═══ 1. PM Chaos: Extended Help ═══ PM Chaos is a simulation of a simple chaotic system: the standard mapping. This is a simple function that takes a point (J,theta) into another point (J',theta'). If you keep applying this function again and again, you get a sequence of points. For certain initial conditions this sequence is a smooth trajectory, and for other initial conditions it is chaotic. To start a calculation, click the left mouse button anywhere within the box of the graph. PM Chaos will begin calculating the sequence of points starting with the initial point you have chosen. The calculation continues until you stop it, either by clicking the left mouse button somewhere else (this will stop the first calculation and start a new one), by selecting Stop! from the Control menu, or by clearing the screen. Related information: o The standard mapping o Using the program ═══ 2. Physics of the standard mapping ═══ The standard mapping is a simplified set of equations that has many of the same qualitative topological properties as do actual physical systems in classical mechanics. The evolution of a classical mechanical system is given by a set of differential equations determined by the Hamiltonian. It is numerically easier, however, to work with a difference equation, that is, one that doesn't evolve continuously, but where, given a point, there is a function that just gives you the next point in the sequence. If you don't know much classical mechanics, you may want to skip the next three paragraphs. You don't need to understand them in order to use the program. Consider a time-independent Hamiltonian with two degrees of freedom, i.e., two canonical coordinates and two canonical momenta. The state of the system is represented by a point in the four-dimensional phase space, and the time evolution of the system from some initial state is a trajectory in phase space. We can choose some arbitrary two-dimensional surface in this phase space, a surface of section, and study the successive intersections of a trajectory with that surface. This gives us a set of difference equations on a two-dimensional reduced phase space. A time-independent Hamiltonian leads to a conserved energy; this means that a trajectory must lie in some three-dimensional subspace of the original four-dimensional phase space. It also means that the mapping given by the surface of section (a canonical transformation) is area-preserving, that is, that any region of the two-dimensional reduced phase space gets mapped to a region of equal area. If the Hamiltonian is integrable (i.e., if there exists a constant of the motion other than the total energy), then motion is actually in a two-dimensional subspace of phase space. In that case, the successive points in the surface of section lie on closed curves in the reduced phase space. This is most easily seen in angle-action variables. In that case, the mapping equation for an integrable system is J' = J theta' = theta + f J. The standard mapping is very similar to this: it is a near-integrable area-preserving mapping in angle-action coordinates. o Equations of the standard mapping o Where to learn more ═══ 3. The standard mapping ═══ The standard mapping is given by the equations J' = J + K sin(theta) theta' = theta + J', mod(2 pi). It is defined for J and theta in the range (0, 2 pi). o Significance of K ═══ 4. The K factor ═══ K is the stochasticity parameter. For K=0, the standard mapping is integrable, and the action variable, J, is a constant of the motion. For K>0 there are some regions of stochasticity, but they are isolated when K is small, and J is nearly constant for many trajectories. For any K<1 there are separatrices dividing the screen into at least two regions. That is: if the motion begins near J=0, it will never end up near J=2 pi. Motion becomes globally stochastic at K=1. At this point, the last separatrix disappears, and there is nothing separating the Jpi regions. For any value of K there are still some islets of stability, but they become more and more isolated, and motion becomes more and more chaotic, as K increases. Note that there are always at least two fixed points: (J,theta) = (0,pi), and (J,theta) = (2pi,pi). For K<4, these fixed points are stable. This program limits K to the range 0 Zoom Out R Set Ranges ═══ 24. On-line help key assignments ═══ Key Action F1 Get help F2 Extended help (from within help) F9 Keys help (from within help) F11 Help index (from within help) Shift+F10 Using help (from within help) ═══ 25. System and window key assignments ═══ Alt+F9 Minimize the window Alt+F10 Maximize the window Alt+Esc Switch to the next program Ctrl+Esc Switch to the Task List Shift+Esc or Alt+Spacebar Go to or from the system menu F10 or Alt Go to or from the action bar Underlined letter Move among the choices on the action bar and pull-down menus Arrow keys Move among the choices on the action bar and pull-down menus ═══ ═══ Actually, the transition isn't exactly at K=1. The approximate value is K=0.9716. See A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion, for more details. ═══ ═══ Presentation Manager is a trademark of the IBM Corporation. ═══ ═══ Presentation Manager is a trademark of the IBM Corporation.