one of the classic examples of a simple discrete-time dynamical system comes from a bouncing ball. if you let a ball go above a table, then, assuming everything is perfectly flat, etc., a simple model for the dynamics is x(n+1)=r x(n), where 0<r<1 is the coefficient of restitution and x(n) is the height of the n-th bounce. very well: that has a simple explicit solution that ends with the ball coming to rest.
but what happens if we give the table a sinusoidal oscillation? let’s take a look…
this is likewise modeled as a dynamical system, but no longer 1-dimensional — it is a 2-dimensional discrete-time dynamical system. without going into a derivation, let me say that the two variables are related to the time-between-bounces and the velocity-upon-impact. indexed by n, the bounce number, this leads to complicated behaviors. of course, everything depends as well on parameters, such as ball size, mass, frequency of oscillation, and initial condition, as you can see from the following simulation.
this system goes straight to the heart of why 2-d discrete-time dynamics can be chaotic. we won’t go into details on this quite yet… but let me say that this is one of the four chaotic dynamical systems that appears in the beginning of the seminal text of guckenheimer & holmes. phil homles (my phd advisor) studied this system in 1982. this simulation is a bit easier to “see” that the static picture appearing in that classic text, but the picture painted by the mathematics is every bit as beautiful & illustrative.
i like pendula in all forms as a way to visualize interesting dynamical systems. i’m most interest in the mathematics behind it all, but occasionally, i let my initial training in mechanical engineering get the best of me. this is one of those times.
consider what happens when you have a double-pendulum, but where the joints are spherical as opposed to planar (or “hinge”) in nature. this is really beautiful.
this can be a little difficult to visualize properly, since there is motion out of the plane of view. let’s take a look at the exacy same simulation, but from below…
so what happens if we build a multi-pendulum with structures other than rigid rods? using hoops (solid tori) is especially mesmerizing, and leads one to believe that the dynamics is very different that the rigid-rod case…
a more careful analysis would show that using rings is equivalent to using rods, as long as you are careful about the equivalent mass (becuase of moment of inertia, the mass of the solid rod and that of the corresponding ring are not the same).
of course, this is be taken to foolish extremes. i do not recommend that you build the following system:
of course, instead of shaking a pendulum horizontally or vertically, one can go meta and hang a pendulum from a pendulum. let’s see what happens with a such a double pendulum.
that looks chaotic, doesn’t it? but what is meant by chaos? that takes a long time to learn properly. for now, let’s focus on one crucial requirement for chaos: sensitive dependence on initial conditions. this is the idea that nearby initial conditions diverge in behavior exponentially quickly. let’s take a look:
why is the double pendulum chaotic when the single pendulum is not (unless shaken in the right way)? it’s all about topology. the configuration space of a single pendulum is two-dimensional — one for the angle and one for the angular velocity. the configuration space is the tangent bundle of a circle, homeomorphic to an annulus. that’s not enough room for a flow to get chaotic. but the double pendulum has twice the dimension — it’s a 4-d tangent bundle of a torus — and that is plenty of room for chaos to creep in.
several commenters on twitter/youtube liked the horizontally shaken pendula and asked about what was being represented. there were many pendula with different initial positions — all initial velocities were set to zero. there were several queries as to what would happen with different lengths or different masses… would resonances with the vibration and the natural frequencies kick in?
dynamical systems studies *interesting* behaviors in systems. one such class of interesting behaviors is equilibria, with stable and unstable varieties. but there’s more. consider a slight variation on the shaken inverted pendulum, where the shaking is happening horizontally. this leads to interesting and varied behaviors, depending (somewhat sensitively) on the initial conditions. some are *periodic* (they repeat over time); others seem to keep evolving and keep getting more & more complex. what is happening? how do we make sense of such? oh, there’s much more to learn about dynamical systems…