systems Chaos
NL are also susceptible to chaos. What is chaos? It is not necessarily random, unpredictable, or erratic motion--such might simply be due to an incomplete understanding of the relevant physics. Characteristics of chaos chaotic trajectories densely fill a system's phase diagram, whereas non chaotic (eg, regular) trajectories tend to be simple curves, such as ellipses chaotic trajectories exhibit a sensitive dependence upon initial conditions, ie, particles having slightly different initial positions wind up in wildly different positions at later times. But if the motion is not chaotic, then initially adjacent particles tend to have adjacent endstates. see Fig. 419 for examples of regular and chaotic trajectories for the NL driven pendulum.
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Fig. 419
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Damped, driven pendulum
Fig. 418
This system is NL, and can exhibit regular & chaotic trajectories. Obtain the EOM by considering the all the torques exerted on mass m: dL dt where L = r p angular momentum z = m 2^ note vectors point in/out of page along ^ z dL z = m 2^ = Nf riction + Ngravity + Ndriving so N = dt where Ngravity = r Fgravity = - mg sin ^ z z and assuming Nf riction = -b^ and Ndriving = Nd cos dt^ z so the EOM is m 2 = -b - mg sin + Nd cos dt Nd b 2 + 0 sin = cos dt or + m 2 m 2 where 0 = g/ = natural oscillation frequency and d = driving frequency recall that torque N = Is this a linear or NL oscillator?
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Dimensionless EOM
Solving this NL system (ie, when is not necessarily small) requires a numerical solution. For numerical work, it is usually convenient to recast the EOM in a dimensionless form. How? And how do you make the time coordinate dimensionless?
2 Divide by 0 to make the EOM dimensionless, and use = 0 t as a dimensionless time coordinate.
b d Nd + cos + sin = 2 2 0 m 2 0 0 m 2 0 0 d 1 d = note that 0 dt d 1 d2 d2 and = 2 0 dt2 d 2 b and setting 2 = = dimensionless damping coefficient m 2 0 Nd and F = 2 = dimensionless torque m 2 0 d and = = dimensionless driving frequency then + 2 + sin = F cos where the derivatives are now with respect to the dimensionless time = 0t. Note that this EOM is a secondorder differential equation (DE).
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Convert to 1st order DE's
Note that most numerical algorithms that solve DE's can only solve systems of 1st order DE's, one common example being the RungeKutta integrator. However you can always convert a single 2nd order DE to two 1st order DE's: set x( ) and y( ) so y and x = = = = ( ) = position x = = velocity = -2y - sin x + F cos y
and this system of equations are easily solved numerically using MAPLE to study chaotic/regular trajectories...
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Poincar e
For Sections regular or nonchaotic trajectories, the phase diagram, x plotted versus x, is usually quite easy to interpret. However the phase diagram for a chaotic trajectory is usually a mess--see Fig. 419. Note that for a 1D system, the usual phase diagram is actually a 3D trajectory (x and x plotted versus t) projected (or flattened) onto a 2D plane:
Fig. 420
For example, note that the left Figs. in 419 is Fig. 420 projected to the (x, t) plane. The middle Figs. in 419 is Fig. 420 projected to the (x, x) plane. However the right Figs. 419 is Fig. 420 plot (x, x) at stroboscopic time intervals t =system's natural time interval (such as the driving period 2/). Such diagrams are called Poincar` sections, and they are used to e
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identify regular and chaotic trajectories. For the driven pendulum problem, one might generate the Poincar` section e by plotting the particle's (x, x) as increments by 2. So what does the Psection for a driven SHO look like? What about quasiperiodic motion of the driven pendulum (eg, a pendulum that briefly circulates, then oscillates)? See Fig. 419. However chaotic motion results in a very intricate and irregular Psection inspection of a system's Psection can be used to identify chaotic/regular trajectories.
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Lyapunov exponent
Another characteristic of chaos is that initially adjacent trajectories tend to diverge away from each other exponentially with time. Suppose you start a article at some initial position x0 and evolve it forward from time=0 to a later time t, and similarly another particle but at a nearby position away, and evo...