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<html><body bgcolor=white><head><center><table><tr><td align=center><b>Rainer Hegger</b></td> <td width=20></td> <td align=center><b>Holger Kantz</b></td> <td width=20></td> <td align=center><b>Thomas Schreiber</b></td></tr></table><title>Exercise 1 using TISEAN Nonlinear Time SeriesRoutines</title></head> <h1>Exercises using TISEAN<br><font color=blue>Part I: Exploring chaos in one-dimensional maps</font></h1></center><hr>Exercise I helps you to make the first steps with TISEAN and illustrates some properties of one dimensional chaotic maps. Due to the single phase space dimension, time series analysis in this case is analysis ofnumerical simulations in phase space. <br><hr><br>The <b>Hénon map</b> <table noborder> <tr><td><font color=blue>x<sub>n+1</sub>=1-ax<sub>n</sub><sup>2</sup>+by<sub>n</sub></font></td></tr><tr><td><font color=blue>y<sub>n+1</sub>=x<sub>n</sub></font></td></tr></table>is a two-dimensional extension of the <b>logistic equation</b>, <fontcolor=blue>x<sub>n+1</sub>=1-ax<sub>n</sub><sup>2</sup></font>.<br> <br>With <font color=blue>b=0</font> and <font color=blue>0 < a < 2</font>, the Hénon map creates a time series of the logistic equation, provided the intial condition for <font color=blue>x</font> is inside the interval [-1,1].<br><br> Theroutine <a href=../docs_f/henon.html> henon </a> allows you togenerate a time series of the Hénon map of arbitrary length, forarbitrary parameters, arbitrary initial conditions, and afterdiscarding transients. Click on <a href=../docs_f/henon.html> henon</a> to see the html-manual page. and type <font color=red>henon-h</font> as a command line in a terminal window of your computer to see the on-linehelp. For proper usage of <ahref=../docs_f/henon.html> henon </a>, you must specify the number of iterates to beproduced by the <font color=orange>-l#</font> option, where <fontcolor=orange>#</font> has to bereplaced by an integer, say, 5000 for 5000 data points to beproduced. Other options can be used to modify thedefaults, and among them are the parameters of the map (the defaultscorrespond to the values originally used by M. Hénon himself, who"invented" the map).<br><br>Use gnuplot for a fast scan through the different scenarios (if you do not have gnuplot, store the data in output files and plot them with your favorite plot-program) in, e.g., the following way, where you have to specify <font color=orange>a</font> in <font color=orange> -Aa </font> of <fontcolor=blue> henon </font> in the <font color=green>plot</font> command:<br><font color=red>yourcomputer:> gnuplot</font><br>gnuplot><font color=green> set yrange [-1:1]</font><br>gnuplot><font color=green> plot '< henon -B0 -A<font color=orange>a</font> -l100' using 0:1 withlinespoints</font>.<br> The plot command combines the successive iterates with lines and helps to guide the eye, but if you want to plotmore than about 500 points you should use <font color=green>withpoints</font> or even <font color=green>with dots</font>, instead. You should observe this way:<br><ul><li> The <b>period doubling bifurcation</b> around a=0.75 (bifurcation toperiod 2), a=1.25 (period 4), a=1.35 (period 8).<li> The <b>two-band chaos</b>: a=1.4 to a=1.55 (approximately).<li> The <b>intermittency</b> close to the birth of the period three orbit bytangent bifurcation at a=1.75 (for intermittency, use a=1.7499).<li> <b>transient chaos</b> on the repeller outside the period 3 orbit:a=1.75. Use <font color=orange> -x0</font> in order to <b> not</b>discard the initial part of the trajectory (the transient, duringwhich the orbit is supposed to settle down on the attractor), and vary the initial condition x (using the <font color=orange> -Xx</font> <ahref=../docs_f/henon.html>option of henon </a>) and study the transients before the trajectory settles down on the period3 orbit. </ul>A not very elegant but simple way to plot a full bifurcationdiagram using gnuplot and the henon routine is to load the file <ahref=henon.gnu>henon.gnu</a> in gnuplot: <font color=green>load'henon.gnu'</font>, where you can easily include more parameter values.<br><br>When a scalar time series is generated by a one-dimensional map, atime delay embedding of lag one shows the graph of the map<font color=blue> x<sub>n+1</sub>=f(x<sub>n</sub>)</font>. The routine <ahref=../docs_c/delay.html> delay</a> by default produces atwo-dimensional delay embedding with unit time lag. The <fontcolor=orange> -d# </font> option sets a different lag.<br>Use <font color=green> plot '< henon -B0 -A<font color=orange>a</font> -l5000 | delay' withdots</font> in the following for several values of <fontcolor=orange>a</font>.<br> <a href=trouble1.html> trouble?</a><ul><li> Convince yourself that when <font color=orange>a</font> is such that the trajectory ischaotic, you see thus a part of the parabolic graph of the map. In particular, for <font color=orange>-A2.0</font> you should see the full parabola.<li> The graph of the p-th iterate can be plotted by using <font color=orange>-dp</font>, i.e. by adjustung the time lag of the embedding to <font color=orange>p</font>. Every intersection of a graph of a 1-d map and the diagonal is a fixedpoint of this map. An intersection point of the graph obtained for<font color=orange>-d2</font> is a fixed point of the second iterate of the map and thus one of thetwo points of a period 2 orbit. Study the orbits of up to period 4 of the logistic map for <fontcolor=orange>a=2 </font> by thismethod. Verify that there are 2 fixed points, 4 period-2 points (one non-trivialorbit and two trivial ones), 8 period-3 points (two non-trivial orbitsand two trivial ones), and 16 period-4 points (what about thecorresponding orbits?). <li> The mechanism of intermittency: Plot the time series for a=1.7499 intime delay coordinates with lag 3 together with the diagonal. Can you identify the reason why thetrajectory is intermittent? <a href=answer_intermittent.html> Answer</a>.</ul><b>The invariant measure:</b><br> <a href=../docs_c/histogram.html>histogram</a>produces a histogram of the input data, where several options can beused for adjusting, e.g., the number of bins.<br>Compute the histograms of the distribution of thevariable <font color=blue>x </font> of the logistic equation forvarious parameter values (e.g.: <font color=green>gnuplot> plot '< henon -B0 -A2 -l10000 | histogram -b100' withhist</font>). When a sufficiently long transient has beendiscarded, such a histogram is the approximation to the invaraintmeasure on the bins of the histogram. Verify numerically:<br><ul><li> The measure corresponding to a periodic orbit constists inequal-height delta-peaks at the locations of the points of this orbit.<li> The measure for <font color=blue>a=2 </font> fulfills <font color=blue>rho(x) = 1/(pi sqrt(1-x<sup>2</sup>))</font>.<li> The measure for chaotic orbits with <font color=blue>a < 2</font> contains a huge (a countable, infinite) number ofsingularities which are images of the singularity at <font color=blue>x=1 </font>.</ul><b>Lyapunov exponent:</b><br> When performing numerical simulations, Lyapunov exponent(s) should <b>only</b> be computed by direct iteration in tangentspace, not by time series analysis! Nonetheless, here we use time series analysis:Use <a href=../docs_c/lyap_k.html> lyap_k </a> and <a href=../docs_c/lyap_r.html> lyap_r </a> to compute the (only) Lyapunov exponent of the logistic equation for <fontcolor=blue>a=2 </font>:<br><font color=red>mycomputer:> henon -l10000 -B0 -A2. | lyap_k -M4 -n1000 -s20 -o lyap_k.dat<br>mycomputer:> henon -l10000 -B0 -A2. | lyap_r -s20 -o lyap_r.dat<br>mycomputer:> gnuplot</font><br>gnuplot><font color=green> plot 'lyap_k.dat' with lines, x*log(2.)-8, 'lyap_r.dat' with lines</font><br>Can you thus confirm the precise value <fontcolor=blue> lambda = ln(2) </font>? Study the resulting plots as afunction of the trajectory length. Also, add noise to the data using <a href=../docs_f/addnoise.html> addnoise </a> or <a href=../docs_c/makenoise.html> makenoise </a>.<br>You should observe that more than about 2% of noise (in root mean squaresense) will destroy the straight lines with slope log(2.). <br><hr></body></html>⌨️ 快捷键说明
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