4. (16 points) Surfaces of Section
(a) (3 points) A plane pendulum with a ball of mass m at the end of a massless, rigid rod of length L is attached to a horizontally moving support that is driven according to x = x0 sin (ω0t). If we introduce a dimensionless time τ = t/T, where T = 1/ω0, and consider the particular case of x0 = L, the Euler-Lagrange equations yield the dimensionless equation
(3)
where overdots represent derivatives with respect to τ. We can turn this second-order differential equation into two first-order ODEs through the standard trick (like we need in order to use ODEint in Python):
(4)
This is an explicitly time-dependent system with one degree of freedom, i.e., two phase space dimensions (θ, ω) with an explicit time dependence. These are the lowest-dimensional systems that can exhibit chaos (time-independent one degree-of-freedom systems cannot).
Using the programming language of your choice, plot numerical solutions for differ-ent initial conditions in (θ, ω) state space, assuming Show that different trajectories cross one another on such a plot. This is due to the explicit time depen-dence (they aren’t crossing at the same time, only at different times). This makes it difficult to visualize the dynamics.
(b) (5 points) We can always convert a time-dependent one degree-of-freedom system to a two degree-of-freedom torus picture like we considered in class by thinking of time as an additional angle θ2:
(5)
i.e. solving the set of differential equations
(6)
Notice that the third ODE is immediately solvable, so this set of differential equa-tions is equivalent to those in part (a). In this picture, θ2 (secretly, time) is the az-imuthal coordinate long-ways around the outside of the doughnut, while our origi-nal θ is the angle around around the small circular cross-section of the doughnut.
Use your programming language of choice to plot a surface of section at θ2 = 0, i.e., plotting a point in θ, ω space only each time θ2 (i.e., time) passes 0, 2π, 4π, etc. Assume again that ω0
2 = g/L. Try different initial conditions and attach a plot that includes quasiperiodic and chaotic trajectories. You should find that different trajec-tories now do not cross one another.
Explain in words what the times θ2 = 0, 2π, 4π, ... physically correspond to in the real system.
(c) (8 points) The differential equations for a double pendulum, where we assume that both balls are of mass m, that the lengths of both pendulums are l = L, and that time has been non-dimensionalized through a dimensionless τ = t/T = ω0t, with are given by
(7)
where ∆θ = θ2 — θ1, and θ1 and θ2 are the angles from the vertical for the top and bottom pendulum, respectively . The dynamics conserve the dimensionless Hamil-tonian
(8)
Making a surface of section for this system is much trickier than the case above (try it!), because you cannot tell the numerical integrator what values of time you want to integrate to ahead of time–you would have to integrate along, and each time you cross the section, go backwards in time and find the exact time at which you crossed. The simplest method would be to narrow down the time through bisection. A nice general trick is to instead switch the independent variable from time to the angle at which you want to take your section (here θ2). We can use the chain rule to convert any time derivative to a derivative with respect to θ2, e.g.,
(9)
This leads to differential equations
(10)
Now that we have everything in terms of θ2 (think of it as time like in part (b), we can again integrate to θ2 = 0, 2π, 4π, ... for our section.
Using your programming language of choice, plot a surface of section at θ2 = 0 (only when passing from negative θ2 to positive) for different initial conditions, all of which have a fixed value of H = 1. Include quasiperiodic and chaotic trajectories.