Re: Long-term behaviour of ODEs

Richard Hayden <r.hayden@xxxxxxxxx> writes:

Hi,

If I have an autonomous locally Lipschitz system of n coupled (non-
linear) ODEs, for which all (unique) solutions are bounded to remain
in some compact subset of \real^n (and thus all solutions exist for
all t >= t_0), am I guaranteed that as t -> \infty, solutions will
converge to some equilibrium solution? If so, where can I find this
result? I have a feeling I read such a result (or something similar) a
while back but can't remember where.

If not, is there usual way to prove that all solutions to an ODE
system do exhibit this behaviour? That they always do is clear
numerically but I've no idea how to go about proving it (if the above
result does not hold). An example might be:

da/dt = -a / (a + b) + c
db/dt = -b / (a + b) + a / (a + b)
dc/dt = -c + b / (a + b)

with initial conditions a_0, b_0, c_0 > 0.

The class of ODEs in question model a fixed population, i.e. satisfy
x'_1(t) + ... + x'_n(t) = 0 for all times t >= t_0 in case this helps.
Similar ODEs I presume arise in biological/chemical context, but I
can't find any useful results.

Since you have the conservation law a + b + c = constant, this reduces your
3 x 3 system to a 2 x 2 system. The Poincare-Bendixson theorem then applies:
the only possible limiting behaviours of a bounded trajectory involve
approaching a fixed point, a limit cycle or a homoclinic orbit. And for
larger n you also have the possibility of strange attractors and chaotic
solutions.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Long-term behaviour of ODEs
    ... If I have an autonomous locally Lipschitz system of n coupled (non- ... linear) ODEs, for which all solutions are bounded to remain ... The class of ODEs in question model a fixed population, ...
    (sci.math)
  • Re: Long-term behaviour of ODEs
    ... Similar ODEs I presume arise in biological/chemical context, ... the only possible limiting behaviours of a bounded trajectory involve ... larger n you also have the possibility of strange attractors and chaotic ...
    (sci.math)