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

dx/dt = y

dy/dt = -x

which has a center at (0,0):

>    with(DEtools):

>    leqs:={diff(x(t),t) = y(t),diff(y(t),t) = -x(t)};

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);

[Maple Plot]

For some ``perturbations'' of the linear system obtained by adding non-linear

terms, we can get spiral sinks at (0,0):

dx/dt = y-(x^2+y^2)*x

dy/dt = -x-(x^2+y^2)*y

>    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)};

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);

[Maple Plot]

Others will have spiral sources:

dx/dt = y+.3*(x^2+y^2)*x

dy/dt = -x+.3*(x^2+y^2)*y

>    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)};

nleqs2 := {diff(y(t),t) = -x(t)+.3e-1*(x(t)^2+y(t)^2)*y(t), diff(x(t),t) = y(t)+.3e-1*(x(t)^2+y(t)^2)*x(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);

[Maple Plot]

>    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)};

nleqs3 := {diff(y(t),t) = -x(t)+.3e-1*(x(t)^2+y(t)^2)*y(t), diff(x(t),t) = y(t)-.3e-1*(x(t)^2+y(t)^2)*x(t)}

Others will have ``skewed'' centers:

dx/dt = y-.3*(x^2+y^2)*x

dy/dt = -x+.3*(x^2+y^2)*y

>    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);

[Maple Plot]

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