Saddle

> A := matrix( [ [ 3, 0 ], [ 1, -2 ] ] );

>

> charpoly( A, lambda );

> eigenvals( A );

Since there is one positive and one negative eigenvalue, the origin is a saddle for this system.

>

> eigenvects( A );

>

The straight-line solutions are:

> Y1 := exp(-2*t) * matrix( 2, 1, [ 0, 1 ] );

> Y2 := exp(3*t) * matrix( 2, 1, [ 5, 1 ] );

>

> ode1 := diff( x(t), t ) = 3*x(t);

> ode2 := diff( y(t), t ) = x(t) - 2*y(t);

> MODEL := { ode1, ode2 }:

>

> VARS := { x(t), y(t) }:

> DOMAIN := t=0..3:

> RANGE := x=-10..10, y=-10..10:

> dirPLOT := DEplot( MODEL, VARS, DOMAIN, RANGE, arrows=MEDIUM ):

> dirPLOT;

>

> IC := [ [ x(0)=0.1, y(0)=1 ], [ x(0)=0.1, y(0)=4 ], [ x(0)=0.1, y(0)=7 ],
[ x(0)=-0.1, y(0)=1 ], [ x(0)=-0.1, y(0)=4 ], [ x(0)=-0.1, y(0)=7 ],
[ x(0)=0.1, y(0)=-1 ], [ x(0)=0.1, y(0)=-4 ], [ x(0)=0.1, y(0)=-7 ],
[ x(0)=-0.1, y(0)=-1 ], [ x(0)=-0.1, y(0)=-4 ], [ x(0)=-0.1, y(0)=-7 ] ]:

> solPLOT := DEplot( MODEL, VARS, DOMAIN, RANGE, IC, arrows=NONE, linecolor=GREEN ):

> #solPLOT;

>

> ICline := [ [ x(0)=0, y(0)=10 ], [ x(0)=0, y(0)=-10 ],
[ x(0)=0.5, y(0)=0.1 ], [ x(0)=-0.5, y(0)=-0.1 ] ]:

> linePLOT := DEplot( MODEL, VARS, DOMAIN, RANGE, ICline, arrows=NONE, linecolor=BLUE ):

> #linePLOT;

>

> display( [ dirPLOT, solPLOT, linePLOT ], title=`Example of Saddle` );

Notice how these solutions move (essentially) straight down the y-axis before moving in the direction of the eigenvector corresponding to the positive eigenvalue.

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