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University of Toronto St. George
Sabine Stanley

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 e▯ects of nonlinearity and the transition from oscillatory motion to chaos. Your computa- tional assignment and problem sets will give you practical experience with this. ▯ We are going to ▯rst look at the qualitative di▯erence 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 di▯erential equations. ▯ In order to study nonlinear systems, we are going to write our equations in a speci▯c form. Any system of ODEs, of WHATEVER order, can be written as series of ▯rst order ODEs for appropriately de▯ned x ’s: i x_1 = F (x1;x1;::2;x ) n x_2 = F (x2;x1;::2;x ) n . . x_ = F (x ;x ;:::;x ) n n 1 2 n ▯ 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 x ’s i 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 sinu- soidal driving force is: 2 ▯ + 2 ▯ + ! 0 sin▯ = (F 0ml)cos! t:d (1) ▯ see python demo ddp for a visual of this system. ▯ De▯ne ▯ ▯ F =(0l! ), 0o  _ 2 2 ▯ + 2 ▯ + ! s0n(▯) = ▯! co0! t: d (2) When ▯ ▯ 1, nonlinear e▯ects kick in because the pendulum can be sent "over the top". (for example, run the system with ▯ = 0:9 in vpython mode and you will see that the pendulum does NOT go over the top. Then run it at ▯ = 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: ▯ De▯ne x ▯ ▯ and x ▯ ▯ = x _ _ , and x ▯ ! t. Then we can write this 1 2 1 3 d system as follows: 9 x_1 = x2 (linear) = 2 2 x_2 = ▯2 x ▯ 2 0sinx 1 ▯! 0cosx 3 (nonlinear) X n = 3X x_3 = ! d (linear) ; ▯ We now have a system of 3 1st order equations for x ;x 1x 2nd 3here 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 ▯rst order derivatives, name your 1st derivatives as a new variable x i So for example, if I had the 5th order equation: 5 d x dt 5 = 10 I could write it as a system of ▯rst order equations by de▯ning: 2 x1= x; x 2 x _1; x3= x_2; x4= x 3 ; 5 = x_4 Then my system of equations is: _ = x (linear) 1 2 _2 = x 3 (linear) _3 = x 4 (linear) _4 = x 5 (linear) _5 = 10 (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 ▯ is varied. ▯ The ▯ = 0:2 case is basically linear. The motion is in step with the forc- ing and the angle stays pretty small. The phase space plot is elliptical. (examine with ddp 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 for1x , one for2x and one for 3 for our system. Notice that x is really just the time variable. So usually, 3 we’ll just look at 2-D slices of this phase space (i.e. just plot 1 vs x2 which is our usual phase space plots of position vs velocity. ▯ We start to see nonlinear e▯ects for ▯ = 0:9. The motion is still in step with the forcing. But the shape of the trace is not purely sinusoidal. Run ddp in vpython mode. Notice how in the phase space, the trajectory moves around at ▯rst until it ▯nally 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 _ function in ddp we see the trace of ▯(t) and ▯(t) and the phase space plot. Aside: You are not responsible for the next bit, but if inter- ested here it is ▯ The plot of ▯(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▯ term in the equation of motion: ▯ Weakly nonlinear e▯ects: Think of Taylor series expansion of sin▯ which gives the nonlinearity in the DDP. 3 sin▯ = ▯ ▯ ▯ + H:O:T: 6 3 Suppose we put the ▯ term on the right-hand-side of the equation, and said that to leading order ▯ ▯ cos! td (Note: I am just trying to show you how subharmonics come up. You aren’t expected to ▯gure this out for yourself. We haven’t gone through the math that leads up to this): ▯ ▯ eid t+ e▯i!dt 3 cos ! d = 2 i3d t i!dt ▯i!dt i3!dt e + 3e + 3e + e
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