ME 406The Lorenz EquationssysidMathematica 4.1.5, DynPac 10.67, 4ê8ê2002intreset;plotreset;‡1. IntroductionThis notebook contains all of the material given in class on the Lorenz equations, and it constitutes section2.5 of the class notes. The Lorenz equations are given by(1)x° = s(y - x) , y° = rx - y -xz , z° = xy - bz .These equations contain three parameters: s, r and b. In what follows, we will always assume that these parame-ters are positive. In all of the numerical calculations below, we take s = 10.0, and b = 8/3. These are the mostused values in the study of the Lorenz equations. We will vary the parameter r over a wide range, and study howthe solutions depend on r.These equations, which are simple in appearance, have solutions with extraordinary properties. They werefirst studied in the 1960's by the M.I.T. meteorologist Edward Lorenz. He developed the equations as a model forthe modal amplitudes in a nonlinear thermal convection problem. Lorenz recognized that the solutions of theequations can exhibit an unusual form of behavior which we now call chaos. It took time for others to realizeexactly what Lorenz had discovered. Lorenz has told the story of the discovery in his book The Essence ofChaos, University of Washington Press, 1993. For a very readable and basic treatment of the equations, seeChapter 9 of Nonlinear Dynamics and Chaos, S.H. Strogatz, Addison-Wesley, 1994. For a general workcontaining a more advanced treatment, see Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields, J. Guckenheimer and P. Holmes, Springer-Verlag, 1983. For a book-length treatment containingmany detailed results, see The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, C. Sparrow,Springer-Verlag, 1982. In our study here, we will continue our primarily experimental approach and use thecomputer to learn about the system. We begin by defining the equations for DynPac.setstate@8x, y, z<D; setparm@8s, r, b<D; sysname = "Lorenz";slopevec = 8s * Hy - xL, r * x - y - x * z, x * y - b * z<;‡2. Basic Propertiesü2.1 SymmetryThe equations are invariant to the transformation(2)(x, y, z) ö (-x, -y, z) .Thus if x(t), y(t), z(t) is a solution, so is -x(t), -y(t), z(t). As we shall see, this symmetry shows up in a number ofways, including the location of the equilibrium points of the system.ü2.2 The z-Axis is InvariantWe see from equations (1) that if x(0) = 0 and y(0) = 0, then x and y remain zero for all t. Thus the z-axisis an orbit, on which(3)z° = -bz , hence z(t) = z(0)e-bt, for x, y = 0. Thus the z-axis is always a part of the stable manifold for the equilibrium at the origin.ü2.3 The System is DissipativeThe divergence of the slopevector isdiv@slopevecD-1 - b - sand this is always negative. As we saw earlier in class, for any given volume V of phase points moving with theflow, we have (4)dVÅÅÅÅÅÅÅÅÅÅÅÅdt= V div[slopevec] = -(1 + b + s)V, hence V(t) = V(0)e-H1+b+sL t . With our canonical values of 10 for s and 8/3 for b, this is V(t) = V(0)e-13.67 t, so that volumes of initial points arereduced by a factor of e in a time 1 ê 13.670.0731529ü2.4 The Solutions are BoundedIt is not hard to prove that the solutions of the Lorenz equations are bounded. Consider first the case whenr < 1. We examine a potential Liapunov function V = x2+ s * Hy2+ z2L;The orbital derivative isSimplify@orbdt@VDD-2 Hx2- H1 + rL x y + y2+ b z2L sThe z-term is negative. We get for the x and y termsx2- H1 + rL x y + y2= Ax -1ÅÅÅÅÅ2 H1 + rL yE2+1ÅÅÅÅÅ4 H1 - rL H3 + rL y2,2 loreq.nband this quantity is postive except for x = y = 0. Thus V° < 0, and this shows that the origin is a global attractor forthe system when r < 1.When r > 1, we have to work a little harder to show the solutions are bounded. We start with a newLiapunov functionV = r * x2+ s * y2+ s * Hz - 2 * rL2;Level surfaces V = V0 are ellipsoids with center at (0, 0, 2r), and semi-axes (è!!!!!!!!!!!V0êr, è!!!!!!!!!!!!V0ês, è!!!!!!!!!!!!V0ês). Theorbital derivative isVdot = Simplify@orbdt@VDD-2 Hy2+ b z2+ r Hx2- 2 b zLL sWe work on this a bit.Vdotmod = Vdot ê H2 s b r2L-y2+ b z2+ r Hx2- 2 b zLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅb r2We can write this as(5)- x2ÅÅÅÅÅÅÅÅÅÅbr- y2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅb r2- Hz - rL2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅr2 + 1The expression in equation (5) is zero on the ellipsoid given by (6) x2ÅÅÅÅÅÅÅÅÅÅbr+ y2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅb r2+ Hz - rL2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅr2 = 1 ,and is positive inside that ellipsoid and negative outside that ellipsoid. By choosing V0 large enough, we canmake the level surface V = V0 lie entirely outside the ellipsoid (6). Any orbit starting outside that level surfacewill cross to the inside, and any orbit starting inside that level surface will remain inside. Thus all of the solutionsare bounded, subject only to the assumption we made at the beginning that s, r, and b are positive. Unlike thecase r < 1, we cannot in this case draw any conclusions about a global attractor or even a stable equilibrium. ü2.5 EquilibriaWe use findpolyeq to look for equilibria.eqpoints = findpolyeq980, 0, 0<, 9-è!!!!!!!!!!!!!!!!!-b + b r , -è!!!!!!!!!!!!!!!!!-b + b r , - 1 + r=,9è!!!!!!!!!!!!!!!!!-b + b r ,è!!!!!!!!!!!!!!!!!-b + b r , -1 + r==We see that the origin is an equilibrium for any values of the parameters. The other two equilibria are real if andonly if r ≥ 1. In fact for r = 1, the three equilibria coincide at {0,0,0}, and we therefore have a pitchfork bifurca-tion at r = 1. We will look at this in more detail later. We name the three equilibria for later reference.loreq.nb 3C0 = eqpoints[[1]]80, 0, 0<C1 = eqpoints[[2]]9-è!!!!!!!!!!!!!!!!!-b + b r , -è!!!!!!!!!!!!!!!!!-b + b r , -1 + r=C2 = eqpoints[[3]]9è!!!!!!!!!!!!!!!!!-b + b r ,è!!!!!!!!!!!!!!!!!-b + b r , -1 + r=Note that the x and y components of C1 and C2 are the negatives of one another, consistent with the symmetrydiscussed earlier. In addition, for each equilibrium we have x = y, which is a direct consequence of x° = s(y - x) atequilibrium.‡3. Bifurcations with Changing