%% 15. Stable and unstable fixed points %% % % \setcounter{page}{185} % \parindent=.6cm % \parskip=-1em % %% % The last chapter dealt with linearization of ODE\kern .5pt s. One of the % most important applications of linearization is the analysis % of fixed points. %% % % Consider an autonomous equation % $${\bf y}'(t) = {\bf f}({\bf y}(t)). \eqno (15.1)$$ % As defined in Chapters~9 and~10, a {\bf fixed point} of % (15.1) is a vector ${\bf y}_*^{}$ such that ${\bf f}({\bf y}_*^{}) = % {\bf 0}$. We saw in those chapters that a fruitful way of % understanding the behavior of an autonomous ODE is % to begin by examining its fixed points in the phase plane % (if $n=2$) or more generally in phase space. % If ${\bf y}_*^{}$ is a fixed point, then the term ${\bf f}_*^{}$ % in Theorem 14.4 vanishes. Equation (14.17) becomes % $$\delta {\bf y}'(t) = {\bf J}_*^{} \delta {\bf y}(t) + % O(\|\delta {\bf y}(t)\|^2), \eqno (15.2)$$ % with $\delta{\bf y}(t) = {\bf y}(t) - {\bf y}_*^{}$ as before. % Near a fixed point, therefore, an ODE behaves approximately like the % equation ${\bf y}' = {\bf A}{\bf y}$ with which % we began the last chapter --- not just % affine, but linear, with ${\bf A}$ being the Jacobian matrix ${\bf J}_*^{}$. % %% % % To illustrate the structure of some linearizations at % fixed points in the phase plane, % here are two plots showing solutions of % (15.2) (without the O'' term) % corresponding to the diagonal matrices % $${\bf J}_*^{} = \pmatrix{-1 & \hphantom{-}0 \cr \hphantom{-}0 & -1} ,~ % \pmatrix{-2 & \hphantom{-}0 \cr \hphantom{-}0 & -1} . % \eqno (15.3)$$ % Each plot shows trajectories emanating from % 16 equally spaced initial points on the unit circle, which % is drawn in black. % ODEformats, th = (pi/8)*(1:16) + .0001; u0 = cos(th); v0 = sin(th); L = chebop(0,1.4); op = @(J) @(t,u,v) [diff(u)-J(1,1)*u-J(1,2)*v; ... diff(v)-J(2,1)*u-J(2,2)*v]; subplot(1,2,1), plot(0,0,'.k',MS,6), hold on c = chebfun('exp(1i*pi*x)'); plot(c,'k',LW,.7) J = [-1 0; 0 -1]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-1.25 1.25 -1.1 1.4]), set(gca,XT,-1:1,YT,-1:1) pp = get(gca,'pos'); pp(1) = pp(1)+.015; set(gca,'pos',pp) text(0,1.15,'stable node',FS,9,HA,CT) axis square, hold off subplot(1,2,2), plot(0,0,'.k',MS,6), hold on, plot(c,'k',LW,.7) J = [-2 0; 0 -1]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-1.25 1.25 -1.1 1.4]), set(gca,XT,-1:1,YT,-1:1) pp = get(gca,'pos'); pp(1) = pp(1)-.015; set(gca,'pos',pp) text(0,1.15,'stable node',FS,9,HA,CT) title(['Fig.~15.1.~~Fixed points for the matrices ' ... '(15.3)\kern -.40in'],FS,11,HA,RT) axis square, hold off %% % \vskip 1.02em %% % % \noindent % In both figures, all trajectories are converging to the origin. % This is because both eigenvalues of ${\bf J}_*^{}$, which in this % case are simply the diagonal entries, are negative. % If all the eigenvalues of the Jacobian at a fixed point are % negative, or more generally satisfy % $\hbox{Re\kern 1pt}\lambda<0$ and thus lie in the open % left half of the complex plane, then ${\bf y}_*^{}$ is called % a {\bf sink}. This implies % that all orbits starting sufficiently close to ${\bf y}_*^{}$ converge to % ${\bf y}_*^{}$ at an exponential rate. % The image on the left above may be the first one that % comes to mind when one thinks about sinks, but the image on % the right is more typical: most trajectories approach a % sink along special directions corresponding to the eigenvectors % associated with eigenvalues of least negative real part. % In this example the eigenvectors are $(1,0)^T$, the direction % of exponential decay at the rate $e^{-2t}\kern 1pt$, and % $(0,1)^T$, the direction of slower decay at the rate % $e^{-t}\kern 1pt$. As $t$ increases, the component in % the $(1,0)^T$ direction becomes negligible compared with the component % in the $(0,1)^T$ direction, so trajectories approach the % origin along the latter, vertical axis. % %% % But still we have diagonal matrices, so these % pictures do not show the general behavior. % To illustrate some further possibilities, here are figures for % two nondiagonal matrices, % $${\bf J}_*^{} = \pmatrix{-1 & \hphantom{-}1 \cr -1 & -1} ,~ % \pmatrix{-2 & 2.5 \cr \hphantom{-}0 & -1} , \eqno (15.4)$$ % whose eigenvalues are $\{-1+i, -1-i\}$ and % $\{-2, -1\}$, respectively. % Again the eigenvalues are in the left half-plane, so % $\bf 0$ is again a sink, but the images are quite different, % showing a combination of rotation mixed with decay. subplot(1,2,1), plot(0,0,'.k',MS,6), hold on, plot(c,'k',LW,.7) J = [-1 1; -1 -1]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-1.35 1.35 -1.2 1.5]), set(gca,XT,-1:1,YT,-1:1) pp = get(gca,'pos'); pp(1) = pp(1)+.015; set(gca,'pos',pp) text(0,1.2,'stable spiral',FS,9,HA,CT) axis square, hold off subplot(1,2,2), plot(0,0,'.k',MS,6), hold on, plot(c,'k',LW,.7) J = [-1 2.5; 0 -1/2]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-1.35 1.35 -1.2 1.5]), set(gca,XT,-1:1,YT,-1:1) pp = get(gca,'pos'); pp(1) = pp(1)-.015; set(gca,'pos',pp) text(0,1.2,'stable node',FS,9,HA,CT) axis square, hold off title(['Fig.~15.2.~~Fixed points for the matrices ' ... '(15.4)\kern -.40in'],FS,11,HA,RT) %% % \vskip 1.02em %% % % \noindent % Note that in the left image, there is no special direction along % which trajectories eventually straighten out. % Such a direction would correspond to an eigenvector, % but for this matrix, the eigenvectors are complex. % The eigenvalues % are $-1+i$ and $-1-i$, both in the left half-plane, which % explains the decay toward the origin. In the right image, % there are real eigenvectors again but % they are far from orthogonal. The effect of this is % that although eventually all the % trajectories decay to the origin, some of them grow % for a while before decaying. This phenomenon is known % as {\em transient growth.} % %% % % All these plots correspond to the same simple case of % a $2$-variable problem with a sink. This is only % the beginning of the many configurations that can arise % in linearized analysis of fixed points. If all the % eigenvalues are in the open right half-plane, that is, with % $\hbox{Re\kern 1pt}\lambda>0$, then ${\bf y}_*^{}$ is a {\bf source} % and the arrows are reversed. If some eigenvalues are in % the left half-plane and the others are in the right half-plane, % then ${\bf y}_*^{}$ is a {\bf saddle point}. % Here are examples of saddle points corresponding to % the matrices % $${\bf J}_*^{} = \pmatrix{-1 & 0 \cr \phantom{-}0 & 1} ,~ % \pmatrix{-1 & 2 \cr \phantom{-}0 & 1} . % \eqno (15.5)$$ % The eigenvalues of both matrices are $\{-1,1\}$. % subplot(1,2,1), plot(0,0,'.k',MS,6), hold on, plot(c,'k',LW,.7) L = chebop(0,0.9); J = [-1 0; 0 1]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-3 3 -2.7 3.3]), set(gca,XT,-3:3,YT,-3:3) pp = get(gca,'pos'); pp(1) = pp(1)+.015; set(gca,'pos',pp) text(0,2.8,'saddle point',FS,9,HA,CT) axis square, hold off subplot(1,2,2), plot(0,0,'.k',MS,6), hold on, plot(c,'k',LW,.7) J = [-1 2; 0 1]; L.op = op(J); for k = 1:16 L.lbc = [u0(k); v0(k)]; [u,v] = L\0; arrowplot(u,v,CO,ivp,MS,3) end axis([-3 3 -2.7 3.3]), set(gca,XT,-3:3,YT,-3:3) pp = get(gca,'pos'); pp(1) = pp(1)-.015; set(gca,'pos',pp) text(0,2.8,'saddle point',FS,9,HA,CT) axis square, hold off title(['Fig.~15.3.~~Fixed points for the matrices ' ... '(15.5)\kern -.40in'],FS,11,HA,RT) %% % \vskip 1.02em %% % % There is a general terminology and theory of behavior near % fixed points that goes beyond these linearized approximations. % We say that a fixed point of a system of the form (15.1) % is {\bf Lyapunov stable} if for any neighborhood % $V$ of~${\bf y}_*^{}$, there is a neighborhood % $U\subseteq V$ such that every trajectory that starts in~$U$ % remains in $V$ for all $t$. % This condition does not require decay, just boundedness. % The fixed point is {\bf asymptotically stable} % if in addition, $U$ can be chosen such that every trajectory % that starts in $U$ converges to ${\bf y}_*^{}$ as % $t\to\infty$.\footnote{Despite this careful terminology, % most of the time we will be more casual and just say % a fixed point is stable if all nearby trajectories converge % to it.} % If~${\bf y}_*$ is not Lyapunov stable, it is {\bf Lyapunov unstable}, implying % that some (not necessarily all) trajectories starting near % ${\bf y}_*$ diverge away as $t$ increases. % The following theorem, which we give without proof, summarizes % some of the relationships between these general notions % and the eigenvalues of~${\bf J}_*^{}$. % %% % % \medskip % {\em % {\bf Theorem 15.1. Stability and eigenvalues of the % Jacobian (\textsf{\textbf{FlAsHI}}).}\ \ Let\/ ${\bf y}_*^{}$ be a fixed point of an % autonomous ODE\/ $(15.1)$ where\/ ${\bf f}$ is twice differentiable % at\/~${\bf y}_*^{}$, and let ${\bf J}_*^{}$ be the associated % Jacobian matrix. % If all the eigenvalues $\lambda$ of\/ ${\bf J}_*^{}$ satisfy % $\hbox{\rm Re\kern 1pt} \lambda < 0$, then\/ ${\bf y}_*^{}$ is % asymptotically stable, % and if at least one of them % satisfies $\hbox{\rm Re\kern 1pt} \lambda > 0$, % then\/ ${\bf y}_*^{}$ is Lyapunov unstable. % } % \medskip % %% % Note that the theorem leaves open the situation in which % all eigenvalues satisfy $\hbox{Re\kern 1pt} \lambda\le 0$ but not % all satisfy $\hbox{Re\kern 1pt} \lambda<0$. In this case % ${\bf y}_*^{}$ is unstable if there % is a defective multiple eigenvalue with $\hbox{Re\kern 1pt}\lambda=0$ (i.e., % associated with a Jordan block of size $\ge 2$). If all % eigenvalues with $\hbox{Re\kern 1pt}\lambda=0$ are nondefective, % then linear analysis is not enough % to determine stability or asymptotic stability; it depends % on the higher-order nonlinear behavior of $\kern 1pt \bf f$. %% % % Let us now consider three examples, systems of ODE\kern .5pt s % we looked at in Chapter~10. The first of these was % the Lotka--Volterra equations (10.4)--(10.5), % $$u' = u - uv, \quad v' = -\textstyle{1\over 5}v + uv,$$ % whose behavior was plotted in Figs.~10.4--10.8 and again % in Figs.~14.5--14.6. % To calculate the fixed points we set $u'=0$, implying $u=0$ or $v=1$, % and $v'=0$, implying $v=0$ or $u=1/5$. So the fixed points % are $(u,v) = (0,0)$ and $(1/5,1)$. The Jacobian of the system at % $(u,v)$ is % $${\bf J} (u,v) = \pmatrix{1-v & -u \cr v & u -1/5},$$ % implying % $${\bf J} (0,0) = \pmatrix{1 & 0 \cr 0 & -1/5}, \quad % {\bf J} (1/5,1) = \pmatrix{0 & -1/5 \cr 1 & 0}.$$ % The first matrix has eigenvalues $1$ and $-1/5$, so % $(0,0)$ is a saddle point. The image on the right % eigenvalues $\pm i/\sqrt 5$, both imaginary, % so $(1/5,1)$ is a center, a point of neutral stability. % This explains the rotation of trajectories that is the conspicuous % feature of Figs.~10.6--10.8. % %% % % The next example from Chapter 10 is the Lorenz equations (10.7), % $$u' = 10(v-u), \quad v' = u(28-w)-v, \quad w' = uv-(8/3)w,$$ % plotted in Figs.~10.9--10.11 and considered further in Chapters~13 % and~14. % To find fixed points we calculate that $u'=0$ implies % $v=u$, $v'=0$ then implies $u=v=0$ or $w=27$, and % $w'=0$ implies $u=v=w=0$ or $u=v = \pm\sqrt{8\cdot 27/3} = \pm 6\sqrt 2$. % So there are three fixed points, and they are % $(u,v,w) = (0,0,0)$ and $(\pm 6\sqrt 2,\kern .8pt \pm 6\sqrt 2,\kern .8pt 27)$. % The Jacobian of the system at $(u,v,w)$ is % $${\bf J} (u,v,w) = \pmatrix{-10 & 10 & 0 \cr 28-w & -1 & -u \cr v & u & -8/3},$$ % implying % $${\bf J} (0,0,0) = \pmatrix{-10 & 10 & 0 \cr 28 & -1 & 0 \cr 0 & 0 & -8/3}$$ % and % $${\bf J} (\pm 6\sqrt 2,\kern .8pt \pm 6\sqrt 2,\kern .8pt 27) = % \pmatrix{-10 & 10 & 0 \cr 1 & -1 & \mp 6\sqrt 2 \cr \pm 6\sqrt 2 &\ % \pm 6\sqrt 2 & -8/3}.$$ % The eigenvalues of $J(0,0,0)$ are about $-22.8$, $-2.7$, and $11.8$, so % this is an unstable saddle point. The other fixed points are % the more interesting ones, with eigenvalues about $-13.9$ and % $0.9 \pm 10.2i$. Thus these fixed points are unstable too, mildly so, but % because of the large imaginary term they have a big rotational component, % as we know well from the Lorenz trajectories of Figs.~10.11 and~13.1. % %% % % The other main example of Chapter 10 is the system % of equations (10.10) of an SIR model % from epidemiology, which with the parameter % choices $\beta =2$ and $\gamma=1$ takes the form % $$S' = -2 SI, \quad I' = (2S-1)I, \quad R' = I.$$ % Behaviors were explored in Figs.~10.13--10.14. % For a fixed point analysis we first set $R'=0$, implying $I=0$. This then % implies $R'=S'=0$, so we see that every choice of $R$ and $S$ gives a % fixed point, so long as $I=0$. The Jacobian matrix is % $${\bf J} (S,I,R) = \pmatrix{-2I & -2S & 0 \cr 2I & 2S-1 & 0 \cr 0 & 1 & 0},$$ % which at a fixed point becomes % $${\bf J} (S,I,R) = \pmatrix{0 & -2S & 0 \cr 0 & 2S-1 & 0 \cr 0 & 1 & 0},$$ % Since this matrix is block lower-triangular, with an upper-left $2\times 2$ % block that is itself upper-triangular, we see that the eigenvalues % are $0$, $0$, and $2S-1$. The latter number has % immediate significance: if $S>1/2$, the matrix has a positive eigenvalue % and the system is unstable, ready to begin an epidemic as % soon as any patient gets infected. % %% % % \begin{center} % \hrulefill\\[1pt] % {\sc Application: transition to turbulence in a pipe}\\[-3pt] % \hrulefill % \end{center} % %% % % Of all the fixed points in the mathematical sciences, perhaps % none has received more attention, or caused % more confusion, than laminar fluid flow in a % pipe.\footnote{There are competitors, though the equations % are not so clear-cut, in climate science. Whenever you hear the % phrase tipping point,'' you can be sure that there is a question of % stability of a fixed point at hand. One tipping point of great % concern involves the melting of the ice in the Arctic: as ice % melts, the earth reflects less light out to space, and the % ice melts faster. See D. Paillard, The % timing of Pleistocene glaciations from a simple % multiple-state climate model,'' {\em Nature} 391 (1998), pp.~378--381.} % (Laminar means smooth and steady.) % This discussion is adapted from % Trefethen, Trefethen, Reddy, and Driscoll, % Hydrodynamic stability without eigenvalues,'' {\em Science,} 1993. % %% % % The problem was made famous by Osborne Reynolds in 1883. % Imagine a long circular pipe with a fluid such as water flowing % through it. The flow is governed by the set of time-dependent PDE\kern .5pt s % known as the Navier--Stokes equations, determining the evolution of the % velocity field ${\bf v}({\bf x},t)$, and an analytical solution corresponding % to laminar flow can be written down easily: $\bf v$ is a time-independent field % pointing along the pipe, corresponding to a steady % flow with a velocity that is maximal at the centerline and % decreases quadratically to zero at the wall. This solution % is a fixed point of the equations, and it is mathematically stable. % We have not discussed PDE\kern .5pt s and their stability, but just % as for ODE\kern .5pt s, the idea is that % any sufficiently small perturbation of the laminar flow velocity % field in a pipe must eventually die away. % So a mathematician would expect that laminar flow of water through a % pipe should be possible at any speed. % %% % % The paradox is that in practice, this is not what is % observed. If the flow in a pipe % is fast enough, it is invariably not laminar but turbulent --- complicated, % apparently chaotic, highly time-dependent. % Clearly the mathematical solution, laminar flow, has something wrong % with it in the laboratory. % What is going on? How can a flow that is stable % mathematically be unstable in practice? % %% % % The explanation is that although % sufficiently small perturbations of the laminar velocity flow field % must eventually decay, the threshold that defines % sufficiently small'' is too small % to be counted upon in practice. % The mathematics of these high-speed flow problems % is such that the minimal amplitude of perturbations that do % {\em not\/} eventually decay is tiny. % The slightest imperfection in the pipe or the % smoothness of the inflow, or the slightest vibration of the laboratory, % may be enough to kick the system into instability. % Geometrically, we say that the {\em basin of attraction % of the laminar state is very narrow.} This makes % the laminar state often effectively unobservable in practice. % %% % % A simple ODE model explains how an extremely narrow % basin of attraction can come about in a set of equations % that seems far from extreme. % Let $R>0$ be a parameter, a % caricature of the Reynolds number, the nondimensional % centerline velocity of the laminar flow. Let % $u$ and $v$ be two dependent variables, caricatures % of the field of velocity perturbations of the laminar flow, % satisfying the equations % $$u' = -R^{-1}u + v - v\sqrt{u^2+v^2}, \quad % v' = -2R^{-1}v + u\sqrt{u^2+v^2}. \eqno (15.6)$$ % Rewriting (15.6) in matrix form reveals the structure more plainly: % $$\pmatrix{u\cr v}' = % \pmatrix{-R^{-1} & 1 \cr 0 & -2R^{-1}} \pmatrix{u\cr v} % + \sqrt{u^2+v^2} \pmatrix{0 & -1 \cr 1 & \hphantom{-}0} \pmatrix{u\cr v}. % \eqno (15.7)$$ % The term on the left is linear, and the term on the % right is quadratic (if $u$ and $v$ are doubled, it multiplies % by 4). % As always, it is the linear term % that governs behavior for sufficiently small $u$ and $v$. Since % the matrix is triangular, its eigenvalues are the diagonal entries, % $-R^{-1}$ and $-2R^{-1}$, and since these numbers are % negative, $(0,0)$ is a sink. % %% % % On the left below is an image of the linear part of the % problem for $R=10$ showing this sink, very much like % the second panel of Figure 15.2. Something physically important % is revealed in this image: a great deal of transient growth % before the eventual decay. (In the fluid mechanics problems % this effect is sometimes called lift-up,'' in which vorticity aligned % with a shear flow excites a growth in local velocity anomalies.) % On the right is an image of the nonlinear problem for the same % value $R=10$. What happens here is that the quadratic term % catches hold of the linear amplification and moves it onto % a new track entirely. Though the initial conditions in this % nonlinear experiment % start at a distance of only $0.02$ from the fixed point, % most of the trajectories spiral up to size $O(1)$ rather than % decaying to the center. For this model, this is the caricature % of transition to turbulence. % th = (pi/8)*(1:16) + .0001; u0 = cos(th); v0 = sin(th); clf, subplot(1,2,1), plot(0,0,'.k',MS,6), hold on, plot(.02*c,'k') N = chebop(0,18); R = 10; N.op = @(t,u,v) [diff(u) + u/R - v; diff(v) + 2*v/R]; for k = 1:16 N.lbc = [.02*u0(k); .02*v0(k)]; [u,v] = N\0; arrowplot(u,v,CO,ivp,MS,3) end axis(.07*[-1 1 -1 1]), set(gca,XT,.05*(-1:1),YT,.05*(-1:1)) text(-.054,-.044,'linearly stable',FS,8), axis square text(-.054,-.0548,'fixed point...',FS,8) pp = get(gca,'pos'); pp(1) = pp(1)+.015; set(gca,'pos',pp) subplot(1,2,2), plot(0,0,'.k',MS,6), hold on, plot(.02*c,'k') N = chebop(0,30); N.op = @(t,u,v) [diff(u) + u/R - v + v*sqrt(u^2+v^2) diff(v) + 2*v/R - u*sqrt(u^2+v^2)]; for k = 1:16 N.lbc = [.02*u0(k); .02*v0(k)]; [u,v] = N\0; big = (norm([u(end) v(end)])>.02); if big, arrowplot(u,v,CO,'r',MS,3) else arrowplot(u,v,CO,ivpnl,MS,3), end end axis(.08*[-1 1 -1 1]), set(gca,XT,.05*(-1:1),YT,.05*(-1:1)) axis square, hold off h1 = text(-.076,-.0405,'...but nonlinearly unstable to',FS,8); h2 = text(-.066,-.052,'small finite perturbations.',FS,8); h3 = text(-.066,-.0635,'Red trajectories grow to $O(1)$.',... FS,8,IN,LT); pp = get(gca,'pos'); pp(1) = pp(1)-.015; set(gca,'pos',pp) title(['Fig.~15.4.~~Stable node with narrow ' ... 'basin of attraction\kern -.6in'],FS,11,HA,RT) %% % \vskip 1.02em %% % % \noindent % Here we zoom out the second plot, showing where the red % trajectories are heading. % axis(1.5*[-1 1 -1 1]), axis square delete(h1), delete(h2), delete(h3) text(0.04,-.3,'...with the',FS,8) text(0.04,-.5,'nonlinear orbits',FS,8) text(0.04,-.7,'shown in full.',FS,8,IN,LT) set(gca,XT,-1:1,YT,-1:1) title(['Fig.~15.5.~~Same dynamics shown on a ' ... 'larger scale\kern -.5in'],FS,11) %% % \vskip 1.02em %% % % Let us draw a plot to visualize this behavior % in another way. For the same equations (15.7) with % $R=10$, Figure 15.6 shows $(u^2+v^2)^{1/2}$ as a function % of\/ $t$ for six trajectories emanating from % initial points $(u,v) = (0, v_0^{})$ with $v_0^{} % = 0,001, 0.0025, 0.005, 0.01, 0.02, 0.04$. The three % initial conditions of lowest amplitude lead to trajectories % that eventually decay to zero, but the other three, shown in % red, increase to $O(1)$. % clf, N = chebop(0,60); N.op = @(t,u,v) [diff(u) + u/R - v + v*sqrt(u^2+v^2) diff(v) + 2*v/R - u*sqrt(u^2+v^2)]; for v0 = [.001 .0025 .005 .01 .02 .04] N.lbc = [0; v0]; [u,v] = N\0; big = (norm([u(end) v(end)])>.2); if big, semilogy(sqrt(u.^2+v.^2),'r') else semilogy(sqrt(u.^2+v.^2),CO,ivpnl), end, hold on end title(['Fig.~15.6.~~ODE model (15.7) ' ... 'of transition to turbulence'],FS,11) xlabel('t',FS,10,IN,LT), ylabel('energy',FS,9), ylim([1e-4 10]) %% % \vskip 1.02em %% % % Unlike turbulence, % the model (15.7) is non-chaotic, and indeed it could not % possibly be chaotic since it is an autonomous first-order system with % just two variables. Similar models with three variables % instead of two, however, % combine the narrow basin of attraction of the % laminar state with chaotic long-time trajectories. % See Baggett, Driscoll, and Trefethen, % A mostly linear model of transition to turbulence,'' % {\em Physics of Fluids,} 1995. % %% % % \smallskip % {\sc History.} The general theory of stability of % ODE\kern .5pt s was developed by Aleksandr Mikhailovich % Lyapunov, a student of Chebyshev, % who was one of the outstanding Russian mathematicians of the era % before the 1917 revolution. Lyapunov's interest in questions of stability % began with fluid and solid mechanics and was set forth in % his great work of 1892, {\em The General Problem of % the Stability of Motion.} His academic descendants include % Smirnov, Sobolev, Kantorovich, Ladyzhenskaya, and other % major figures of twentieth century Russian mathematics. % %% % % \smallskip % {\sc Our favorite reference.} % The idea of a tipping point was made famous by % Malcolm Gladwell's bestseller % {\em The Tipping Point: How Little Things Can Make a % Big Difference,} which first appeared in 2000. % Gladwell emphasizes tipping points related % to the mathematics of epidemiology --- which, as he vividly shows, % applies to many more areas than just epidemiology. % %% % % \begin{displaymath} % \framebox[4.7in][c]{\parbox{4.5in}{\vspace{2pt}\sl % {\sc Summary of Chapter 15.} The starting point of analysis of an % autonomous system of ODE\kern .5pt s ${\bf y}' = {\bf f}({\bf y})$ % is determination and classification of its fixed points % ${\bf y}_*^{}$, defined by the % condition ${\bf f}({\bf y}_*^{}) = {\bf 0}$. % For ${\bf y}\approx {\bf y}_*^{}$, the difference $\delta {\bf y}(t) = % {\bf y}(t) - {\bf y}_*^{}$ evolves approximately according to the % linear equation $(\delta {\bf y})' = {\bf J}_*^{} (\delta {\bf y})$, % where ${\bf J}_*^{} = (\partial{\bf f}/\partial{\bf y})({\bf y}_*^{})$ is % the Jacobian matrix at ${\bf y}_*^{}$. A fixed point is stable if all % the eigenvalues of\/ ${\bf J}_*^{}$ lie in the left half of the % complex plane and unstable if at least one eigenvalue is in the % right half-plane. % \vspace{2pt}}} % \end{displaymath} % %% % \smallskip\small\parskip=1pt\parindent=0pt % {\em Exercise $15.1$. Unstable or stable?} % Classify the fixed point $y = y' = y''=0$ for the ODE\kern .5pt s % {\em (a)} $y''+5y' + 4y = 0$, % {\em (\kern .7pt b)} $y'''+6y''+11y' + 6y = 0$. % \par % {\em Exercise $15.2$. A system with four fixed points.} % Consider the system % $u' = {1\over 3}(u^{}-v^{})(1-u^{}-v^{})$, $v' = u^{}(2-v^{})$. % {\em (a)} Give a formula for % the Jacobian matrix {\bf J} as a function of $u$ and $v$. % {\em (\kern .7pt b)} Determine the fixed points and evaluate the % Jacobian at these points. % {\em (c)} Find the eigenvalues analytically, and classify the nature % of each fixed point. % \par % {\em Exercise $15.3$. Eulerian wobble.} % Determine the fixed points of (10.9) and analyze their stability. % \par % {\em Exercise $15.4$. Fixed points are unattainable.} % Let ${\bf y}'(t) = {\bf f}({\bf y})$ be an autonomous ODE satisfying % the continuity hypotheses of Theorems 11.2 or Theorem 11.3, % and suppose ${\bf y}_*^{}$ is a fixed point. Let ${\bf y}(t)$ % be the solution for $t>0$ with % initial value ${\bf y}(0)$. Prove that if % ${\bf y}(0) \ne {\bf y}_*^{}$, % then ${\bf y}(t) \ne {\bf y}_*^{}$ for all $t$. % \par % {\em \underline{Exercise $15.5$}. A cyclic system of three ODE\kern .5pt s} % (adapted from Guckenheimer and Holmes, % Structurally stable heteroclinic cycles, {\em Mathematical Proceedings of % the Cambridge Philosophical Society,} 1988). % Consider the system of ODE\kern .5pt s $u' = u(1-u^2-bv\kern .3pt ^2-c\kern .3pt w^2)$, % $v' = v(1-v^2-b\kern .3pt w^2-c\kern .3pt u^2)$, % $w' = w(1-w^2-b\kern .3pt u^2-c\kern .3pt v^2)$, % where $b$ and $c$ are parameters. % {\em (a)} Plot the solution $u(t)$ for $t\in [\kern .3pt 0,800\kern .3pt]$ % with $b=0.55$, $c=1.5$ and % initial conditions $u(0)=0.5$, $v(0)=w(0)=0.49$. % Make similar plots of $v(t)$ and $w(t)$, and also of the % whole trajectory in $u$-$v$-$w$ space, and comment on these shapes. % {\em (\kern .7pt b)} What are the four fixed points of this system that the plots just % drawn come close to? % For large $t$, the trajectory moves approximately in a cycle from % one fixed point, to another, to a third, and then back again. (It is % approximating a {\em heteroclinic cycle}.) % Which fixed point is the trajectory near at $t=800\kern .5pt$? % {\em (c)} Find the eigenvalues of the appropriate matrix at one of % these fixed points. What does this tell us about the structure of % this fixed point? How does that fit with the observed % trajectory? % \par % {\em Exercise $15.6$. Oregonator.} % Exercise 10.2 presented the nonlinear equations known as the Oregonator. % Here, continue with the parameters as presented in that exercise. % {\em (a)} The origin $u=v=w=0$ is a fixed point. Determine the relevant % eigenvalues and classify the linearized behaviour of the system there. % {\em (\kern .7pt b)} There is another fixed point with $u=w>0$. Find it, determine % its eigenvalues, and classify it. %