ME 406 Example of Stable and Unstable Manifolds

Transcription

1 ME 46 Eample of Stable and Unstable Manifolds intreset; plotreset; ssid Mathematica 4..2, DnPac.66, 3ê8ê22 Introduction In this notebook, we stud an eample given in Differential Equations and Dnamical Sstems, Lawrence Perko, second edition, Springer-Verlag, 996, p.. The equations are = - - 2, = 2 +. () This sstem has an equilibrium point at (,). We will construct both the linear and nonlinear stable and unstable manifolds in the vicinit of the origin. Then we will construct the global stable and unstable manifolds. Sstem Definition We define the sstem for DnPac. setstate[{,}]; setparm[{}]; slopevec = , + 2 <; parmval = {}; ssname = "nlsstem"; intreset; plotreset; Equilibrium Point We start b analzing the equilibrium at the origin. We first use classif2d, and then we use eigers to obtain the eigenvectors and eigenvalues at the origin.

2 2 manifold.nb classif2d[{,}] Abbreviations used in classif2d. L = linear, NL = nonlinear, R2 = repeated root. Z = one zero root, Z2 = two zero roots. This message printed once. unstable - saddle eigers = eigss@8, <D 88, <, 88, <, 8, <<< Stable and Unstable Manifolds for Nonlinear Sstem We will use blue for stable manifolds, red for unstable manifolds. The stable manifold corresponds to the negative eigenvalue. To construct it, we integrate backwards in time, starting ver near the equilibrium, displaced from the equilibrium along the eigenvector associated with the eigenvalue. The solution going backwards in time could grow ver rapidl, so we avoid the possibilit of overflow and crash b using range checking. We limit the range to a bo from to in both variables. We add an arrow to the mid-point of each orbit. We remove the aes (and put a frame around the picture) so that the tangencies can be better seen when we compare linear and nonlinear sstems. ranger = {{.4,.4},{.4,.4}}; rangeflag = True; t =.; h = -.2; nsteps = ; initset = {.5*{,},-.5*{,}}; arrowflag = True; arrowvec = 8 ê 2<; asprat =.; plrange = 88.,.<, 8.,.<<; aon = False; frameon = True; setcolor@8blue<d;

3 manifold.nb 3 stableport = portrait[initset,t,h,nsteps,,2]; nlsstem Now we construct the unstable manifold. setcolor@8red<d; initset = {.5*{,},-.5*{,}}; h =.2;

4 4 manifold.nb unstableport = portrait[initset,t,h,nsteps,,2]; nlsstem Now we combine these

5 manifold.nb 5 nlman = show@stableport, unstableportd; nlsstem Stable and Unstable Manifolds for Linearized Sstem Now we construct the linearized sstem and then go through the same manifold constructions for it. We first save the nonlinear sstem under the name nlsstem. savess[nlsstem]; We continue to use {,} for the state vector, and both parmval and parmvec remain empt lists. Thus for the linear sstem, we must redefine onl the slope function. It is the derivative matri at equilibrium times the state vector. slopevec = Dot[dermatval[{,}],{,}] 8-, < ssname = "linsstem"; Now we construct the linear stable manifold. ranger = {{.4,.4},{.4,.4}};

6 6 manifold.nb rangeflag = True; t =.; h = -.2; nsteps = ; initset = {.5*{,},-.5*{,}}; arrowflag = True; arrowvec = 8 ê 2<; asprat =.; plrange = 88.,.<, 8.,.<<; aon = False; setcolor@8blue<d; linstableport = portrait[initset,t,h,nsteps,,2]; linsstem Now we construct the unstable manifold. setcolor@8red<d; initset = {.5*{,},-.5*{,}};

7 manifold.nb 7 h =.2; linunstableport = portrait[initset,t,h,nsteps,,2]; linsstem We combine the last two pictures.

8 8 manifold.nb lman = show@linstableport, linunstableportd; linsstem Now for the finale, showing both the linear and nonlinear manifolds and illustrating the tangencies at the equilibrium point.

9 manifold.nb 9 show@lman, nlmand; linsstem Global Manifolds for Nonlinear Sstem Now we restore the nonlinear sstem, and we tr to find the global stable and unstable manifolds. We will get some surprises in that effort. restoress@nlsstemd; We enlarge our plotting window, and we begin with the construction of the stable manifold. We start on the eigenvectors ver near the origin and integrate backwards in time. eps =.; initset = 8eps * 8, <, eps * 8, <<; t =.; h =-.2; nsteps = ; plrange = 88-2., 2.<, 8-2., 2.<<; asprat =.; rangeflag = True; ranger = plrange; bothdirflag = False; setcolor@8black<d;

10 manifold.nb labon = "Stable Manifold"; arrowflag = True; arrowvec = 8 ê 2<; globestab = portrait[initset,t,h,nsteps,,2]; 2 Stable Manifold Now we carr this out for the unstable manifold. For this, we start near the equilibrium on the eigenvectors and integrate forward in time. initset = 8eps * 8, <, eps * 8, <<; h =.2; labon = "Unstable Manifold";

11 manifold.nb globeunstab = portrait[initset,t,h,nsteps,,2]; 2 Unstable Manifold The biggest surprise here is the homoclinic loop rooted in the origin: it is part of both the global unstable and global stable manifold! A little thought shows that this doesn't violate the rules, whether or not we imagined such a thing could happen. If we start on the loop and go forward in time, we arrive at the origin, hence it is part of the stable manifold. If we start on the loop and go backward in time, we also arrive at the origin, so it is part of the unstable manifold. We can show the two manifolds together: labon = "Unstable and Stable Manifolds";

12 2 manifold.nb globeunstabd; 2 Unstable and Stable Manifolds Although the homoclinic loop is not periodic, it is prett clear that there is a periodic solution or solutions in its interior, and also an equilibrium point. The equilibrium point is eas to find -- it is {,}, also confirmed b DnPac: findpoleq 88, <, 8, <, 8HL ê3, -HL 2ê3 <, 8-HL 2ê3, HL ê3 << We classif the new equilibrium. classif2d@8, <D stable HLL, indeterminate HNLL - center Let's tr a few initial conditions inside the loop. initset = 88,.4<, 8, -.7<<; t =.; h =.2; nsteps = 5; arrowvec = 8 ê 2<; labon = "Periodic Solutions";

13 manifold.nb 3 periodic = portrait[initset,t,h,nsteps,,2]; 2 Periodic Solutions -2 Thus the equilibrium at {,} is a nonlinear center. We put all of this together labon = "Phase Portrait";

14 4 manifold.nb globeunstab, periodicd; 2 Phase Portrait We add two more orbits for completeness. Two in the upper right, and a third which curves around the outside of the homoclinic loop. initset = 882., <, 82.,.4<, 82., <<;

15 manifold.nb 5 moreport = portrait[initset,t,h,nsteps,,2]; 2 Phase Portrait

16 6 manifold.nb globeunstab, periodic, moreportd; 2 Phase Portrait