MATH 304 -- Ordinary Differential Equations
Comparing Nonlinear and Linearized Systems
November 10, 2004
The following examples illustrate that the type of an equlibrium
for a linear system can change when we add non-linear terms,
in ``unstable'' cases such as a linear system with a repeated real
root, or a linear system with a center. We consider the case
of a center first. We start by plotting solutions of
which has a center at (0,0):
> | with(DEtools): |
> | leqs:={diff(x(t),t) = y(t),diff(y(t),t) = -x(t)}; |
> | phaseportrait(leqs,[x(t),y(t)],t=0..8,[[x(0)=0,y(0)=1],[x(0)=2,y(0)=0]],stepsize=.1,linecolor=black); |
For some ``perturbations'' of the linear system obtained by adding non-linear
terms, we can get spiral sinks at (0,0):
> | nleqs1:={diff(x(t),t) = y(t)-(x(t)^2+y(t)^2)*x(t),diff(y(t),t) = -x(t)-(x(t)^2+y(t)^2)*y(t)}; |
> | phaseportrait(nleqs1,[x(t),y(t)],t=0..8,[[x(0)=0,y(0)=1],[x(0)=2,y(0)=0]],stepsize=.1,linecolor=black); |
Others will have spiral sources:
> | nleqs2:={diff(x(t),t) = y(t)+(0.03)*(x(t)^2+y(t)^2)*x(t),diff(y(t),t) = -x(t)+(0.03)*(x(t)^2+y(t)^2)*y(t)}; |
> | phaseportrait(nleqs2,[x(t),y(t)],t=0..4,[[x(0)=0,y(0)=1],[x(0)=2,y(0)=0]],stepsize=.1,linecolor=black); |
> | nleqs3:={diff(x(t),t) = y(t)-(0.03)*(x(t)^2+y(t)^2)*x(t),diff(y(t),t) = -x(t)+(0.03)*(x(t)^2+y(t)^2)*y(t)}; |
Others will have ``skewed'' centers:
> | phaseportrait(nleqs3,[x(t),y(t)],t=0..20,[[x(0)=0,y(0)=1],[x(0)=2,y(0)=0]],stepsize=.1,linecolor=black); |
> |