MATH 6410 - 1 MAPLE PLOTS OF SYSTEM TRAJECTORIES

> with(linalg):with(DEtools):with(plots):# Warning, the protected names norm and trace have been redefined and unprotected

Warning, the previous binding of the name adjoint has been removed and it now has an assigned value

Warning, the name changecoords has been redefined

> restart:

> with(plots):with(linalg):with(DEtools):Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the previous binding of the name adjoint has been removed and it now has an assigned value

> F:= (xx,yy)-> xx-yy-xx^2+xx*yy;G:=(xx,yy)->-yy-xx^2;

F := xx, yy( ) ® xx - yy - xx 2 + yy xx

G := xx, yy( ) ® -yy - xx 2

>

This time, enter a nonlinear system of equations and a list of initial data.

> deqtn:={diff(x(t),t)=F(x(t),y(t)),

> diff(y(t),t)=G(x(t),y(t))};

deqtn := d dt

x t( ) = x t( ) - y t( ) - x t( )2 + y t( ) x t( ), d dt

y t( ) = -y t( ) - x t( )2ìíî

üýþ

> ICs:=[[x(0)=-1,y(0)=3],[x(0)=0,y(0)=3],[x(0)=.9,y(0)=3],[x(0)=.6,y(0)=3],[x(0)=.8,y(0)=3], [x(0)=1,y(0)=3],[x(0)=2,y(0)=3],[x(0)=3,y(0)=3],[x(0)=3,y(0)=0],[x(0)=3,y(0)=-1.5], [x(0)=0.05,y(0)=0], [x(0)=-.05,y(0)=0],[x(0)=0,y(0)=-.05], [x(0)=0,y(0)=.05], [x(0)=1.5,y(0)=-3], [x(0)=1,y(0)=-3], [x(0)=0,y(0)=-3],[x(0)=-1,y(0)=-3],[x(0)=-2,y(0)=-3],[x(0)=-1.1,y(0)=-1], [x(0)=-1,y(0)=-.9], [x(0)=-.9,y(0)=-1],[x(0)=-1,y(0)=-1.1],[x(0)=-1,y(0)=-.95], [x(0)=-1,y(0)=-1.05]]:

>

> DEplot(deqtn,[x(t),y(t)],t=0..15,ICs,x=-3..3,y=-3..3,arrows=small,

> stepsize=.1,color=navy,linecolor=red,title=`Nonlinear System`);

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Nonlinear System

>

To find the critical points, enter the right side. then find the vanishing points of the vector field.

> solve({F(u,v)=0,G(u,v)=0},{u,v});

u = 0, v = 0{ }, v = -1, u = 1{ }, v = -1, u = -1{ }

> x0:=-1;y0:=-1;x0 := -1

y0 := -1

Plot of solutions of the nonlinear equation near the critical point (-1,-1)

> IC2:=[seq([x(0)=-1.+0.01*k,y(0)=-1.+0.01*k],k=0..10)]:

> DEplot(deqtn,[x(t),y(t)],t=0..15,IC2,x=-1.1..-0.9,y=-1.1..-0.9,arrows=small,

> stepsize=.1,color=navy,linecolor=red,scaling=constrained,title=`Nonlinear System near (-1,-1)`);

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Nonlinear System near (-1,-1)

Look near the equilibrium point (-1,-1). First, near that point, by the 2-dimensional Taylor's Theorem, the system is approximated by the linear system whose matrix is [F(-1+u,-1+v),G(-1+u,-1+v)]=[0 +diff(F(x,y),x)*u +diff(F(x,y),y)*v, 0+ diff(G(x,y),x)*u+diff(G(x,y),y)*v].

> A:=subs({x=x0,y=y0},matrix([ [diff(F(x,y),x),diff(F(x,y),y)],[diff(G(x,y),x),diff(G(x,y),y)]]));

A := 2 -2

2 -1

éêêë

ùúúû

> eigenvects(A);

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I 7 , 1éêë

ùúû

ìíî

üýþ

éêêë

ùúúû

, 12

- 12

I 7 , 1, 34

- 14

I 7 , 1éêë

ùúû

ìíî

üýþ

éêêë

ùúúû

Define a linear system using the matrix.

> ODE1:={diff(x(t),t)=(A[1,1]*x(t)+A[1,2]*y(t)),diff(y(t),t)=A[2,1]*x(t)

> +A[2,2]*y(t)};

ODE1 := d dt

y t( ) = 2 x t( ) - y t( ), d dt

x t( ) = 2 x t( ) - 2 y t( )ìíî

üýþ

> ICL2:=[seq([x(0)=0.1*k,y(0)=0.1*k],k=0..10)]:

> phaseportrait(ODE1,[x(t),y(t)],0.0..11.0,ICL2, x=-1.1..1.1,

> y=-1.1..1.1, scaling=constrained,stepsize=.05,title=`Linearization at

> (-1,-1)`,linecolor=black);

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Linearization at(-1,-1)

The real parts are positive, so the local picture is that of a spiral source.

Plot of solutions of the nonlinear equation near the critical point (-1,-1)

> x0:=1;y0:=-1;

x0 := 1

y0 := -1

> IC2:=[seq([x(0)=1.0+0.1*cos(k*Pi/6),y(0)=-1.0+0.1*sin(k*Pi/6)],k=0..11)]:

> DEplot(deqtn,[x(t),y(t)],t=0..3,IC2,x=0.9..1.1,y=-1.1..-0.9,arrows=small,

> stepsize=.03,color=navy,linecolor=red,scaling=constrained,title=`Nonli

> near System near (1,-1)`);

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Nonlinear System near (1,-1)

Find the linearization near (1,-1) as before and plot the solutions of the linearized equations.

> A:=subs({x=x0,y=y0},matrix([ [diff(F(x,y),x),diff(F(x,y),y)],

> [diff(G(x,y),x),diff(G(x,y),y)]]));

A := -2 0

-2 -1

éêêë

ùúúû

> eigenvects(A);-1, 1, 0, 1[ ]{ }[ ], -2, 1, 1, 2[ ]{ }[ ]

> ODE2:={diff(x(t),t)=(A[1,1]*x(t)+A[1,2]*y(t)),diff(y(t),t)=A[2,1]*x(t)+A[2,2]*y(t)};

ODE2 := d dt

y t( ) = -2 x t( ) - y t( ), d dt

x t( ) = -2 x t( )ìíî

üýþ

> ICl2:=[seq([x(0)=cos(k*Pi/6),y(0)=sin(k*Pi/6)],k=0..11)]:

> phaseportrait(ODE2,[x(t),y(t)],0.0..4.0,ICl2, x=-1.1..1.1,

> y=-1.1..1.1, scaling=constrained,stepsize=.05,title=`Linearization at

> (1,-1)`,linecolor=black);

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Linearization at(1,-1)

3. STABLE LIMIT CYCLES

Rayleigh's Clarinet Equation

Finally, we illustrate by way of examples, that a planar system may have a stable limit cycle, a closed periodic trajectory that attracts nearby trajectories. Consider the equation proposed by Lord Rayleigh for the vibration of a clarinet reed. mx'' = -kx + ax' -b(x')^3 which is stiffer than linear drag but soggier for small displacements. As in the text, we convert to a first order system with a=b=m=k=1.

> ODE3:={diff(x(t),t)=y(t),diff(y(t),t)=-x(t)+y(t)-y(t)^3};

ODE3 := d dt

x t( ) = y t( ), d dt

y t( ) = -x t( ) + y t( ) - y t( )3ìíî

üýþ

> IC3:=[[x(0)=-2,y(0)=3],[x(0)=0,y(0)=3],[x(0)=2,y(0)=3],

> [x(0)=-2,y(0)=-3],[x(0)=0,y(0)=-3],[x(0)=2,y(0)=-3],

> [x(0)=-.2,y(0)=.3],[x(0)=0,y(0)=.3],[x(0)=.2,y(0)=.3],

> [x(0)=-.2,y(0)=-.3],[x(0)=0,y(0)=-.3],[x(0)=.2,y(0)=-.3]]:

> DEplot(ODE3,[x(t),y(t)],t=0..10,IC3,x=-3..3,y=-3..3,arrows=small,

> stepsize=.1,color=black,linecolor=t,title=`Clarinet Equation Phase

> Portrait`);

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Clarinet Equation PhasePortrait

The Brusselator

A model for a hypothetical chemical interaction of two species, which was proposed by researchers from Brussels, is given by the following equation

> a:= 1.0; b:= 3.0;a := 1.0

b := 3.0

> ODE4:={diff(x(t),t)=a-(b+1)*x(t)+x(t)^2*y(t),diff(y(t),t)=b*x(t)-x(t)^

> 2*y(t)};

ODE4 := d dt

y t( ) = 3.0 x t( ) - x t( )2 y t( ), d dt

x t( ) = 1.0 - 4.0 x t( ) + x t( )2 y t( )ìíî

üýþ

> IC4:=[[x(0)=0.2,y(0)=0.2],[x(0)=2.2,y(0)=0.2],[x(0)=4.2,y(0)=0.2],[x(0

> )=0.2,y(0)=3.2],[x(0)=0.2,y(0)=5.2],[x(0)=2.2,y(0)=6.2],[x(0)=2,y(0)=3

> ],[x(0)=1,y(0)=4],[x(0)=1,y(0)=2.6]]:

> DEplot(ODE4,[x(t),y(t)],t=0..7,IC4,x=0..4,y=0..6,arrows=small,

> stepsize=.0475,color=black,linecolor=[red,coral,yellow,green,cyan,blue

> ,navy,maroon,magenta],title=`Brusselator Equation Phase Portrait`);

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Brusselator Equation Phase Portrait

Predator Prey models

The simplest model is the Lotka-Volterra for the (nondimensionalized) prey x(t) and the predator y(t)

> a:= 0.7;a := 0.7

> ODE5:={diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),t)=a*y(t)*(x(t)-1)};

ODE5 := d dt

y t( ) = 0.7 y t( ) x t( ) - 1( ), d dt

x t( ) = x t( ) 1 - y t( )( )ìíî

üýþ

> IC4:=[[x(0)=0.2,y(0)=0.2],[x(0)=2.2,y(0)=0.2],[x(0)=4.2,y(0)=0.2],[x(0

> )=0.2,y(0)=3.2],[x(0)=0.2,y(0)=5.2],[x(0)=2.2,y(0)=6.2],[x(0)=2,y(0)=3

> ],[x(0)=1,y(0)=4],[x(0)=1,y(0)=2.6]]:

> DEplot(ODE5,[x(t),y(t)],t=0..15,IC4,x=0..6,y=0..6,arrows=small,

> stepsize=.0475,color=black,linecolor=[red,coral,yellow,green,cyan,blue

> ,navy,maroon,magenta],title=`Brusselator Equation Phase Portrait`);

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Brusselator Equation Phase Portrait

More realistically, the prey growth is not unbounded but logistic, predator carrying capacity is proportional to prey density

> a:= 0.75; b:= 0.2;c:=0.05;a := 0.75

b := 0.2

c := 0.05

> b-(a-sqrt((1.0-a-c)^2+4.0*c))*(1.0+a+c-sqrt((1.0-a-c)^2+4.0*c));-0.1407602310

> ODE6:={diff(x(t),t)=x(t)*(1-x(t)-a*y(t)/(c+x(t))),diff(y(t),t)=b*y(t)*(1-y(t)/x(t))};

ODE6 := d dt

x t( ) = x t( ) 1 - x t( ) - 0.75 y t( )0.05 + x t( )

æçè

ö÷ø

, d dt

y t( ) = 0.2 y t( ) 1 - y t( )x t( )

æçè

ö÷ø

ìíî

üýþ

> IC5:=[[x(0)=0.2,y(0)=0.2],[x(0)=0.4,y(0)=0.2],[x(0)=0.5,y(0)=0.2],[x(0

> )=0.6,y(0)=0.2],[x(0)=0.2,y(0)=0.4],[x(0)=0.2,y(0)=0.5],[x(0)=0.4,y(0)=1.0

> ],[x(0)=1,y(0)=0.5],[x(0)=0.4,y(0)=0.4],[x(0)=0.39,y(0)=0.39]]:

> DEplot(ODE6,[x(t),y(t)],t=0..60.0,IC5,x=0..0.9,y=0..0.5,arrows=small,

> stepsize=.0475,color=black,linecolor=[red,coral,yellow,green,cyan,blue

> ,navy,maroon,magenta,black],title=`Brusselator Equation Phase Portrait`);

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Brusselator Equation Phase Portrait