From Periodic Motion to Unbounded Chaos: Investigations of the Simple Pendulum

The simplest example of the onset of chaos in a Hamiltonian system is provided by the “standard” or Chirikov-Taylor model. As a nonlinearity parameter, k , is increased the long term behavior of the momentum, p , is examined. At k = 0, p is conserved. For k < k,, for all startingpoints,p is of bounded variation. For some starting points its behavior is periodic, for others quasi-periodic, for others chaotic. At some critical value of k , unbounded chaotic variation first appears. A scaling analysis to describe this onset is described. The problem we will discuss in this lecture has a long history. The basic work in the problem was done by Kolmogorov, Arnold and Moser [l-31; more recent work has been done by J. Greene, B. V. Chirikov, D. Escande, F. Doveil, and R. MacKay. I have worked on this problem in collaboration with S . J. Shenker; parts of this and related work were done in collaboration with M. Widom, A. Zisook, M . Feigenbaum, D. Bensimon and Subir Sarkar. The phenomenology of Hamiltonian systems is quite different from that of dissipative systems. In this lecture we shall analyze in detail the breakdown of a KAM curve and the onset of unbounded chaotic motion in a particular map. First, let us give three physical systems to motivate the study of this map. First consider a pendulum (Fig. 1) ml2ntj = -mgsin(2ne) (1) in which we choose units of 0 so that 0 = 1 corresponds to 360”. Let us act upon this system periodically by modulating g, the force due to gravity (say, by wiggling the support of the pendulum) g = go + g, sin (ut) (2) We get equations i. = -k(t)sin(2nB) B = r Here k ( t ) is a periodic function of line with frequency w and r is the velocity of the pendulum. We can now use a trick due to PoincarB to transform this differential equation into a map. Observe the pendulum once each period of the force; let rj = r ( t j ) and B j = 8(t j ) where tj = ( 2 n / ~ ) ~ . Since the phase of the external force at time tj is independent of j , one can integrate the equations of motion (3) over this period, expressing the new state of the pendulum in terms of its state one period earlier: (3) * From lectures originally delivered at Erice in 1983. The writeup comes from notes by R. de la Llave and J. Sethna. (4) In general, F and G will be some nonlinear functions periodic in 8 . The simplest model which seems to capture the physics of this system is the standard map = rj (k/2n) sin (2~0,) (5) = e, +rj+] also known as the Chirikov-Taylor model [4]. (Fradkin and Huberman [ 51 have studied this periodically modulated pendulum and have indicated how eq. (4) can be converted to eq. (5) in several limiting situations.) The second system we will use to motivate this map is an accelerator model (Fig. 2). Envision a particle moving around a circular track, accelerated each time it enters a small box; the acceleration is provided by an a.c. field in the box at the time the particle enters; the eqs. (5) describe the state of the particle as it enters the box for the j + 1st time in terms of its state as it entered the time before. One can reexpress eq. (5) in the form ej+l 28, + e j + = (k/2n) sin ( 2 ~ 0 , ) (6) which as a discrete version of eq. (1) perhaps makes the connection to the pendulum problem more clear. Finally, consider a solid state model of a one-dimensional array of atoms adsorbed on a periodic substrate (Fig. 3). The j t h atom feels a force from the springs connecting it to its two neighbors, and from the gradient of the potential energy at its position on the periodic substrate. If we choose 0, = xj/a to be the position of the j th atom xi divided by the substrate lattice constant a, then kspring[(ej B j + l ) + (ej -ej-1)] = ksubstratesin (2nej) (7) System i s wiggled u p cind down t