig. 4.4   Motion of DDP for . After 4 cycles (4 s) the motion settles down to a regular oscillation that looks like sinusoidal with a period equal to the driving frequency. However, the regular oscillations are not sinusoidal but the curve is flatter at the extremes.

 

Fig. 4.5   DDP spectrum for . Most of the energy supplied by the driving force to the DDP system excites the fundamental frequency (driving frequency fD = 1.00 Hz). A small amount of energy also excites the 3rd harmonic at a frequency of 3.00 Hz.

 

 

The boundary between weak and strong driving stimulus is around  .

 

When we increase the driving strength by a small amount, we find more harmonics excited.

 

Strong driving strength             

Fig. 4.6   Motion of DDP for . After 4 cycles (4 s) the motion settles down to a regular oscillation with a period equal to the driving frequency that has a DC component.

  

 

Fig. 4.7   As the driving strength increases more energy goes into the harmonics. After one cycle of the driving signal, all the harmonics will also have undergone multiple complete cycles. Hence, the motion made up of all the harmonics will still oscillate at the driving frequency.

 

 

The motion for strong stimuli is very different from the weak stimuli. In the first 5 s of the motion, the pendulum swings through about and two and half counter clockwise rotations. Then in the next 2 s it swings from  to   then settles down to oscillate almost sinusoidally around  which is really . This means that the pendulum has made one complete revolution since t = 0 before oscillating backwards and forwards about .

 

From the graphs we cannot be completely sure that the motion is exactly periodic. One way to test this is to examine the peaks in the angular displacement plot more carefully. This is easily done using the Matlab function findpeaks. The results are displayed as a table in the Command Window for the times of the peaks and their magnitudes.

 

% Find peaks for calc of period if flagP = 1 

  flagP = 1;

  if flagP == 1

     [pks, loc] = findpeaks(X,t);

     TP = (loc(end) - loc(end-2))/2;

  end

 

 for c = 1 : length(pks)

    fprintf('   %2.3f %2.3f  \n', loc(c), pks(c))

 end

 

 

 

 

From the table of results, it is clear that after the initial transient oscillations have decayed, the oscillation period is 1.0 s and a constant peak height of ~14.75. So, there is strong evidence (not a proof) that a periodic attractor is approach with a period exactly equal to the driving force.

 

 

5   PERIOD DOUBLING

    atmDDP.m

 

The most noticeable feature for a larger value of the drive strength  is that the trajectory is very different if you use different initial conditions. After the initial irregular transient oscillations in angular displacement, they appear to be at least approximately sinusoidal with a period equal to the drive period. However, on close examination you will notice that alternate peaks (or troughs) are not all of the same height.

 

Fig. 5.1   Angular displacement   .

 

Fig. 5.2   Angular displacement   .

 

Fig. 5.3   Expanded view of angular displacement  in figure 5.2. The attractor is approximately sinusoidal with alternate peaks having different heights, repeating themselves every two drive cycles (2 s). 

 

 

 

Fig. 5.4   Angular velocity  .  Note: double peaks.  After the initial fluctuations, the oscillations settle down to an attractor that is approximately sinusoidal.

 

 

Fig. 5.6   Frequency spectrum of angular displacement  .

 

 

Table 1   Time between adjacent peaks of different height is 1.0 s and the pattern repeats itself every 2.0 s.

 

 

You can see from the enlargement shown in figure 3 and Table 1, the peaks alternate between two distinct heights and this pattern persists. This pattern no longer repeats itself at the period of the driver, but rather the period of the motion is equal to twice the period of the driver. This is referred to as period doubling. Therefore, there is a subharmonic of the drive frequency .  The Fourier transform shown in figure 5.6 shows a dominant peak at the drive frequency (1.0 Hz) and small peaks at the subharmonic (0.5 Hz) and the 2nd harmonics 2 (2.0 Hz), 3rd harmonic (3.0 Hz), and 4th harmonic (4.0 Hz).

 

 

 

Fig. 5.7   Phase space orbits.  It shows the periodic attractor with period 2 (2 s). The bottom graph shows the two distinct loops each lasting one drive cycle (1 s) in the orbit for the time from 40 s to 50 s when the transient oscillations have fully decay.

 

                         

When the transient oscillations have fully decayed, a closed orbit evolves in the phase space plot that has two distinct orbits can be seen, so the motion repeats itself every two cycles, that is, it has period 2.

 

 

6    Transition to non-sinusoidal motion

      atmPendulumMF.m

 

     

 

In Example 5, even though there was a doubling of the period due to the alternating peaks, the dominant motion was still approximately sinusoidal at the frequency of the driving force. When the driving strength is slightly increased, a subharmonic term dominates and the motion is no longer sinusoidal. The motion settles down to the periodic attractor which has a period of three times the driver period (3 s).

Fig. 6.1   DDP  . After the initial transience, the motion settles down to a periodic attractor with a period of three times the drive period (3 s).

 

          

  

Fig. 6.2.  Strong driver strength . The subharmonic 1/3 is excited. The motion is non-sinusoidal but periodic with a period three times the drive period. he 1/3 Hz subharmonic dominates.

 

Fig. 6.3.  Two solution for the DDP with different initial conditions  but with the same drive strength. The motions are entirely different.

 

For the case of linear DDP, the attractor is independent of the initial conditions. This is not the case for nonlinear DDP. For a nonlinear oscillator, different initial conditions can lead to very different trajectories. For the motion with the initial condition  the period of oscillation is 3 s. When the initial condition was   the period is actually 2 s with the peaks and troughs having slightly different heights.

 

 

7    Period doubling cascade

      atmPendulumMF.m

 

Figure (7.1) shows a sequence of four plots for the DDP with increasing drive strength  for the same initial conditions .

 

  Motion settles down to steady oscillations at the same frequency of the drive excitation. The attractor has a period of 1.00 s. The expanded view shows that all the peaks have the same height.

 

  Motion settles down to steady oscillations at the same frequency of the drive excitation. But now, the peaks are not all the same height. The maxima alternate between two fixed heights, so the attractor has a period of 2.0 s.

 

 

  Motion settles down to steady oscillations at the same frequency of the drive excitation. But now, the peaks are not all the same height. The maxima alternate between four fixed heights, so the attractor has a period of 4.0 s.

 

 

 

  Motion settles down to steady oscillations at the same frequency of the drive excitation. But now, the peaks are not all the same height. The maxima alternate between eight fixed heights, so the attractor has a period of 8.0 s. the pattern repeats itself every 8 drive cycles.

 

Fig. 7.1   A period doubling cascade due to increasing drive strength .

              

 

These four pictures show a period-doubling cascade. Increasing the drive strength would produce further doublings of the period 16, 32,  … ,  .  It can be difficult to find the threshold drive frequencies between doublings. The doublings that occur get faster and faster as the drive strength  is increased. This period-doubling in many nonlinear systems has been observed, for example, electrical circuits, chemical reactions, and balls bouncing on oscillating surfaces.

 

 

8    CHAOS and sensitivity to initial conditions

      atmPendulumMF.m

 

For drive strengths greater than about a critical value of , the solution is now not even periodic at all! In linear theory, a damped system driven periodically must eventually respond periodically at the driving frequency and the oscillations independent upon the initial conditions.  In this case for drive strengths greater than the critical value, the DDP nonlinear system will oscillate forever without ever repeating – it is chaotic.

 

Fig. 8.1   Chaos. Initial conditions  

 

The case for  with initial conditions , is shown in figure 8.1. The DDP is obviously trying to oscillate at the drive frequency fD = 1.00 Hz. Nevertheless, the actual oscillations wander around erratically without ever repeating themselves. This erratic and non-periodic motion is one of the chief features of chaotic motion.

 

Fig. 8.2   Chaos. Initial conditions

             

              

 

The other defining feature of chaos is that a trajectory is extremely sensitive to the initial conditions as shown in figure 8.2 where the initial displacement is decreased by only 0.5%. For the duration of about 7 drive cycles, the motions are almost identical, then they follow different trajectories. This makes it an impossibility to predict the trajectories for chaotic motion.

 

 

 9   Lyapunov Exponents

     atmPendulum2A.m

 

Figures 9.1 shows two motions of the pendulum with a small driving force  with different initial conditions  . The upper graph shows the angular displacements as functions of time and the lower graph, the difference in the angular displacements . is plotted on a logarithmic scale.

 

 Fig. 9.1   Logarithmic plot of , the separation of two identical DDPs with a weak driving force  that were released with different initial angular displacements .

 

It is clear from figure 9.1 the maxima in the log scale for  decrease linearly, hence  decays exponentially, dropping 5 orders of magnitude in the first five drive cycles. In the linear regime, the separation  of two identical DDPs with different initial conditions, decreases exponentially with time. Linear oscillators are insensitive to the initial conditions. So, to make accurate predictions, you only need to know the initial conditions to the same accuracy.

 

Fig. 9.2   Logarithmic plot of , the separation of two identical DDPs with a moderate driving force  that were released with different initial angular displacements .

 

With a moderate drive strength  the motions are periodic and their trajectories converge with  decaying exponentially. For the motion governed by a moderate driving signal, the sharp dips in the  plot occur when one of the pendulums reaches a turning point, following which  will vanish since the two pendulums will cross each other. You will notice that decreases steady with time such that (ignoring the dips). This means that the motion of the pendulum is predictable. Knowledge of the motion of the first pendulum enables you to predict the motion of the second pendulum even though you don’t know the second pendulum’s initial conditions when the forcing is small to moderate.

 

 

Fig. 9.3   Logarithmic plot of , the separation of two identical DDPs with a strong driving force  that were released with an extremely small difference in initial angular displacements . The motion for the large excitation initially shows that the two motions are almost identical then suddenly they diverge as  increases rapidly and irregularly with time. Hence, the motions of the two pendulums diverge from one another with time.

 

For the first few drive cycles, the two pendulums move with very similar trajectories then the crests shown in figure 9.3 increase linearly, showing that  increases exponentially for some time. So, for chaotic damped driven motion of the pendulum, we simple do not know where the pendulum will be in the future, since even for minute differences in the initial conditions it will lead to trajectories that are very different.

 

The mean line drawn (red) to the logarithmic scale in  implies , hence

           

 

where the parameter  is known as the Lyapunov exponent. The above results for pendulums which start with nearly identical initial conditions indicate that for small or medium forcing, the motions will converge exponentially  whereas for high forcing, the trajectories diverge exponentially . The pendulum system while obeying deterministic laws may still exhibit irregular and unpredictable behaviour due to an extreme sensitivity to initial conditions.      

 

 

Chaotic motion exhibits extreme sensitivity to initial conditions and it is this sensitivity makes accurate predicts impossible.

 

 

10    Large drive strengths

        atmDDP.m

 

So far, we have observed the changes in motion of the damped driven pendulum as we have increased the drive strength    

      Small           linear regime – pure sinusoidal

      Moderate     nearly linear regime addition of harmonics, subharmonics, period cascading

      Large          nonlinear regime – chaotic motion

 

But happens when we increase drive strength further ???

 

We get another surprising result - the motion is again periodic, but there is a drift along with the oscillation. For  as shown in figure (10.1), the pendulum is oscillating but also consistently rotating all the way around until it reverses its direction of rotations at irregular intervals. The phase space plot shows a series of almost closed orbits and the general drift in the angular displacement.

 

Fig. 10.1     Large drive strength .  

 

Fig. 10.2    Large drive strength .  Expanded view of the angular displacement showing the erratic superimposed oscillations at the driving frequency as the pendulum rotates.    

 

For the motion for  as shown in figure 10.1, during the first 8 s the pendulum rotates anticlockwise approximately through 13 revolutions and then starts to rotate in the other sense and again reverses direction around 15 s and 25. The pattern of movement never repeats itself. The motion can be described as a rolling motion as the pendulum swings approximately through 2(1 revolution) each drive cycle. The motion is chaotic. This chaotic motion is emphasized in the phase space plot in figure 10.2 as the orbit fails to repeat itself or to close o itself.

 

If we decrease drive strength slightly to  as shown in figure 10.3 the motion is periodic with the angular velocity  returning to the same value once each cycle.

 

Fig. 10.3   Large drive strength . The motion is periodic with the angular displacement decreasing 2 rad each drive cycle. The angular velocity  is clearly periodic with period equal to the drive period of 1 s.

 

The DDP executes a “rolling” motion, making a complete a clockwise revolution each drive cycle. The motion is again periodic with  decreasing by 2 each drive cycle.

 

You will that that  increases (figures 10.4 and 10.5)  the DDP actually alternatives between intervals of chaotic motion separated by intervals of nonchaotic – periodic motion.

 

 

Fig. 10.4      Chaotic motion. Superimposed on the drift in the angular displacement are the erratic oscillations at the drive frequency of 1 Hz.

 

 

Fig. 10.5     Periodic motion. Note: strong fundamental frequency (1 Hz) and prominent harmonics 2 (2 Hz), 3 (3 Hz), and 4 (4 Hz).

 

 

The damped driven pendulum system is so-so simple, but the response is anything but simple. As the amplitude of the driving signal is increased, the motion changes from periodic to chaotic to periodic with drift. This complex response is typical of nonlinear systems. The motion of the pendulum for different model parameters is perplexing considering the fact that the system is completely deterministic but with irregular behaviour.

 

 

11   BIFURCATION DIAGRAM

      atmPendulumBDS.m            atmPendulumBD.m

 

The purpose of a bifurcation diagram is to show in a single plot, the changing periods, alternating periodicity, and chaos as the drive strength  varies. It is a plot of or a plot of  . Figure 11.1 shows the bifurcation diagrams from Taylor’s book Classical Dynamics.

 

 

 

 

Fig. 11.1   Bifurcation diagrams from Taylor’s book Classical Mechanics.

 

It is not an easy task to plot the bifurcation diagram.  The required steps are:

1        Choose a large number of evenly spaced values for   from  to .

2        Choose the initial conditions  and .

3        Solve the equation of motion for DDP given by equation (1) for each value of                      for t = 0 to t = tmax.

4        Check for periodicity or non-periodicity when all the transience has disappeared by examining  or from tmin to tmax in one-cycle intervals with period T

                           

5        Plot these values of  or from tmin to tmax against each value of .  For larger values of  the pendulum can start to make many revolutions, so it is necessary to restrict  to the range  . This can be done using the Matlab statement  X = atan2(tan(X),1)  where X is the Matlab variable for .

 

Fig. 11.2 shows the bifurcation diagrams of the angular displacement  and the angular velocity  verses the drive strength  .  The bifurcation diagram is very dependent upon choice of parameters and initial conditions. The diagrams show the period doubling cascade effect. The angular displacement plot shown figure 11.2 is “poor” compared with Taylor’s plot shown in figure 11.1. Don’t know why the two diagrams are so different in quality. Someone can have a look at my code and improve the Script atmPendulumBDS.m.

Fig. 11.2   Bifurcation diagrams for DDP.   atmPendulumBDS.m

 

Figure 11.3 shows the plot  which shows the period doubling. So, an alternative way to construct the bifurcation diagram is the find the peaks in the angular displacement  after the transient oscillations have fully decayed for each value of the drive strength.

 

Fig. 11.3   Period doubling occurs for  since alternate peaks vary in height.   atmDDP.m

 

Figure 11.4 shows the bifurcation diagram constructed by finding the peaks in the angular displacement using the Matlab function findpeaks.  The section of the Script atmPendulumBD.m  that solves the ODE and finds the peak values is

 

% Solutions X and w stored as a [2D] array

 for k = 1 : Lgamma

     K(3) = gamma(k)*wN^2;

     [t, sol] = ode45(@(t,s) EM(t,s,K), tSpan,s0);

   

% Angular displacement and Velocity

     X = sol(:,1);

     w = sol(:,2);

% Find peak values

    [a, b] = findpeaks(X(tL-10000:tL),t(tL-10000:tL));

    pks(k,:) = a;

% Restrict angular displacement to range -pi to +pi   

    pks = atan2(tan(pks),1);

end

 

The bifurcation diagram in figure 11.4 is an improvement in the plot shown in figure 11.2. Period doubling from 1 to 2 occurs around ; 2 to 4 around ; 4 to 8 not resolved; and chaos when . The details in figure 11.4 and much less distinct than in Taylor’s plot shown in figure 11.1.

Fig. 11.4   Bifurcation diagram constructed by find the peaks in the angular displacement after the transient oscillations have fully decayed.   It took 73 seconds for Matlab to perform all the operation to output .   atmPendulumBD.m

 

Figure 11.5 shows bifurcation plots for a larger range of drive strengths. Again, my plots are not well defined as Taylor’s plot in figure 11.1. But, figure 11.5 shows a number of transitions between periodic motion and chaotic motion.

Fig. 11.5   Bifurcation diagrams for drive strength  from 1.000 to 1.500. The Script atmPendulumBDS.m was used. The Script atmPendulumBD.m could not be used because for the pendulum motion became chaotic and no peaks can be identified using the function findpeaks.

 

 

12   POINCARE SECTION

atmPendulumPS.m

 

There is a way to study chaotic motion that is better than simply plotting the trajectory in phase space because after many cycles it contains too much information to be useful. Consider the phase space plot where points are not plotted at every time step but only at times given by

              

 

 

 

Such a phase space plot is called a Poincare section. If the pendulum oscillates at the driving frequency, then only one point will appear motion in the Poincare section. If the oscillation has twice the frequency of the driving force, then the Poincare section will have two points. If the motion is not periodic and chaotic the Poincare section will consist of a pattern of points called the attractor. The attractor has a structure that is frequently beautiful even though the motion unpredictable and chaotic, yet at the same time preserve a coherent global structure.

 

Period 1 oscillations 

For  the motion is periodic with the period equal to the drive period (1.00 s).

 

 

Fig. 12.1   The phase plot (red: orbit after transient motion decayed away) and Poincare section for . Because the oscillation is described as period 1 motion, there is only one dot in the Poincare section.

 

However, if you zoom in on the Poincare section, there is a line of distinct points as shown in figure 12.2.

Fig. 12.2   The Poincare section shows a line of points in the expanded view of figure 12.1 instead of a single point.  WHY !!!

 

 

Period 2 oscillations 

For  the motion is periodic with the period equal to twice drive period (1.00 s).

Fig. 12.3   The phase space plot for the DDP system . The red trajectory shows the orbit for the last 100 time steps. The motion is period 2 since there are two loops that make up one period of the oscillation. The magenta dots show the point that are displayed on the Poincare section.

 

Fig. 12.4   Poincare section . The two points shown in the Poincare section indicate the motion is period 2.

 

 

Next, we can look at the Poincare section for chaotic motion. In Taylor’s Classical Mechanics his simulation produced the plot shown in figure 12.5.

 

 

Fig. 12.5   Taylor’s Poincare section for chaotic motion.

 

I could not reproduce the curves shown in figure 12.5 using the same parameters as Taylor.  ??? My plot is shown in figure 12.6.

 

Fig. 12.6   Poincare section for chaotic motion using the Script atmPenulumPS.m.

 

 

   

REFERENCES

Basic of Atmospheric Science   A. Chandrasekar

Classical Mechanics   J.R. Taylor

 

Collection of notebooks on dynamical systems  http://www2.me.rochester.edu/courses/ME406/webexamp5/

 

http://galileoandeinstein.phys.virginia.edu/7010/CM_22a_Period_Doubling_Chaos.html

 

https://www.cfm.brown.edu/people/dobrush/am34/live/part3.html

 

GitHub - springer-math/Dynamical-Systems-with-Applications-using-MATLAB: Source code for 'Dynamical Systems with Applications using MATLAB®, 2ed' by Stephen Lynch http://www.springer.com/book/9783319068190

 

Oscillations in Damped Driven Pendulum: A Chaotic System

https://www.arcjournals.org/pdfs/ijsimr/v3-i10/5.pdf