ME 406 Trapping a Limit Cycle

Transcription

1 orbtrap.nb 1 ME 6 Trapping a Limit Ccle In[3]:= ssid Mathematica.1, DnPac 1.65, ê13ê In[33]:= In[3]:= intreset; plotreset; ü Introduction In this notebook, we localize a limit ccle b orbit trapping. The eample we consider, given below, is taken from Nonlinear Differential Equations and Dnamical Sstems, nd edition, Ferdinand Verhulst, Springer, 1996, eample.8, page 8, with a correction of a sign error. = - H L, = - - H L. We now define the sstem for Mathematica. We include a parameter a so that variations of the sstem can be studied easil. For our sstem, a = 1. In[35]:= In[36]:= In[37]:= setstate@8, <D; slopevec = 8 - a * * H^+ ^- * - 3L, - - a * * H^ + ^ - * - 3L<; setparm@8a<d; In[38]:= parmval = 81<; In[39]:= ssname = "LimCc.8"; In[33]:= plrange = 88-5, 5<, 8-5, 5<<; In[331]:= labshift = 1; ü Equilibrium Point We look for equilibrium states using findpoleq. In[33]:= Out[33]= findpoleq -Â - 3a 98, <, 9 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1-3 Â a =, 9 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Â-3a a a a, ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Â a == a

2 orbtrap.nb Mathematica has found three equilibria of which onl one is real. As shown in class, there are no other real equilibria. (Important caution: we never know for sure without independent confirmation whether a root-finder such as findpoleq has found all of the equilibrium states.) Let's look at the stabilit of the equilibrium at the origin: In[333]:= classifd@8, <D Abbreviations used in classifd. L = linear, NL = nonlinear, R = repeated root. Z1 = one zero root, Z = two zero roots. This message printed once. unstable - spiral This is tpical of what we would find inside a limit ccle. In a search for periodic solutions, we move net to the Bendison test. ü Application of Bendison Test From the Bendison test, we know that periodic solutions can occur onl in regions where the divergence of the slope vector changes sign or is identicall zero. We use the function signcontour to map the algebraic sign of the divergence here. The function dival returns the divergence with current values of the parameters substituted. In[33]:= Out[33]= bendi = dival@slopevecd - H + L - - H L We turn off the graphics aes and turn on the frame. In[335]:= aon = False; frameon = True;

3 orbtrap.nb 3 In[336]:= graph1 = signcontour@bendid; LimCc.8 8a<=8 1.< - - The divergence is positive in the white region (which can be shown to be a circle with center at = 3/, = and radius è!!!!!! 33 /). B the Bendison criterion, an periodic solution will have to enclose both white and gre regions. This is onl a necessar condition, and we now move to orbit trapping techniques to establish that there is a periodic solution. ü Orbit Trapping Because the are eas, we will start with circles. We define our trapping curve as a circle of radius b with center at the origin. We first define an arc and then a curve from the arc. In[337]:= In[338]:= trapperarc@b_d := 88b * Cos@uD, b* Sin@uD<, 8u,, * Pi<<; trappercurve@b_d := 8trapperarc@bD< Now we use the routine orbcross to see whether a given circle traps orbits. We start with a radius b =.5. In[339]:= orbcross@trappercurve@.5dd

4 orbtrap.nb Thus the circle with radius.5 traps orbits outside. If we can find a larger circle which traps orbits inside, we will have established the eistence of a periodic solution. We tr b = 5.. In[3]:= orbcross@trappercurve@5.dd We have been luck. We now know that there is a periodic solution somewhere between b =.5 and b = 5.. Let's refine our guesses. In[31]:= orbcross@trappercurve@1.dd This is optimum because it still traps, but there are tangencies. For a slightl larger b, we would get crossings in both directions. Let's verif that: In[3]:= orbcross@trappercurve@1.1dd Now we locate more precisel an outer trapping curve, working back from b = 5.. In[33]:= orbcross@trappercurve@.dd In[3]:= orbcross@trappercurve@3.dd This also appears optimum, and we verif it b a slightl smaller b:

5 orbtrap.nb 5 In[35]:= orbcross@trappercurve@.9dd Now we have a periodic orbit trapped between b = 1. and b = 3.. Let's plot these two curves in red: In[36]:= In[37]:= In[38]:= In[39]:= In[35]:= In[35]:= displa = False; setcolor@8red<d; graph = plotcurve@trappercurve@1.dd; graph3 = plotcurve@trappercurve@3.dd; displa = True; graph = show@graph1, graph3, graphd; LimCc.8 8a<=8 1.< - - Now let's add the limit ccle in blue. In[353]:= nsteps = 5;

6 orbtrap.nb 6 In[35]:= h =.5; In[355]:= t = ; In[356]:= initvec = 8, <; In[357]:= In[358]:= sol1 = limcc@initvec, t, h, nstepsd; arrowflag = True; In[359]:= arrowvec = 81 ê <; In[36]:= In[361]:= setcolor@8blue<d; graph5 = phaser@sol1d; LimCc.8 8a<=8 1.< - Now we put it all together: -

7 orbtrap.nb 7 In[36]:= graph6 = show@graph, graph5d; LimCc.8 8a<=8 1.< - - As a final calculation, we add a few orbits approaching the limit ccle, also in blue. In[363]:= initset = 88, <, 8, <, 83, 3<, 8-3, -3<, 83, -3<, 8-3, 3<, 81, 1<, 81, -1<<; In[36]:= nsteps = 1; In[365]:= arrowvec = 81 ê 8<;

8 orbtrap.nb 8 In[366]:= graph7 = portrait@initset, t, h, nsteps, 1, D; LimCc.8 8a<=8 1.< - -

9 orbtrap.nb 9 In[367]:= graph8 = show@graph6, graph7d; LimCc.8 8a<=8 1.< - - The approach orbits show clearl the stabilit of the limit ccle.