ME 406Orbital StabilitysysidMathematica 6.0.3, DynPac 11.01, 1ê13ê2009‡1. IntroductionIn this notebook we develop some examples which illustrate the concept of orbital stability. We beginby reviewing the "standard" definition of stability, which is usually called Liapunov stability. The system underconsideration is an autonomous set of n first order differential equations:X° = F(X) ,where X is the state vector and F is the slope function. Here is the definition of Liapunov stability. Given a solution X*(t) with a given initial value X*( t0), wesay that X*is stable if, given any e > 0, we can find a d > 0 such that for any solution X(t) satisfying »» X*( t0) - X(t0)»» < d, it is true that »» X*( t) - X( t)»» < e for all t r t0. We call a stable solution X* strictly stable if there existsan h > 0 such that »» X*( t0) - X( t0)»» < h implies that »» X*( t) - X( t)»» goes to zero as tz ¶ . This way of defining stability means that neighboring solutions evaluated at the same time must remainneighboring. We shall see below that for periodic solutions this is in general too restrictive a definition ofstability. A more appropriate definition of stability for such systems will be given after our first example.‡2. Example 1 - A Nonlinear CenterWe consider Duffing's equation with a hard spring, a parameter a, and no damping: x– +x + a x3= 0 . We define the system for DynPac.setstate@8x, y<D; setparm@8a<D;slopevec = 9y, -x - a x3=; sysname = "Duffing";We set a parameter value of 0.5.parmval = 80.5<;As we already know, all of the solutions of this equation for a ¥ 0 are periodic. As an example, let's findand plot the solution with intial conditions (1.5, 0).t0 = 0.0; h = 0.02; nsteps = 300; initvec = 81.5, 0<;sol1 = integrate@initvec, t0, h, nstepsD;asprat = 1.0; plrange = 88-2.2, 2.2<, 8-2.2, 2.2<<; imsize = 250;graph1 = phaser@sol1D-2-112x-2-112yDuffing 8a< = 8 0.50<We find the period of the solution:[email protected] consider a nearby solution, obtained by altering the initial conditions slightly.initvec = 81.6, 0<; sol2 = integrate@initvec, t0, h, nstepsD;2 orbstab.nbgraph2 = phaser@sol2D-2-112x-2-112yDuffing 8a< = 8 0.50<We show the two graphs together.graph3 = show@graph1, graph2D-2-112x-2-112yDuffing 8a< = 8 0.50<The orbits are everywhere close, and we could have made them even closer had we chosen the initial values of xto be even closer. This is a situation which would be reasonable to call stable. However, by our earlier defini-tion of stability, these solutions are unstable. It is because of the period. For the second solution we have orbstab.nb 3The orbits are everywhere close, and we could have made them even closer had we chosen the initial values of xto be even closer. This is a situation which would be reasonable to call stable. However, by our earlier defini-tion of stability, these solutions are unstable. It is because of the period. For the second solution we have [email protected] the periods differ slightly. Points on neighboring orbits will not remain close, because they go around theorbits in different times. Let's make a movie to illustrate that more graphically. We use the function phaseseq.Double click on the graph below to start the movie. In the printed version of this notebook, only the first frameof the movie is shown.setcolor@8Red, Black, Black<D;[email protected], 0<, 81.6, 0<<, t0, h, 10, 160, 1, 2, graph3D-2-112x-2-112yDuffing 8a< = 8 0.50<The situation just illustrated clearly calls for an alternative definition of stability. Even though the twosolutions at any given time do not remain close, the two orbits do remain close in space. The concept of orbitalstability is appropriate to describe this situation. A solution X*with orbit G* is said to be orbitally stable if givenany e > 0, we can find a d such that for any solution X starting at t0a distance less than d from G*, it is true thatfor all t ¥ t0, the distance between the solution X and G*remains less than e. Although this is intuitively clear,one needs to define precisely the concept of the distance between a solution and an orbit. This is done, forexample, in Nonlinear Ordinary Differential Equations, by D.W. Jordn and P. Smith, second edition, OxfordUniversity Press, 1987, Chapter 8, or Dynamics and Bifurcations, J. Hale and H. Koçak, Springer-Verlag,1991, Chapter 11. It is clear that by this new definition, solutions of Duffing's equation are stable.‡3. Limit Cycles4 orbstab.nb‡3. Limit CyclesWith limit cycles, the concept of Liapunov stability may apply, but not strict Liapunov stability. To seethat, consider two initial points both on the cycle. When the points have gone through one cycle, they will beexactly the same distance apart as when they started, so the distance between the two solutions will not go tozero as t z ¶. With slight variations, the same situation is true for one initial point on the cycle and one just offthe cycle. The orbit of the intial point just off the cycle will approach the cycle for large t, but in general therewill always be a phase difference between the two points. Hale and Koçak (Chapter 11 -- cited above) give anexcellent discussion of these concepts, along with definitions of three levels of orbital stability. We will contentourselves here with constructing two movies which illustrate some of the concepts for stable limit cycles. Weuse the van der Pol cycle. We will first construct a movie showing the motion of two phase points which startwith a small separation on the limit cycle. Then we will construct a movie showing one phase point starting onthe cycle, and one phase point starting just inside the cycle. We begin by defining the system for DynPac.setstate@8x, y<D; setparm@8m<D; slopevec = 9y, -x - m * Ix2- 1M * y=;sysname = "van der Pol"; parmval = 81<;We construct a graph of the limit cycle.initvec = 82, 0<; t0 = 0.0; h = 0.02; nsteps = 337;sol4 = limcyc@initvec, t0, h, nstepsD;plrange = 88-3, 3<, 8-3, 3<<; asprat = 1.0; imsize = 250;orbstab.nb 5graph4 = phaser@sol4D-3-2-1123x-3-2-1123yvan der Pol 8m< = 81<[email protected] look at the last 10 points in the solution.points = Take@sol4, -10D8813.3, 2.0084, 0.0304849<, 813.32, 2.00859, -0.0102867<,813.34, 2.008, -0.0486543<, 813.36, 2.00666, -0.0847496<,813.38, 2.00462, -0.118704<, 813.4, 2.00193, -0.150645<,813.42, 1.99861, -0.180698<, 813.44,