Lecture 17: Nonlinearity & Chaos
Lecture 17: Nonlinearity & Chaos • relevant reading: these notes and slides are your main resource. For further reading, see the recommended text by Baker & Gollub: parts of chapters 1, 2, 3.
Introduction • In this section of the course, we will consider the effects of nonlinearity and the transition from oscillatory motion to chaos. Your computational assignment and problem sets will give you practical experience with this. • We are going to first look at the qualitative difference between regular oscillator motion and chaotic motion. You will be doing some of this work in your computational assignment. • Chaos refers to the irregularity or unpredictability of certain motions. Its important to note that this “unpredictability” is NOT because we don’t understand the equations the govern the system. You can know those perfectly and still end up with chaos. This type of chaos is called “Deterministic chaos”, and is a feature of many systems of ordinary differential equations. • In order to study nonlinear systems, we are going to write our equations in a specific form. Any system of ODEs, of WHATEVER order, can be written as series of first order ODEs for appropriately defined xi ’s: x˙ 1 = F1 (x1 , x2 , ..., xn ) x˙ 2 = F2 (x1 , x2 , ..., xn ) .. . x˙ n = Fn (x1 , x2 , ..., xn ) • Necessary condition for chaos: n ≥ 3, and the system be nonlinear. • Notice there is no t as an independent variable on the right hand side of the above equations. We will see that t can become one of these xi ’s in the system of equations.
1
• Consider the damped/driven nonlinear pendulum (i.e. don’t make small angle approx). The equation governing the motion for a sinusoidal driving force is: θ¨ + 2γ θ˙ + ω02 sin θ = (F0 /ml) cos ωd t.
(1)
• see python demo ddp nonlinear.py for a visual of this system. • Define β ≡ F0 /(mlω02 ), so θ¨ + 2γ θ˙ + ω02 sin(θ) = βω02 cos ωd t.
(2)
When β ∼ 1, nonlinear effects kick in because the pendulum can be sent ”over the top”. (for example, run the system with β = 0.9 in vpython mode and you will see that the pendulum does NOT go over the top. Then run it at β = 1.2 and you will see that it does). • So we are going to write this second order equation as a system of 1st order equations. Here is how: • Define x1 ≡ θ and x2 ≡ θ˙ = x˙ 1 , and x3 ≡ ωd t. Then we can write this system as follows:
x˙ 1 = x2 x˙ 2 = −2γx2 − ω02 sin x1 + βω02 cos x3 x˙ 3 = ωd
(linear) (nonlinear) X n = 3X (linear)
• We now have a system of 3 1st order equations for x1 , x2 , x3 and there is nonlinearity in the system (because of the cos and sin terms). So there is a possibility of chaos. • In general, the way to form these 1st order equations is always the same. If you have higher than first order derivatives, name your 1st derivatives as a new variable xi . So for example, if I had the 5th order equation: d5 x = 10 dt5 I could write it as a system of first order equations by defining: 2
x1 = x, x2 = x˙ 1 , x3 = x˙ 2 , x4 = x˙ 3 , x5 = x˙ 4 Then my system of equations is: x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5
= = = = =
x2 x3 x4 x5 10
(linear) (linear) (linear) (linear) (linear)
Notice that although the n ≥ 3 requirement is met, the nonlinear requirement is not met so there is no chaos possible in this system. • With a coupled pendulum system (remember lectures 15 and 16), we have two second order ODEs, giving n = 4, and a high degree of nonlinearity. So this system has a lot of potential for chaos.
Development of Chaos in the DDP • Let us examine the behaviour of the Damped Driven Pendulum (DDP) as β is varied. • The β = 0.2 case is basically linear. The motion is in step with the forcing and the angle stays pretty small. The phase space plot is elliptical. (examine with ddp nonlinear.py in vpython mode). • NOTE: Technically, the full phase space would be a plot of all the dependent variables in our system. So in this case we would want to plot trajectories in 3D (i.e. one axis for x1 , one for x2 and one for x3 for our system. Notice that x3 is really just the time variable. So usually, we’ll just look at 2-D slices of this phase space (i.e. just plot x1 vs x2 which is our usual phase space plots of position vs velocity. • We start to see nonlinear effects for β = 0.9. The motion is still in step with the forcing. But the shape of the trace is not purely sinusoidal. Run ddp nonlinear.py in vpython mode. Notice how in the phase space, the trajectory moves around at first until it finally settles down on a closed curve. This ’moving around’ is part of the transient epoch 3
when the homogeneous solution is still playing apart (because it hasn’t damped out yet). Once the homogeneous solution decays, we are left with periodic motion, although not purely sinusoidal like in the linear case. • If we use the pylab plotting function instead of the vpython plotting ˙ and the function in ddp nonlinear.py we see the trace of θ(t) and θ(t) phase space plot. Aside: You are not responsible for the next bit, but if interested here it is • The plot of θ(t) gives shallower peaks compared to sinusoidal, and the phase space ellipse has “sharpened” sides. What is happening here is that ”harmonics” of the driving frequency are appearing. Harmonics are multiples of the original frequency. Why do they appear? A nice reasoning can be given if you consider the Taylor series expansion of the sin θ term in the equation of motion: • Weakly nonlinear effects: Think of Taylor series expansion of sin θ which gives the nonlinearity in the DDP. sin θ = θ −
θ3 + H.O.T. 6
Suppose we put the θ3 term on the right-hand-side of the equation, and said that to leading order θ ∼ cos ωd t. (Note: I am just trying to show you how subharmonics come up. You aren’t expected to figure this out for yourself. We haven’t gone through the math that leads up to this): 3 eiωd t + e−iωd t cos ωd t = 2 i3ωd t e + 3eiωd t + 3e−iωd t + ei3ωd t = 8 cos 3ωd t + 3 cos ωd t = . 4 So we get a term of frequency 3ωd — a harmonic — forcing the equation of motion! This flattens the peaks relative to sinusoidal. 3
4
End of Aside: You are responsible for the next bit • We can plot the motion in phase space. To see the solution after the transient decays, we only plot after some initial time period. (in ddp nonlinear.py this is done with the parameter numcyclestart). Then our phase space shows only an ellipse with “sharpened” sides, which represents the slowing down of the oscillator as it approaches the top. We’ve started plotting at a time at which the pendulum’s motion is already fixed on this ellipse. At this β, after every period, the motion completes 1 full ellipse. We therefore say that this system has an attractor (since the trajectory was attracted to this ellipse) and that in this case, the attractor is a single period cycle. • Another way of representing this motion is a Poincar´e section, which is a snapshot of the motion in phase space taken every driving period τd = 2π/ωd (also only after the transient decays). • On the left are the phase space and the Poincar´e section (running ddp nonlinear.py in pylab mode will make a similar figure):
5
• Notice that because we return to the same place after every driving period, we only get a single dot on the Poincare section. So for an attractor that is a single period cycle, the Poincare section is a single dot. • What happens as we increase β? If we try 1.07 we get the phase space plot on the right in the figure above after the transient decays. • Notice that the maxima in θ bounce back and forth between 2 values. You can characterize this motion with 2 periods: (1) The time it takes to cover 2π in angle around the graph and (2) The time it takes to get back to exactly the same spot as before. In the Poincare section, the first period is the time it takes to go from 1 dot to the other dot (this is the driving frequency). The second period is the time it takes to go from 1 dot to the other dot then back to the first dot. In this example that is 2 × the driving period (or equivalently, 0.5 × the driving frequency). This attractor is known as a “two period cycle” because you bounce between 2 cycles. If I were to sketch what is happening in time I would get:
In ddp noonlinear.py this looks like:
6
Notice how the max change height every second one, similar to the hand drawn sketch above (although the difference in magnitude is less here). • So we can get subharmonics with frequencies ωm = ωd /m of our driving frequency (for our example above, the period of the subharmonic was 2x the driving period, so the frequency is half the driving frequency and m = 2). • What if we increase β further? For β = 1.15 you get the following phase space plot and Poincare section: (note: if you try this value using ddp nonlinear.py you will get a small sampling of this figure. You need to increase the number of time points a lot to get the figure below. I haven’t tried it, but if you are game, go for it. If it crashes, I’m sorry)
7
• This is where the advantage of the Poincar´e section is seen clearly. The phase space trajectories are too dense to make sense. The Poincar´e section provides a look at an ordered structure underneath the mess of the phase portrait. Notice the complicated layered structure that occupies a compact region of phase space in the Poincare section. This is no longer a single period cycle, or a two period cycle or a 10 period cycle. The attractor in this case is “stranger” and hence, this attractor is known as a “strange attractor”. • The attractor has a self-similar structure: reappearance of certain patterns at smaller scales. You can see this by zooming in on the attractor in the Poincare section. For the same system as above, if I let the program run longer, then my attractor has more structure and looks like:
8
If I magnify a region in the top right quadrant it looks like:
If I magnify the region near of the figure above near the bottom right I get:
So the structure you see in the figure looks the same no matter what the resolution you have. This is a standard characteristic of a “strange attractor”. A shape like this is said to have a “fractal dimension”. • When you have a strange attractor, your system is chaotic. This is 9
because (as we will see next time), it is hard to predict the location of your system at a particular time. This is because the attractor is so complicated, that you could be at any of an infinite number of points. The problem then is if I start my system with 2 initial conditions that are SLIGHTLY different, then can end up far apart at a later time. Since its hard (actually impossible) for me to measure my exact initial condition, I don’t know which scenario I actually started with. This “sensitivity to initial conditions” is the hallmark of chaotic systems. • So right now you have a picture in your head that as we increased β we went from a single period attractor, to a 2 period attractor to a strange attractor. So the question now is, what happens if we increase β further. The answer may surprise you. You can actually return to another single period cycle attractor! • For example with β = 1.35 we get:
This is again, a single period attractor although the type of motion is 10
quite different. Using ddp nonlinear.py in vpython mode we see that the pendulum is continuously spinning around the top in 1 direction. As we continue to increase β again, we start getting period doubling just like we did before. First we get a two period cycle at β = 1.45 (represented by 2 points on the Poincare section, set numcycstart=30), then at β = 1.47 we get a 4 period cycle (note: the actual β values might be slightly different on your computer):
(Note: at the lower β values when we first saw our two period cycle, there was also a 4 period cycle at slightly higher β although we didn’t show it). • The period keeps doubling as β is increased until we get to another chaotic regime:
11
• This period doubling is known as a “subharmonic cascade” to chaos. • The behaviour is summarized in the “bifurcation diagram”. What you do is plot the θ value from your Poincare section as a function of β. So for β values where you have a one period cycle, you have 1 dot on the curve, for β values where you have a 2 period cycle, you have 2 values on the curve at the same β value and so on. The picture you get is below. The bifurcation diagram also exhibits a self-similar structure
12
(each picture below the first one is a zoom in on a part of the previous picture):
13
Note: You could have also made your bifurcation diagram by plotting θ˙ from your Poincare section instead of θ. The shape of the figure would look the same, but the numbers would change slightly.
14
Introduction • In this section of the course, we will consider the effects of nonlinearity and the transition from oscillatory motion to chaos. Your computational assignment and problem sets will give you practical experience with this. • We are going to first look at the qualitative difference between regular oscillator motion and chaotic motion. You will be doing some of this work in your computational assignment. • Chaos refers to the irregularity or unpredictability of certain motions. Its important to note that this “unpredictability” is NOT because we don’t understand the equations the govern the system. You can know those perfectly and still end up with chaos. This type of chaos is called “Deterministic chaos”, and is a feature of many systems of ordinary differential equations. • In order to study nonlinear systems, we are going to write our equations in a specific form. Any system of ODEs, of WHATEVER order, can be written as series of first order ODEs for appropriately defined xi ’s: x˙ 1 = F1 (x1 , x2 , ..., xn ) x˙ 2 = F2 (x1 , x2 , ..., xn ) .. . x˙ n = Fn (x1 , x2 , ..., xn ) • Necessary condition for chaos: n ≥ 3, and the system be nonlinear. • Notice there is no t as an independent variable on the right hand side of the above equations. We will see that t can become one of these xi ’s in the system of equations.
1
• Consider the damped/driven nonlinear pendulum (i.e. don’t make small angle approx). The equation governing the motion for a sinusoidal driving force is: θ¨ + 2γ θ˙ + ω02 sin θ = (F0 /ml) cos ωd t.
(1)
• see python demo ddp nonlinear.py for a visual of this system. • Define β ≡ F0 /(mlω02 ), so θ¨ + 2γ θ˙ + ω02 sin(θ) = βω02 cos ωd t.
(2)
When β ∼ 1, nonlinear effects kick in because the pendulum can be sent ”over the top”. (for example, run the system with β = 0.9 in vpython mode and you will see that the pendulum does NOT go over the top. Then run it at β = 1.2 and you will see that it does). • So we are going to write this second order equation as a system of 1st order equations. Here is how: • Define x1 ≡ θ and x2 ≡ θ˙ = x˙ 1 , and x3 ≡ ωd t. Then we can write this system as follows:
x˙ 1 = x2 x˙ 2 = −2γx2 − ω02 sin x1 + βω02 cos x3 x˙ 3 = ωd
(linear) (nonlinear) X n = 3X (linear)
• We now have a system of 3 1st order equations for x1 , x2 , x3 and there is nonlinearity in the system (because of the cos and sin terms). So there is a possibility of chaos. • In general, the way to form these 1st order equations is always the same. If you have higher than first order derivatives, name your 1st derivatives as a new variable xi . So for example, if I had the 5th order equation: d5 x = 10 dt5 I could write it as a system of first order equations by defining: 2
x1 = x, x2 = x˙ 1 , x3 = x˙ 2 , x4 = x˙ 3 , x5 = x˙ 4 Then my system of equations is: x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5
= = = = =
x2 x3 x4 x5 10
(linear) (linear) (linear) (linear) (linear)
Notice that although the n ≥ 3 requirement is met, the nonlinear requirement is not met so there is no chaos possible in this system. • With a coupled pendulum system (remember lectures 15 and 16), we have two second order ODEs, giving n = 4, and a high degree of nonlinearity. So this system has a lot of potential for chaos.
Development of Chaos in the DDP • Let us examine the behaviour of the Damped Driven Pendulum (DDP) as β is varied. • The β = 0.2 case is basically linear. The motion is in step with the forcing and the angle stays pretty small. The phase space plot is elliptical. (examine with ddp nonlinear.py in vpython mode). • NOTE: Technically, the full phase space would be a plot of all the dependent variables in our system. So in this case we would want to plot trajectories in 3D (i.e. one axis for x1 , one for x2 and one for x3 for our system. Notice that x3 is really just the time variable. So usually, we’ll just look at 2-D slices of this phase space (i.e. just plot x1 vs x2 which is our usual phase space plots of position vs velocity. • We start to see nonlinear effects for β = 0.9. The motion is still in step with the forcing. But the shape of the trace is not purely sinusoidal. Run ddp nonlinear.py in vpython mode. Notice how in the phase space, the trajectory moves around at first until it finally settles down on a closed curve. This ’moving around’ is part of the transient epoch 3
when the homogeneous solution is still playing apart (because it hasn’t damped out yet). Once the homogeneous solution decays, we are left with periodic motion, although not purely sinusoidal like in the linear case. • If we use the pylab plotting function instead of the vpython plotting ˙ and the function in ddp nonlinear.py we see the trace of θ(t) and θ(t) phase space plot. Aside: You are not responsible for the next bit, but if interested here it is • The plot of θ(t) gives shallower peaks compared to sinusoidal, and the phase space ellipse has “sharpened” sides. What is happening here is that ”harmonics” of the driving frequency are appearing. Harmonics are multiples of the original frequency. Why do they appear? A nice reasoning can be given if you consider the Taylor series expansion of the sin θ term in the equation of motion: • Weakly nonlinear effects: Think of Taylor series expansion of sin θ which gives the nonlinearity in the DDP. sin θ = θ −
θ3 + H.O.T. 6
Suppose we put the θ3 term on the right-hand-side of the equation, and said that to leading order θ ∼ cos ωd t. (Note: I am just trying to show you how subharmonics come up. You aren’t expected to figure this out for yourself. We haven’t gone through the math that leads up to this): 3 eiωd t + e−iωd t cos ωd t = 2 i3ωd t e + 3eiωd t + 3e−iωd t + ei3ωd t = 8 cos 3ωd t + 3 cos ωd t = . 4 So we get a term of frequency 3ωd — a harmonic — forcing the equation of motion! This flattens the peaks relative to sinusoidal. 3
4
End of Aside: You are responsible for the next bit • We can plot the motion in phase space. To see the solution after the transient decays, we only plot after some initial time period. (in ddp nonlinear.py this is done with the parameter numcyclestart). Then our phase space shows only an ellipse with “sharpened” sides, which represents the slowing down of the oscillator as it approaches the top. We’ve started plotting at a time at which the pendulum’s motion is already fixed on this ellipse. At this β, after every period, the motion completes 1 full ellipse. We therefore say that this system has an attractor (since the trajectory was attracted to this ellipse) and that in this case, the attractor is a single period cycle. • Another way of representing this motion is a Poincar´e section, which is a snapshot of the motion in phase space taken every driving period τd = 2π/ωd (also only after the transient decays). • On the left are the phase space and the Poincar´e section (running ddp nonlinear.py in pylab mode will make a similar figure):
5
• Notice that because we return to the same place after every driving period, we only get a single dot on the Poincare section. So for an attractor that is a single period cycle, the Poincare section is a single dot. • What happens as we increase β? If we try 1.07 we get the phase space plot on the right in the figure above after the transient decays. • Notice that the maxima in θ bounce back and forth between 2 values. You can characterize this motion with 2 periods: (1) The time it takes to cover 2π in angle around the graph and (2) The time it takes to get back to exactly the same spot as before. In the Poincare section, the first period is the time it takes to go from 1 dot to the other dot (this is the driving frequency). The second period is the time it takes to go from 1 dot to the other dot then back to the first dot. In this example that is 2 × the driving period (or equivalently, 0.5 × the driving frequency). This attractor is known as a “two period cycle” because you bounce between 2 cycles. If I were to sketch what is happening in time I would get:
In ddp noonlinear.py this looks like:
6
Notice how the max change height every second one, similar to the hand drawn sketch above (although the difference in magnitude is less here). • So we can get subharmonics with frequencies ωm = ωd /m of our driving frequency (for our example above, the period of the subharmonic was 2x the driving period, so the frequency is half the driving frequency and m = 2). • What if we increase β further? For β = 1.15 you get the following phase space plot and Poincare section: (note: if you try this value using ddp nonlinear.py you will get a small sampling of this figure. You need to increase the number of time points a lot to get the figure below. I haven’t tried it, but if you are game, go for it. If it crashes, I’m sorry)
7
• This is where the advantage of the Poincar´e section is seen clearly. The phase space trajectories are too dense to make sense. The Poincar´e section provides a look at an ordered structure underneath the mess of the phase portrait. Notice the complicated layered structure that occupies a compact region of phase space in the Poincare section. This is no longer a single period cycle, or a two period cycle or a 10 period cycle. The attractor in this case is “stranger” and hence, this attractor is known as a “strange attractor”. • The attractor has a self-similar structure: reappearance of certain patterns at smaller scales. You can see this by zooming in on the attractor in the Poincare section. For the same system as above, if I let the program run longer, then my attractor has more structure and looks like:
8
If I magnify a region in the top right quadrant it looks like:
If I magnify the region near of the figure above near the bottom right I get:
So the structure you see in the figure looks the same no matter what the resolution you have. This is a standard characteristic of a “strange attractor”. A shape like this is said to have a “fractal dimension”. • When you have a strange attractor, your system is chaotic. This is 9
because (as we will see next time), it is hard to predict the location of your system at a particular time. This is because the attractor is so complicated, that you could be at any of an infinite number of points. The problem then is if I start my system with 2 initial conditions that are SLIGHTLY different, then can end up far apart at a later time. Since its hard (actually impossible) for me to measure my exact initial condition, I don’t know which scenario I actually started with. This “sensitivity to initial conditions” is the hallmark of chaotic systems. • So right now you have a picture in your head that as we increased β we went from a single period attractor, to a 2 period attractor to a strange attractor. So the question now is, what happens if we increase β further. The answer may surprise you. You can actually return to another single period cycle attractor! • For example with β = 1.35 we get:
This is again, a single period attractor although the type of motion is 10
quite different. Using ddp nonlinear.py in vpython mode we see that the pendulum is continuously spinning around the top in 1 direction. As we continue to increase β again, we start getting period doubling just like we did before. First we get a two period cycle at β = 1.45 (represented by 2 points on the Poincare section, set numcycstart=30), then at β = 1.47 we get a 4 period cycle (note: the actual β values might be slightly different on your computer):
(Note: at the lower β values when we first saw our two period cycle, there was also a 4 period cycle at slightly higher β although we didn’t show it). • The period keeps doubling as β is increased until we get to another chaotic regime:
11
• This period doubling is known as a “subharmonic cascade” to chaos. • The behaviour is summarized in the “bifurcation diagram”. What you do is plot the θ value from your Poincare section as a function of β. So for β values where you have a one period cycle, you have 1 dot on the curve, for β values where you have a 2 period cycle, you have 2 values on the curve at the same β value and so on. The picture you get is below. The bifurcation diagram also exhibits a self-similar structure
12
(each picture below the first one is a zoom in on a part of the previous picture):
13
Note: You could have also made your bifurcation diagram by plotting θ˙ from your Poincare section instead of θ. The shape of the figure would look the same, but the numbers would change slightly.
14
