DYNAMICAL SYSTEMS Tutorial 17

Transcription

1 map.nb DYNAMICAL SYSTEMS Tutorial 7 Iterated Maps sysid Mathematica 4..2, DynPac.67, 3ê9ê22 plotreset; intreset; ü Functions and Variables Used in This Tutorial asprat, bifurcmap, bifurc3dmap, bimap, borat, classifymap, cobweb, eigsysmap, eigvalmap, findpolyfi, imsize, intreset, iterate, jacob, jacobval, mapcomp, mapval, nfindfi, nfindpolyfi, parmval, periodmap, phaser, phaser3d, plotreset, plrange, plrange3d, pointcon, portraitmap, portrait3dmap, ptsize, rangeflag, ranger, residualfi, setback, setcolor, setmap, setde, setparm, setstate, slopevec, stripsol, sysid, sysreport, timeplot, and viewmap. ü Description of Systems Used in This Tutorial In this tutorial, our objective is to illustrate the use of the functions defined for iterated mappings. For eamples, we will use the logistic map for a D 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 eample, if we are studying a map f[], then one way of finding orbits of period two is to look for fied points of f[f[]]. 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 eamples of this below. ü Logistic Map The logistic map is discussed in many references. A very complete and readable discussion is given in Chapter of Nonlinear Dynamics and Chaos by Steven Strogatz, Addison-Wesley, 994. Many of the interesting properties of the map were discovered by the mathematical biologist Robert May ("Simple Mathematical Models with Very Complicated Dynamics," Nature 26, 459, 976.) The basic form of the map is n+ = r n ( - n ). As is well-known this map ehibits a wide and interesting range of behavior as r is varied. We define the system for DynPac, starting by the setmap command.

2 map.nb 2 setmap; setstate@8<d; setparm@8r<d; parmval = 83.2<; slopevec = 8r * * H - L<; sysreport SYSTEM DEFINITION H.67L System name sysname = System State vector statevec = 8< State units stateunits = 8< Slope vector slopevec = 8r * H - L * < Parameter vector parmvec = 8r< Parameter values parmval = 83.2< Parameter units vector parmunits = 8< Time unit timeunit = System Type = mapping We 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; sysreport SYSTEM DEFINITION H.67L System name sysname = System State vector statevec = 8< State units stateunits = 8< Slope vector slopevec = 8r * H - L * < Parameter vector parmvec = 8r< Parameter values parmval = 83.2< Parameter units vector parmunits = 8< Time unit timeunit = System Type = differential equation We return to the map setting.

3 map.nb 3 setmap; We start by viewing the map. viewmap@d; r H - L, 8r<=8 3.2< The picture suggests that there are two fied points -- one at and one between.6 and.8. We find these. Because the mapping is a polynomial, we can use findpolyfi or nfindpolyfi. We can also use the more general nfindfi, which requires an initial guess. findpolyfi@d 98<, 9 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - + r == r This gives the answer in terms of the parameter r. To find numerical values we can use nfindpolyfi or nfindfi: nfindpolyfi@d 88.<, << nfindfi@.5d < We can check the accuracy of the fied point by finding the residual with residualfi. residualfi@8.6875<d 8.< Alternatively, we can evaluate the map at the fied point. mapval@8.6875<d <

4 map.nb 4 We check the stability of these two fied points. classifymap@8<d unstable classifymap@8.6875<d unstable Thus both of the fied points are unstable. This could also be determined by the eigenvalues at those points. Any eigenvalue greater than one in magnitude indicates instability. eigvalmap@8<d 83.2< eigvalmap@8.6875<d 8-.2< As neither fied 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 3. We ask for 2 iterates with none thrown away. sol = iterate@3,., 2, D 88., 3<, 8.,.56672<, 82., <, 83.,.5387<, 84.,.79527<, 85.,.5229<, 86.,.79857<, 87.,.54736<, 88.,.79935<, 89.,.53333<, 8.,.79943<, 8.,.539<, 82., <, 83.,.5352<, 84., <, 85.,.5346<, 86., <, 87.,.5345<, 88., <, 89.,.5345<, 82., << 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 epect 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 =. We try this. stripsol@sol, D 883<, <, <, <, <, <, <, <, <, <, <, 8.539<, <, <, <, <, <, <, <, <, << It is clear from the solution list that we have an orbit of period 2. DynPac can tell us this also. periodmap@sold Solution contains a periodic orbit; period = 2 Another approach to finding this periodic orbit is to consider the fied points of the first iterated mapping. First we graph it.

5 map.nb 5 viewmap@2d; Comp 2 of r H - L, 8r<=8 3.2< We see four fied points. Of course two will be the fied 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. nfindpolyfi@2d 88.<, <, <, << We see the same two values that showed up eplicitly in the orbit calculated above. We check the stability of the period two orbit by checking the stability of these as fied points of the second iterated mapping. classifymap@8.5345<, 2D strictly stable classifymap@ <, 2D strictly stable Thus the period two orbit is stable. The last few function evaluations have provided eamples of applying functions to higher compositions of the map -- in this case the second composition. It is the optional last argument 2 that causes this. We haven't yet looked eplicitly at the second composition, although we can do that easily with mapcomp[n], which returns the nth composition. mapcomp@2d 8r 2 H - L H - r H - L L< Now we look at something new. We construct a cobweb plot to show the approach to the stable orbit of period 2. The function to do this is cobweb[initval,niter,ntoss], where initval is the starting point for the iteration, niter is the number of iterations to plot, and ntoss is the the number to calculate and throw away first (to eliminate transients). We start with an initial condition 3, and we ask for iterations, throwing none away before plotting.

6 map.nb 6 cobweb@83<,, D; r H - L, 8r<=8 3.2< We see the eventual approach to the orbit. We can get the pure orbit by throwing away the transients. We perform the same calculation, only now throwing away points first.

7 map.nb 7 cobweb@83<,, D; r H - L, 8r<=8 3.2< Now we get the pure orbit. Let's increase the value of r to 3.5, and carry out a short iteration. parmval = 83.5<; sol2 = iterate@83<,., 3, D 88., 3<, 8.,.6985<, 82., <, 83.,.55936<, 84., <, 85.,.38336<, 86.,.8272<, 87.,.529<, 88.,.875<, 89.,.38283<, 8., <, 8.,.5896<, 82., <, 83.,.38282<, 84.,.82694<, 85.,.5884<, 86., <, 87.,.38282<, 88.,.82694<, 89.,.5884<, 82., <, 82.,.38282<, 822.,.82694<, 823.,.5884<, 824., <, 825.,.38282<, 826.,.82694<, 827.,.5884<, 828., <, 829.,.38282<, 83.,.82694<< A close inspection shows an orbit of period 4. We verify that. periodmap@sol2d Solution contains a periodic orbit; period = 4 The points on the 4-orbit should be fied points of the fourth composition of the map.

8 map.nb 8 nfindpolyfi@4d 88.<, Â<, Â<, Â<, Â<, <, 82857<, <, Â<, Â<, <, <, <, <, Â<, Â<< We find lots of roots. In the interpretation, it helps to look also at the fied points of the basic mapping and the mapping iterated once. nfindpolyfi@2d 88.<, 82857<, <, << nfindpolyfi@d 88.<, << By comparing these, we conclude that () the basic map has fied points f =.; f2 =.74286; (2) the second composition has in addition a period two orbit f2 = 2857; f22 =.85743; and (3) the fourth composition has a period four orbit f4 =.38282; f42 =.5884; f43 =.82694; f44 = ; Thus for r = 3.5, the map has two fied points, a period two orbit and a period four orbit. Yet all we saw in the iteration was the period four orbit. This strongly suggests that it is the only stable attractor. We check the stability of all of these now. classifymap@8f<d unstable classifymap@8f2<d unstable classifymap@8f2<, 2D unstable classifymap@8f22<, 2D unstable classifymap@8f4<, 4D strictly stable classifymap@8f42<, 4D strictly stable

9 map.nb 9 classifymap@8f43<, 4D strictly stable classifymap@8f44<, 4D strictly stable Let's look at a cobweb of this stable period four orbit. cobweb@83<,, D; r H - L, 8r<=8 3.5<.8.6 Now we look at the pure orbit..6.8

10 map.nb D; r H - L, 8r<=8 3.5<.8.6 We increase r again, this time to 3.7. parmval = 83.7<;.6.8

11 map.nb sol3 = iterate@83<,,, D 88, 3<, 8,.65527<, 82, <, 83,.57788<, 84, <, 85, 57393<, 86,.77225<, 87,.7664<, 88, <, 89,.82672<, 8,.5339<, 8,.9266<, 82, 6746<, 83, <, 84,.73873<, 85,.7459<, 86,.75533<, 87, <, 88, <, 89,.59294<, 82,.89369<, 82,.35736<, 822, <, 823, 887<, 824, <, 825, 5836<, 826,.78958<, 827, <, 828,.66826<, 829,.8235<, 83, <, 83,.97383<, 832, 843<, 833,.74669<, 834, <, 835,.77755<, 836,.6499<, 837,.8545<, 838, 67984<, 839,.9227<, 84, 68562<, 84,.72685<, 842, <, 843,.7227<, 844, <, 845,.74995<, 846,.76955<, 847, <, 848, <, 849,.5555<, 85, <, 85, 57983<, 852,.78283<, 853, <, 854,.66672<, 855,.82283<, 856, <, 857,.9926<, 858, 7468<, 859,.7375<, 86,.7786<, 86,.75632<, 862,.69258<, 863, <, 864,.68579<, 865, <, 866, 294<, 867, <, 868,.34749<, 869, <, 87,.592<, 87, <, 872, 5674<, 873,.75986<, 874,.7688<, 875, <, 876,.8383<, 877,.5976<, 878, <, 879, 696<, 88,.7358<, 88,.75628<, 882,.6824<, 883,.823<, 884,.58687<, 885,.89778<, 886,.3469<, 887,.83287<, 888,.5672<, 889, <, 89, 59933<, 89,.7762<, 892,.7598<, 893, <, 894,.89546<, 895,.5747<, 896,.96625<, 897,.33226<, 898, <, 899,.698<, 8,.88727<< Now there is no obvious repetition. We check it for periodicity anyway. periodmap@sol3d Solution does not contain a periodic orbit. Are there fied points? nfindpolyfi@d 88.<, << nfindpolyfi@2d 88.<, <, <, << nfindpolyfi@4d 88.<, Â<, Â<, Â<, Â<, <, <, Â<, Â<, <, <, <, <, <, Â<, Â<< There are two fied points, a period two orbit and a period four orbit. Presumably they are all unstable, since they don't show up in our iteration. Are there orbits of higher period? If we try nfindpolyfi[8], Mathematica goes away for a very long time and returns with a large list of roots, mostly comple, and missing most of the real roots that we already know from above. Let's look at the stability of the orbits we have found. It is sufficient to check just one point on each orbit if that one point is unstable. classifymap@8<d unstable

12 map.nb 2 classifymap@ <d unstable classifymap@8.3922<, 2D unstable classifymap@ <, 4D unstable If this system with r = 3.7 has a stable attractor, we haven't found it yet. We try a cobweb plot. cobweb@83<, 2, D; r H - L, 8r<=8 3.7< This could be chaotic. Let's do a fancy cobweb plot now with color, showing the sensitive dependence on initial conditions. We take 26 initial points in the range [,5], and we assign a gradually varying color from red to blue. First we construct the color list. collist = Table@RGBColor@H - il,, id, 8i,,,.4<D; setcolor@collistd; Now we construct a list of initial conditions.

13 map.nb 3 initvec = Table@ + i, 8i,,.5,.2<D; Now we construct the cobweb plot. plrange = 88, <, 8, <<; asprat = ; cobweb@initvec, 6, D; r H - L, 8r<=8 3.7< We can also make a plot of versus time by using timeplot. We construct a solution with 5 iterates and then plot it as a function of time. plrange = 88, 5<, 8, <<; asprat =.7; setcolor@8black<d; soltime = iterate@83<,,5,d; pointcon = True;

14 map.nb 4 timeplot@soltime, D; Comp 2 of r H - L, 8r<=8 3.7< t pointcon = False; plrange = 88, <, 8, <<; Both the time plot and the cobweb plot show the chaotic nature of this orbit. In particular the cobweb plot shows clearly the spread of an initially compact set of initial conditions. We can get an overview of the behavior of the system with a bifurcation diagram. This produces a graph in which the abcissa is the parameter being varied, and along the ordinate, the iterates are plotted. If we throw away initial transients, the result is a plot of the attractor of the system as a function of the parameter. The function which does this is bimap[npts,ntoss,nparm,name,range,initvec,pname,prange,ncomp]. The arguments are npts, the number of iterates plotted at each parameter value npts and ntoss, the number of transient iterates thrown away before plotting. The number of parameter values plotted is nparm. The name of the state variable plotted is name, and its plotting range is range. The initial condition for the iteration is initvec, which may contain parameter symbols. The name of the parameter being varied is pname, and it is varied through the range prange. The final argument ncomp is optional and specifies the level of function composition of the basic map. Generally for the logistic map about 2 iterates have to be thrown away to get good results, and about to 2 iterates have to be plotted, at about to 2 parameter values. We carry this out for r in the range 2.8 to 4. We ask for a larger image size, for a background color of Wheat, and points plotted in Blue. setback@wheatd; imsize = 4; setcolor@8blue<d; ptsize =.2; asprat =.7;

15 map.nb 5 biout = bimap@2, 2, 2,, 8, <, 83<, r,82.8, 4.<D; 8r H - L < r We see clearly the bifurcation from a stable fied point to a stable orbit of period 2 at r = 3, and then the bifurcation from period two to period four at r between 3.4 and 3.5. The further period doublings occur at decreasing increments in r, and the orbit becomes chaotic for r ª Note the intriguing window just beyond 3.8. Let's eplore this briefly. We set r to We iterate and throw away initial points in an effort to get rid of the transients. parmval = 83.83<; sol4 = iterate@83<,, 3, D 88,.95747<, 8,.5649<, 82,.54666<, 83,.95747<, 84,.5649<, 85,.54666<, 86,.95747<, 87,.5649<, 88,.54666<, 89,.95747<, 8,.5649<, 8,.54666<, 82,.95747<, 83,.5649<, 84,.54666<, 85,.95747<, 86,.5649<, 87,.54666<, 88,.95747<, 89,.5649<, 82,.54666<, 82,.95747<, 822,.5649<, 823,.54666<, 824,.95747<, 825,.5649<, 826,.54666<, 827,.95747<, 828,.5649<, 829,.54666<, 83,.95747<< A surprising result -- a period 3 orbit! periodmap@sol4d Solution is periodic; period = 3 Because we saw it, it surely is stable, but we can check that. classifymap@ <, 3D strictly stable

16 map.nb 6 classifymap@ <, 3D strictly stable classifymap@8.5649<, 3D strictly stable Let's look at the cobweb plot for this. asprat = ; plrange = 88, <, 8, <<; cobweb@83<, 5, D; r H - L, 8r<=8 3.83< Now we throw away the transients and look at the periodic orbit.

17 map.nb 7 cobweb@83<, 5, D; r H - L, 8r<=8 3.83<.8.6 We look at the third iterated mapping..6.8

18 map.nb 8 viewmap@3d; Comp 3 of r H - L, 8r<=8 3.83< nfindpolyfi@d 88.<, << nfindpolyfi@3d 88.<, <, <, <, 8.524<, <, <, << The third iterated map has 8 fied points. Two of these are unstable fied points of the basic map. The other si turn out to be the components of two period 3 orbits, one stable and one unstable. From the iteration carried out above, we know that the components of the stable orbit are.5649,.54666, and The unstable period 3 orbit then is {.6357,.524, and }. Let's start on this orbit and iterate, and see what happens.

19 map.nb 9 sol6 = iterate@8.6357<,.,, D 88.,.6357<, 8.,.524<, 82., <, 83.,.6357<, 84.,.524<, 85., <, 86.,.6357<, 87.,.5242<, 88., <, 89.,.6357<, 8.,.5242<, 8., <, 82.,.6357<, 83.,.5243<, 84., <, 85.,.63572<, 86.,.5245<, 87., <, 88.,.63573<, 89.,.5248<, 82., <, 82.,.63575<, 822.,.5243<, 823., <, 824.,.63578<, 825.,.5242<, 826.,.95529<, 827.,.63583<, 828.,.52435<, 829., <, 83.,.63592<, 83.,.52457<, 832., <, 833.,.6366<, 834.,.52493<, 835., <, 836.,.6363<, 837.,.52454<, 838., <, 839.,.63669<, 84., <, 84., <, 842.,.63734<, 843., <, 844.,.95525<, 845.,.63844<, 846.,.52476<, 847.,.95562<, 848.,.6429<, 849.,.52582<, 85.,.9557<, 85.,.64346<, 852., <, 853.,.9549<, 854.,.6494<, 855.,.52743<, 856.,.95468<, 857.,.65925<, 858.,.5349<, 859.,.95442<, 86.,.6793<, 86.,.53565<, 862., <, 863.,.72368<, 864., <, 865., <, 866.,.84467<, 867.,.5768<, 868., <, 869., 3859<, 87.,.68226<, 87.,.8346<, 872.,.53925<, 873.,.956<, 874.,.764<, 875., <, 876.,.9453<, 877.,.9836<, 878.,.68273<, 879.,.926<, 88.,.35483<, 88.,.82584<, 882., <, 883.,.9394<, 884., 46233<, 885.,.7858<, 886.,.78725<, 887.,.64554<, 888.,.88756<, 889., 2246<, 89.,.929<, 89., 78987<, 892.,.7747<, 893.,.67743<, 894., <, 895., <, 896., <, 897.,.62773<, 898.,.52944<, 899., <, 8.,.6237<< Although it isn't very strongly unstable, we can see that it is drifting off the orbit. Here's a longer run. sol7 = iterate@8.6357<,.,, D; last < Thus after a steps starting on the unstable period 3 orbit, we end up on the stable period 3 orbit. ü Hénon Map The Hénon map is a two-dimensional map developed by Michel Hénon to study chaos and strange attractors ("A Two-Dimensional Mapping with a Strange Attractor," Commun. Math. Phys. 5, 69, 976). It is discussed in many books on dynamical systems -- for eample in section 2.2 of Nonlinear Dynamics and Chaos by Steven Strogatz, Addison-Wesley, 994. The Hénon mapping provides a computationally straightforward way to study the compleities of chaos. The mapping has two parameters a and b. It is given by We define the system for DynPac. n+ = y n + - a n 2, y n+ = b n. setstate@8, y<d; setparm@8a, b<d; sysname = "Henon"; setmap; slopevec = 8y + - a 2,b<; parmval = 8.4,.3<;

20 map.nb 2 sysreport SYSTEM DEFINITION H.67L System name sysname = Henon State vector statevec = 8, y< State units stateunits = 8, < Slope vector slopevec = 8 - a * ^2 + y, b * < Parameter vector parmvec = 8a, b< Parameter values parmval = 8.4,.3< Parameter units vector parmunits = 8, < Time unit timeunit = System Type = mapping These parameter values were found by Hénon to produce a chaotic attractor. We begin by calculating the Jacobian of the map. jacob -b jacobval -.3 Thus the map will be dissipative (giving contracting areas in phase space) when»b» <. We construct a very lengthy iteration -- 5 points. We throw away the first points. One reason for calculating so many points is that we plan to zoom in on the structure of the attractor produced. Notice how much quicker this is than solving a differential equation for the same number of time steps. solhen = iterate@8, <,., 5, D; We use staterange to determine a suitable plotting window. staterange@solhend 88, , 449.<, , 299.<<, 8y, , 449.<, , 292.<<< plrange = 88-.5,.5<, 8-.5,.5<<; aon = False; frameon = True; asprat =.7;

21 map.nb 2 hengraph = phaser@solhend; Henon 8a, b<=8.4,.3< y This graph shows the chaotic attractor for the map. There is an incredible fine structure which does not show up on this scale. We zoom in by choosing a plotting window near the point {.5, }. In this zoom process, we follow closely the presentation in Strogatz' book of Hénon's work. We define a graphics zoombo that will put a bo on a graph to show the window used in the following graph. zoombo@rng_d := Graphics@ 8GrayLevel@.7D, Rectangle@First@rngD, Last@rngDD<, DisplayFunction -> IdentityD

22 map.nb , <<D, hengraphd; Henon 8a, b<=8.4,.3< The grey rectangle is the entire window for the net zoomed graph. plrange = 88.54,.7<, 8.5, <<;

23 map.nb 23 hengraph2 = phaser@solhend; Henon 8a, b<=8.4,.3<.9 y We zoom in once again show@zoombo@88.62,.85<, 8.64,.9<<D, hengraph2d; Henon 8a, b<=8.4,.3<

24 map.nb 24 Once again, the grey rectangle is the window for the net (and last) plot. plrange = 88.62,.64<, 8.85,.9<<; phaser@solhend;.9 Henon 8a, b<=8.4,.3<.9.89 y The unfolding of structure is amazing. In each case what appears to be three lines, becomes, when viewed more closesly, a set of 6 lines, grouped as three, two and one. The fractal nature of the attractor is such that this unfolding continues forever. Of course whether or not we see it in our graphs depends on how many iterates we have calculated. With our 5 points we can't go much further than we have here. We can eplore the sensitive dependence on initial position by looking at two orbits starting close together. We use red and blue for the orbit colors, and we carry out the construction using the function portraitmap. W throw away the first iterations. setcolor@8red, Blue<D; ptsize =.8; plrange = 88-.5,.5<, 8-.5,.5<<;

25 map.nb 25 <, 8.,.<<,.,,,, 2D; Henon 8a, b<=8.4,.3< y We can see from the colors that both orbits are spread over the attractor. We can also see the sensitive dependence from a time plot. plrange = 88, 5<, 8-.52,.52<<; solt = iterate@8, <,., 5, D; solt2 = iterate@8.,.<,., 5, D; pointcon = True;

26 map.nb 26 solt2<, D;.5 Henon 8a, b<=8.4,.3< t plrange = 88-.5,.5<, 8-.5,.5<<; pointcon = False; It is of interest to see how the iterates of the Hénon map depend on the parameter a. We leave b fied and vary a through 5 values, constructing a phase plot for each value. We accomplish this in an automated way by using the function bifurcmap[intlist,t,niter,ntoss,i,j,parmlist,ncomp]. The argument intlist is the list of initial conditions to be used in each graph. In this case, we use only one initial condition. The initial time is t and the number of interations is niter, with ntoss being thrown away first. The components i and j are to be plotted. The list of parameter values to be used is in parmlist. The final argument ncomp is optional. It is the level of function composition to be used with the basic map, and the default is. parmlist = 88,.3<, 8.6,.3<, 8.8,.3<, 8.,.3<, 8.2,.3<, 8.4,.3<<; ptsize =.8; setcolor@8black<d; bifurcmap@88, <<,.,,,, 2, parmlistd; Bifurcation sequence for parmlist = 88,.3<, 8.6,.3<, 8.8,.3<, 8.,.3<, 8.2,.3<, 8.4,.3<<

27 map.nb 27 Henon 8a, b<=8,.3< y Henon 8a, b<=8.6,.3< y

28 map.nb 28 Henon 8a, b<=8.8,.3< y Henon 8a, b<=8.,.3< y

29 map.nb 29 Henon 8a, b<=8.2,.3< y Henon 8a, b<=8.4,.3< y An interesting sequence. Looks like stable period two orbits for a =,.6, and.8, followed by a period four orbit for a =., and then chaotic orbits for a =.2 and.4. Let's look in more detail at the period four orbit. parmval = 8.,.3<;

30 map.nb 3 henfour = iterate@8, <,., 2, 2D 882., , <, 82.,.95695, <, 822., , 8558<, 823.,.27498, <, 824., , <, 825.,.95695, <, 826., , 8558<, 827.,.27498, <, 828., , <, 829.,.95695, <, 82., , 8558<, 82.,.27498, <, 822., , <, 823.,.95695, <, 824., , 8558<, 825.,.27498, <, 826., , <, 827.,.95695, <, 828., , 8558<, 829.,.27498, <, 822., , << Clearly period 4, as DynPac also tells us: periodmap@henfourd Solution is periodic; period = 4 We name the four points of the orbit. pt = Drop@First@henfourD, D , < pt2 = Drop@henfour@@2DD, D , < pt3 = Drop@henfour@@3DD, D , 8558< pt4 = Drop@henfour@@4DD, D , < We check by hand the orbit. mapval@ptd , < mapval@pt2d , 8558< mapval@pt3d , < mapval@pt4d , < From the calculation above, It is clear that the orbit is stable, but we check one point on it anyway. classifymap@pt, 4D strictly stable

31 map.nb 3 eigvalmap@pt, 4D , < eigsysmap@pt, 4D , <, , <, , <<< From what we have found, it appears that the Hénon map goes through a period-doubling sequence to chaos. We construct a view of this with a bifurcation diagram, varying a from to.4, and plotting versus a. We throw away the first 3 points and then plot 2, for each of 2 values of the parameter. asprat =.7; ptsize =.2; setcolor@8blue<d; biouthen = bimap@2, 3, 2,, 8-.5,.5<, 8, <, a, 8,.4<D; a 2 + y, b < a This diagram suggests lots of other things to eplore later, but we leave it for now. One final point on the Hénon map. Not all of the solutions are bounded. For some initial conditions and some parameter values, the iterates run off to infinity. We eplore this briefly. parmval = 8.7,.3<;

32 map.nb 32 <,., 2, D 88.,, <, 8.,.3,.3<, 82.,.47,.9<, 83., ,.344<, 84., , <, 85., , -6787<, 86., , -7725<, 87.,.78777, <, 88., -.637, 34233<, 89.,.2279, -.834<, 8., ,.36837<, 8., , <, 82., , <, 83., , <, 84., , <, 85., µ, -55.3<, 86., µ 2, µ <, 87., µ 43, µ 2 <, 88., µ 87, µ 42 <, 89., µ 74, µ 86 <, 82., µ 348, µ 73 << This is clearly heading for an overflow! To prevent overflows and possible crashes, we can use range checking, in which we specify a bo (named ranger) to contain the solution. The iteration is stopped as soon as the solution leaves the bo. To turn on range checking, we set rangeflag = True. rangeflag = True; ranger = 88-, <, 8-, <<; iterate@8, <,., 2, D 88.,, <, 8.,.3,.3<, 82.,.47,.9<, 83., ,.344<, 84., , <, 85., , -6787<, 86., , -7725<, 87.,.78777, <, 88., -.637, 34233<, 89.,.2279, -.834<, 8., ,.36837<, 8., , <, 82., , <, 83., , << We see that the iteration was stopped when a solution point first left the bo. The last point inside the bo is stored in the variable outbound. outbound 884., , << rangeflag = False; ü A Three-Dimensional Map Most published work on iterated maps deals with one- and two-dimensional maps. Such maps can ehibit the full range of compleities, unlike the situation with sets of autonomous differential equations where dimension three is the minimum dimension for chaos. All of this is by way of saying that there is no natural or well-studied eample to use here. In order to illustrate some of the 3D plotting features for maps, we define a somewhat artificial 3D map consisting of the logistic map in one dimension and the Hénon map in the other two. The mapping is defined by We define this system for Dynpac. n+ = y n + - a n 2, y n+ = b n, z n+ =rz n ( - z n ). setstate@8, y, z<d; setparm@8a, b, r<d; sysname = "Hybrid"; parmval = 8.8,.3, 2.8<; slopevec = 8y + - a 2, b, r zh - zl<; setmap;

33 map.nb 33 sysreport SYSTEM DEFINITION H.67L System name sysname = Hybrid State vector statevec = 8, y, z< State units stateunits = 8,,< Slope vector slopevec = 8 - a * ^2 + y, b *, r * H - zl * z< Parameter vector parmvec = 8a, b, r< Parameter values parmval = 8.8,.3, 2.8< Parameter units vector parmunits = 8,,< Time unit timeunit = System Type = mapping On the basis of work done earlier, we epect for the present parameter values to get an orbit of period 2, in which z is stationary. Let's try it.

34 map.nb 34 solhy = iterate@8,,.5<,., 75, D 88.,,,.5<, 8.,,,.7<, 82.,,.3,.588<, 83.,.268,.6,.67837<, 84., ,.384,.6969<, 85.,.33945, ,.66552<, 86., , 834, <, 87.,.9929, -.595,.65744<, 88., , ,.63595<, 89.,.2872, , <, 8., -568,.3867,.635<, 8.,.24792, , <, 82., , , <, 83.,.26455, -.56, <, 84., , ,.63973<, 85.,.25742, -.725, <, 86., , ,.6485<, 87.,.2648, -.465, <, 88., ,.37843,.64574<, 89.,.2596, , <, 82., , ,.64237<, 82.,.25973, ,.6435<, 822., ,.37799, <, 823.,.25949, , <, 824., , ,.64252<, 825.,.25959, ,.64325<, 826., , , <, 827.,.25955, ,.64329<, 828., , ,.64272<, 829.,.25957, , <, 83., ,.37787, <, 83.,.25956, -.537, <, 832., , ,.6428<, 833.,.25956, ,.64292<, 834., , ,.64282<, 835.,.25956, , <, 836., , , <, 837.,.25956, , <, 838., , , <, 839.,.25956, , <, 84., , , <, 84.,.25956, , <, 842., , ,.64285<, 843.,.25956, , <, 844., , , <, 845.,.25956, ,.64286<, 846., , , <, 847.,.25956, , <, 848., , , <, 849.,.25956, , <, 85., , , <, 85.,.25956, , <, 852., , , <, 853.,.25956, , <, 854., , , <, 855.,.25956, , <, 856., , , <, 857.,.25956, , <, 858., , , <, 859.,.25956, , <, 86., , , <, 86.,.25956, , <, 862., , , <, 863.,.25956, , <, 864., , , <, 865.,.25956, , <, 866., , , <, 867.,.25956, , <, 868., , , <, 869.,.25956, , <, 87., , , <, 87.,.25956, , <, 872., , , <, 873.,.25956, , <, 874., , , <, 875.,.25956, , << periodmap@solhyd Solution contains a periodic orbit; period = 2 Let's plot the approach to this in 3D. plotreset; pointcon = False; borat = 8,, <; ptsize =.; plrange3d = 88-.5,.5<, 8-.5,.5<, 8, <<;

35 map.nb 35 Hybrid 8a, b, r<=8.8,.3, 2.8<.8 z y A projection on the y plane: frameon = True; asprat = ; aon = False;

36 map.nb 36 2D; Hybrid 8a, b, r<=8.8,.3, 2.8<.3 y To get the pure orbit, we iterate again, this time throwing away the first iterates. solhy2 = iterate@8,,.5<,., 2, D; periodmap@solhy2d Solution is periodic; period = 2

37 map.nb 37 2D; Hybrid 8a, b, r<=8.8,.3, 2.8<.3 y We see the two points of the orbit in diagonally opposite corners. Now we try a portrait in 3D with several different initial conditions. We also change a so that the Hénon map is chaotic. parmval = 8.4,.3, 2.8<; plotreset; ptsize =.8; pointcon = False; borat = 8,, <; setcolor@8red, Blue<D; plrange3d = 88-.5,.5<, 8-.5,.5<, 8, <<;

38 map.nb 38 8.,.,.5<<,.,,,, 2, 3D; Hybrid 8a, b, r<=8.4,.3, 2.8<.8 z y Finally we will do a short bifurcation sequence, letting both r and a vary, using the command bifurc3dmap. setback@wheatd; parmlist = 88.,.3, 2.8<, 8.2,.3, 3.5<, 8.4,.3, 3.7<<; bifurc3dmap@88,, 3<, 8.,., 5<<,., 2, 2,, 2, 3, parmlistd; Bifurcation sequence for parmlist = 88.,.3, 2.8<, 8.2,.3, 3.5<, 8.4,.3, 3.7<<

39 map.nb 39 Hybrid 8a, b, r<=8.,.3, 2.8<.8 z y

40 map.nb 4 Hybrid 8a, b, r<=8.2,.3, 3.5<.8 z y

41 map.nb 4 Hybrid 8a, b, r<=8.4,.3, 3.7<.8 z y This last graph shows a kind of double chaos, and perhaps that is as good a place to quit as any.