DYNAMICAL SYSTEMSTutorial 17Iterated MapssysidMathematica 4.1.2, DynPac 10.67, 3ê9ê2002plotreset;intreset;ü Functions and Variables Used in This Tutorialasprat, bifurcmap, bifurc3Dmap, bimap, boxrat, classifymap, cobweb, eigsysmap, eigvalmap, findpolyfix, imsize, intreset, iterate, jacob, jacobval, mapcomp, mapval, nfindfix, nfindpolyfix, parmval, periodmap, phaser, phaser3D, plotreset, plrange, plrange3D, pointcon, portraitmap, portrait3Dmap, ptsize, rangeflag, ranger, residualfix, setback, setcolor, setmap, setde, setparm, setstate, slopevec, stripsol, sysid, sysreport, timeplot, and viewmap.ü Description of Systems Used in This TutorialIn this tutorial, our objective is to illustrate the use of the functions defined for iterated mappings. For examples, we will use the logistic map for a 1D case, the Henon map for a 2D case, and a combination of the two for a 3D case. In many cases, it is useful to apply some of the functions directly to iterates of the map. For example, if we are studying a map f[x], then one way of finding orbits of period two is to look for fixed points of f[f[x]]. Most of the functions used in DynPac for mappings allow an optional final argument which is the level of composition desired. We will see a number of examples of this below. ü Logistic MapThe logistic map is discussed in many references. A very complete and readable discussion is given in Chapter 10 of Nonlinear Dynamics and Chaos by Steven Strogatz, Addison-Wesley, 1994. Many of the interesting properties of the map were discovered by the mathematical biologist Robert May ("Simple Mathematical Models with Very Complicated Dynamics," Nature 261, 459, 1976.) The basic form of the map isxn+1 = rxn(1 - xn) .As is well-known this map exhibits a wide and interesting range of behavior as r is varied. We define the system for DynPac, starting by the setmap command. map.nb 1setmap;setstate@8x<D; setparm@8r<D; parmval = 83.2<; slopevec = 8r* x * H1 - xL<;sysreportSYSTEM DEFINITION H10.67LSystem name sysname = SystemState vector statevec = 8x<State units stateunits = 8<Slope vector slopevec = 8r* H1 - xL * x<Parameter vector parmvec = 8r<Parameter values parmval = 83.2<Parameter units vector parmunits = 8<Time unit timeunit =System Type = mappingWe could use this same function as the slope for a differential equation. The command set de switches back to differential equation mode. The primary difference in the two modes is the actual stepping algorithm used in constructing solutions -- a Runge-Kutta step for a differential equation, and a map iteration for the mapping. It is only at that basic level of code that the two modes differ.setde;sysreportSYSTEM DEFINITION H10.67LSystem name sysname = SystemState vector statevec = 8x<State units stateunits = 8<Slope vector slopevec = 8r* H1 - xL * x<Parameter vector parmvec = 8r<Parameter values parmval = 83.2<Parameter units vector parmunits = 8<Time unit timeunit =System Type = differential equationWe return to the map setting.map.nb 2setmap;We start by viewing the map.viewmap@D;0.2 0.4 0.60.81x0.20.40.60.81r H1 - xL x, 8r<=8 3.20<The picture suggests that there are two fixed points -- one at 0 and one between 0.6 and 0.8. We find these. Because the mapping is a polynomial, we can use findpolyfix or nfindpolyfix. We can also use the more general nfindfix, which requires an initial guess.findpolyfix@D980<, 9-1 + rÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅr==This gives the answer in terms of the parameter r. To find numerical values we can use nfindpolyfix or nfindfix:nfindpolyfix@D880.<, 80.6875<<[email protected]<We can check the accuracy of the fixed point by finding the residual with [email protected]<D80.<Alternatively, we can evaluate the map at the fixed [email protected]<D80.6875<map.nb 3We check the stability of these two fixed points.classifymap@80<[email protected]<DunstableThus both of the fixed points are unstable. This could also be determined by the eigenvalues at those points. Any eigenvalue greater than one in magnitude indicates instability.eigvalmap@80<D83.2<[email protected]<D8-1.2<As neither fixed point is stable for this value of r, there might be a periodic orbit. Let's perform a short iteration with a more or less arbitrary intial condition of 0.23. We ask for 20 iterates with none thrown away.sol1 = [email protected], 0.0, 20, 0D880., 0.23<, 81., 0.56672<, 82., 0.785755<, 83., 0.538701<, 84., 0.795207<, 85., 0.521129<,86., 0.798571<, 87., 0.514736<, 88., 0.799305<, 89., 0.513333<, 810., 0.799431<,811., 0.513091<, 812., 0.799452<, 813., 0.513052<, 814., 0.799455<, 815., 0.513046<,816., 0.799455<, 817., 0.513045<, 818., 0.799455<, 819., 0.513045<, 820., 0.799455<<The answer is in the form of a list pairs, with the first element in each pair being the time coordinate, the second the iterate value. All of the functions in DynPac expect solution lists in this form. If it is desired to form a list without the time coordinate, this can be accomplished by the function stripsol[sol,n], which removes the nth state variable from the list. The time coordinate is associated with n = 0. We try this.stripsol@sol1, 0D880.23<, 80.56672<, 80.785755<, 80.538701<, 80.795207<, 80.521129<, 80.798571<,80.514736<, 80.799305<, 80.513333<, 80.799431<, 80.513091<, 80.799452<, 80.513052<,80.799455<, 80.513046<, 80.799455<, 80.513045<, 80.799455<, 80.513045<, 80.799455<<It is clear from the solution list that we have an orbit of period 2. DynPac can tell us this also.periodmap@sol1DSolution contains a periodic orbit; period = 2Another approach to finding this periodic orbit is to consider the fixed points of the first iterated mapping. First we graph it.map.nb 4viewmap@2D;0.2 0.4 0.60.81x0.20.40.60.81Comp 2 of r H1 - xL x, 8r<=8 3.20<We see four fixed points. Of course two will be the fixed points of the original map, but the other two should be the points on the period-two orbit of the original map. We check this.nfindpolyfix@2D880.<, 80.513045<, 80.6875<, 80.799455<<We see the same two values that showed up explicitly in the orbit calculated above. We check the stability of the period two orbit by checking the stability of these as fixed points of the second iterated [email protected]<,2Dstrictly [email protected]<,2Dstrictly stableThus the period two orbit is stable. The last few function evaluations have provided examples of applying functions to higher compositions of the map -- in this