31.5. EXERCISES 609

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Definition 31.4.3 A saddle point x0 for x′ = f (x) is an equilibrium point (f (x0) = 0)which is not stable, but which has the property that in in every set of the form

{x : r > |x−x0|> 0}

there are points for which the solution having these as initial conditions converges to x0as t → ∞. These saddle points occur for example if you have a negative and a positiveeigenvalue for Df (x0).

The following very interesting theorem can be obtained from the above.

Theorem 31.4.4 If you have a periodic orbit of a solution to an autonomous two dimen-sional differential equation, x′ = f (x) , then it must go around some equilibrium point.If there is only one equilibrium point inside the periodic orbit, then it cannot be a saddlepoint.

31.5 Exercises1. The Van der Pol equation describes nonlinear oscillations. It is

x′′+(x2−1

)x′+ x = 0 (31.1)

Show that it has a non constant periodic solution. Do as follows. First write as a firstorder system

x′ = yy′ =−

((x2−1

)y+ x

)Have MATLAB or some other computer algebra system give a graph of solutions forthe above system corresponding to various initial conditions in a way to show theperiodic solution. Try the following:

f=@(t,x)[x(2);-((x(1)ˆ2-1)*x(2)+x(1))];d=pi/4; r=.3;hold onfor n=1:9[t,x]=ode45(f,[0:.05:10],[n*r*cos(n*d);n*r*sin(n*d)]);plot(x(:,1),x(:,2),’LineWidth’,1.5)end