Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

Differential equations in the plane

Because a solution path of a differential equations can not cross itself, the dynamical in the place can not become too complicated. It is possible to have equilibrium points or limit cycles. Every orbit which does not start at an equilibrium point or on limit cicle either converges to one of these or it escapes to infinity. A famous problem of David Hilbert from 1900 asks how many limit cycles a polynomial differential equation can have.
d/dt x=ann xn yn+ ... + a11 xy + a10 x + a01 y + a0
d/dt y=bnn xn yn+ ... + a10 xy + b10 x + b01 y + b0
in the plane can have.

Are there only finitely many limit cycles? Is there a bound on their number which only depends on n?
A special class of differential equations are Lienard systems

d2/dt2 x + F'(x) d/dt x + G'(x) = 0

which is with y=d/dt x equivalent to

d/dt x = y 
d/dt y = -G'(x)-F'(x) y

or with y=d/dx +F, G'(x)=g equivalent to

d/dt x = y-F(x)
d/dt y = -g(x)

Example: van der Pool equation

d/dt x = y 
d/dt y = -x -(x2-1) y

Example: Duffing oscillator

d/dt x = y
d/dt y = -x -x3 

Last Wednesday, news broke that a student Elin Oxenhielm made progress on the Hilbert problem, claiming that a Lienard system with g(x)=x and F(x) a polynomial of degree 2k+1 has at most k limit cycles. The paper containing a sketch of a proof is here. Critics doubt that the proof will hold.

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