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View/Sys/Gal: Ode " Comments" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (0)

This gallery contains various classic systems that were used to test OdeFactory. It is an older gallery that was updated 4/4/17.

The Duffing and Lotka-Volterra systems date back to 1918 and 1910 respectively. They both grew out of ode models of physical systems and they both have an extensive literature.

The other systems are predominately nonautonomous and from contemporary ode textbooks as opposed to dynamical systems textbooks. The older ode textbook emphasis was generally on non-computer based math analysis.

The "solutions" discussed in this gallery are computer based.

View/Sys/Gal: IMap "2D linear system" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (x-y,x+y)

This system of odes is defined by the equations:

dx/dt = x-y,

dy/dt = x+y

Image 1: Ode view with colored V fld nullclines and trajectories.

Image 2: IMap view with orbits.

View/Sys/Gal: Ode "Duffing oscillator" in "Classic."
Range: (vMax,vMin) = (2.000,-2.000), (hMin,hMax) = (-2.000,18.850)
VFld: ( y, -.2*y-x*abs(x)+1.5*cos(2*t)+.5)

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = -.2*y-x*abs(x)+1.5*cos(2*t)+.5

This is one of the various forms of the Duffing oscillator system.

Image 1: The solution curve for ICs (t,x,y) = 0,0,0).

See: "Numerical Solution of Differential Equations" by Isaac Fried, 1979, p. 221, ex 10.

View/Sys/Gal: EMap "Duffing oscillator, 2D nonautonomous" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y,-x*(x^2-k)-b*y+a*cos(c*t)), k = 1; b = .14; a = .30; c = .6

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = -x*(x^2-k)-b*y+a*cos(c*t)

in the (t,x,y) coordinate system.

The 1st order 2D system corresponds to the 2nd order ode

y'' = -y*(y^2-k)-b*y'+a*cos(c*x)

in the (x,y,y') coordinate system.

Parameters are:

k = 1; b = .14; a = .3; c = .6

This is a 2D nonautonomous system.

Image 1: Ode 3D/(t,x,y) view.

Image 2: Ode 3D/(x,y) PMap view with k = 1; b = .2; a = .3; c = 1. The ICs are (x,y) = (0,0) and t-final is about 3000. The orbit is chaotic.

Image 3: IMap view for k = 1.7, b = .13, a = .22, c = .59. There are three attractors: one in quadrants 1 (blue), one in quadrant 3 (green) and one 2-part attractor (red) in quadrants 2 and 4.

Image 4: EMap view. Prisoner set is black.

There is an extensive literature regarding the Duffing equation. You might want to start here.

View/Sys/Gal: Ode "Duffing oscillator, 3D autonomous" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y,-x*(x^2-k)-b*y+a*cos(c*z),1), k = 1; b = .14; a = .30; c = .6

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = -x*(x^2-k)-b*y+a*cos(c*z),

dz/dt = 1.

Parameters are:

k = 1; b = .14; a = .30; c = .6

You can always convert a nonautonomous system to an equivalent autonomous system in a phase space of one additional dimension.

Starting with the previous 2D nonautonomous system, t was replaced by z and dz/dt = 1 was added to get this equivalent 3D autonomous system.

Image 1: Ode 3D/(t,x,y) view.

View/Sys/Gal: Ode "Lotka-Volterra equations" in "Classic."
Range: (vMax,vMin) = (14.000,-6.000), (hMin,hMax) = (-6.000,14.000)
VFld: (-a*x+b*x*y, c*y-d*x*y), a = 1; b = 1; c = 1; d = 1

This system of odes is defined by the equations:

dx/dt = -a*x+b*x*y,

dy/dt = c*y-d*x*y

Parameters are:

a = 1; b = 1;

c = 1; d = 1

Reference.

Mattuck lecture 33.

The parameters are all positive and the physical region is the 1st quadrant. It takes 50 minutes but Mattuck proves that the fixed point is a stable center.

In WA, enter:

plot e^(ln(xy)-x-y), {x,0,5}, {y,0,5}

to see the contour curves discussed by Prof. Mattuck.

In OdeFactory click the Flow button to see the flow in the phase space.

Image 1: Ode trajectories.

ICs

period

(2,2)

6.94

(3,3)

8.283

(4,4)

9.94

(5,5)

11.78

View/Sys/Gal: EMap "Lotka-Volterra equations as an EMapCT3" in "Classic."
Range: (vMax,vMin) = (8.062,-1.938), (hMin,hMax) = (-4.962,5.038)
VFld: (-a*x+b*x*y, c*y-d*x*y), a = 1.10; b = 1.50; c = .50; d = .00;

This system of odes is defined by the equations:

dx/dt = -a*x+b*x*y,

dy/dt = c*y-d*x*y

Parameters are:

a = 1.10; b = 1.50;

c = .50; d = .00;

Some "art" image examples follow.

The images are interesting EMap views of the Lotka-Volterra system created by varying parameters, axis limits and color tables.

Image 1: EMap view.

View/Sys/Gal: EMap "Lotka-Volterra equations as an EMapCT3 ver 2" in "Classic."
Range: (vMax,vMin) = (0.016,-0.017), (hMin,hMax) = (-0.015,0.014)
VFld: (-a*x+b*x*y, c*y-d*x*y), a = 1.100; b = 1.000; c = 1.100; d = .000;

This iteration is defined by:

x <- -a*x+b*x*y,

y <- c*y-d*x*y.

Parameters are:

a = 1.100; b = 1.000;

c = 1.100; d = .000;

EMap CT: 3

Image 1: Another EMap image.

View/Sys/Gal: EMap "Lotka-Volterra equations as an EMapCT3 ver 3" in "Classic."
Range: (vMax,vMin) = (3.572,-1.428), (hMin,hMax) = (-2.500,2.500)
VFld: (-a*x+b*x*y, c*y-d*x*y), a = 1.100; b = 1.000; c = 1.100; d = 1.000;

This iteration is defined by:

x <- -a*x+b*x*y,

y <- c*y-d*x*y.

Parameters are:

a = 1.100; b = 1.000;

c = 1.100; d = 1.000;

EMap CT: 3

Image 1: Yet another EMap image>

View/Sys/Gal: EMap "Lotka-Volterra equations, time scaled as EMap" in "Classic."
Range: (vMax,vMin) = (0.641,-0.641), (hMin,hMax) = (0.478,2.215)
VFld: ((1+s*.01)*(-a*x+b*x*y), (1+s*.01)*(c*y-d*x*y)), a = .90; b = 1.30; c = 1.20; d = .89; s = 4.30;

This iteration is defined by:

x <- (1+s*.01)*(-a*x+b*x*y),

y <- (1+s*.01)*(c*y-d*x*y).

Parameters are:

a = .90; b = 1.30;

c = 1.20; d = .89;

s = 4.30;

EMap CT: 0

Image 1: Parameter s scales time.

Try CT 3.

Try zooming out.

View/Sys/Gal: EMap "Lotka-Volterra equations, time scaled as EMapCT3" in "Classic."
Range: (vMax,vMin) = (0.577,0.000), (hMin,hMax) = (1.460,2.271)
VFld: ((1+s*.01)*(-a*x+b*x*y), (1+s*.01)*(c*y-d*x*y)), a = .910; b = 1.000; c = 1.300; d = .890; s = .040;

This iteration is defined by:

x <- (1+s*.01)*(-a*x+b*x*y),

y <- (1+s*.01)*(c*y-d*x*y).

Parameters are:

a = .910; b = 1.000;

c = 1.300; d = .890;

s = .040;

EMap CT: 3

Image 1: A final EMap image.

View/Sys/Gal: Ode "p. 60, Example 2, from KKO" in "Classic."
Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,2.500)
VFld: (-2*t*x+t)

This ode is defined by the equation:

dx/dt = -2*t*x+t

in the (t,x) coordinate system.

The 1st order 1D system corresponds to the 1st order ode

y' = -2*x*y+x

in the (x,y) coordinate system.

This is a 1st order linear, normal ode. It has the standard form:

dx/dt+P(t)*x=Q(t).

The general solution is:

x(t)=1/2+c*e^(-t^2).

Image 1: Three solution curves.

All solutions go to x = 1/2 as t gets large. All solutions are also symmetric about the t axis.

c = 0 gives the straight line through x = 1/2

c = 1 gives the upper curve through x = 3/2

c = -1 gives the lower curve through x = -1/2

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "p. 60, Example, from KKO, using fns P and Q" in "Classic."
Range: (vMax,vMin) = (2.560,-2.560), (hMin,hMax) = (-2.560,2.560)
VFld: (-P*x+Q), P = 2*t; Q = t

This ode is defined by the equation:

dx/dt = -P*x+Q

in the (t,x) coordinate system.

The 1st order 1D system corresponds to the 1st order ode

y' = -P*y+Q

in the (x,y) coordinate system.

Functions are:

P = 2*t; Q = t

NOTE: User defined functions must always be defined in (t,x,y,z,w) coordinates.

Image 1: Some solution curves.

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "p. 61, eqn (2-20), from KKO" in "Classic."
Range: (vMax,vMin) = (0.062,-0.062), (hMin,hMax) = (-0.062,0.062)
VFld: (-x/t+1)

This ode is defined by the equation:

dx/dt = -x/t+1

in the (t,x) coordinate system.

The 1st order 1D system corresponds to the 1st order ode

y' = -y/x+1

in the (x,y) coordinate system.

The general soln is:

x(t) = c/t+t/2 for t > 0 and

x(t) = -c/t+t/2 for t < 0.

c = 0 gives x(t) = t/2

Image 1: Some solution curves.

Zoom in to see what is happening near (0,0).

Image 2: Image 1 zoomed in.

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "p. 62, eqn (2-23), from KKO" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (-x+(t*x)^2)

Bernoulli's equation has the form:

y'+p(x)*y=q(x)*y^n

It is nonlinear for all n other than 0 and 1.

An example is:

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

in (t,x) coordinates, which can also be written in the y(x) form as:

y' = -y+(x*y)^2,

in (x,y) coordinates

In (t,x) coordinates, the solutions are:

x(t) = 1/(2+2*t+t^2+c*e^t) and

x(t) = 0.

Image 1: Solution curves.

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "p. 64, ex 33, Riccati eqn, from KKO" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-4.000,6.000)
VFld: ((sin(t)*x)^2-x/(sin(t)*cos(t))-cos(t)^2)

This ode is defined by the equation:

dx/dt = (sin(t)*x)^2-x/(sin(t)*cos(t))-cos(t)^2

in the (t,x) coordinate system.

The 1st order 1D system corresponds to the 1st order ode

y' = (sin(x)*y)^2-y/(sin(x)*cos(x))-cos(x)^2

in the (x,y) coordinate system.

It is a Riccati equation of the general form:

y'+a2(x)*y^2+a1(x)*y+a0(x)=0

A particular soln is:

x(t) = cos(t)/sin(t)

Image 1: Representative solution curves.

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "p. 84, eqn (2-52), from KKO" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y,-(tan(t)-2/tan(t))*y)

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = -(tan(t)-2/tan(t))*y

in the (t,x,y) coordinate system.

The 1st order 2D system corresponds to the 2nd order ode

y'' = -(tan(x)-2/tan(x))*y'

in the (x,y,y') coordinate system.

Since tan(0) = 0, the 2D system is not defined at t = 0.

Image 1: Solution curves in the (t,x) plain with ICs (t,x,y) = (1,1,2) and (-1,1,2).

Image 2: Solution curves in the (t,y) plain with ICs (t,x,y) = (1,1,2) and (-1,1,2).

This is an example from: "Elementary Differential Equations" by Kreider, Kuller and Ostberg

View/Sys/Gal: Ode "two species competing for the same prey" in "Classic."
Range: (vMax,vMin) = (1.000,-1.000), (hMin,hMax) = (-1.000,1.000)
VFld: (x*(1-x)-x*y,2*y*(1-y/2)-3*x*y)

This system of odes is defined by the equations:

dx/dt = x*(1-x)-x*y,

dy/dt = 2*y*(1-y/2)-3*x*y

It is a model describing two species competing for the same prey. For more details see;

http://www.sosmath.com/diffeq/system/qualitative/qualitative.html

Applying WA to

0 = x*(1-x)-x*y, 0 = 2*y*(1-y/2)-3*x*y

gives fixed points at

(0,2), (.5,.5), (1,0) and (0,0).

Image 1: Trajectories in the R2+ view.

View/Sys/Gal: EMap "two species competing for the same prey, EMapCT3 view" in "Classic."
Range: (vMax,vMin) = (0.407,-1.212), (hMin,hMax) = (-0.237,1.222)
VFld: (x*(c-x)-x*y,a*y*(d-y/2)-b*x*y), a = 1.70; b = 2.10; c = 1.10; d = 1.00;

This iteration is defined by:

x <- x*(c-x)-x*y,

y <- a*y*(d-y/2)-b*x*y.

Parameters are:

a = 1.70; b = 2.10;

c = 1.10; d = 1.00;

EMap CT: 3

Image 1: An EMap art image.

View/Sys/Gal: Ode "y'' = (2*y + ln(x))/x^2" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y, (2*x + ln(t))/t^2)

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = (2*x + ln(t))/t^2

in the (t,x,y) coordinate system.

The 1st order 2D system corresponds to the 2nd order ode

y'' = (2*y + ln(x))/x^2

in the (x,y,y') coordinate system.

The 2D system is not defined at t = 0 and the 2nd order ode is not defined at x = 0.

The solution curves in the (t,x) plain can cross because the system is nonautonomous.

Image 1: Some x(t) solution curves.

View/Sys/Gal: IMap "y'' = -sin(y)-b*y'+a*cos(c*x)" in "Classic."
Range: (vMax,vMin) = (12.000,-8.000), (hMin,hMax) = (-10.000,10.000)
VFld: (y,-sin(x)-b*y+a*cos(c*t)), a = 1.47; b = .5; c = .67

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = -sin(x)-b*y+a*cos(c*t)

in the (t,x,y) coordinate system.

The 1st order 2D system corresponds to the 2nd order ode

y'' = -sin(y)-b*y'+a*cos(c*x)

in the (x,y,y') coordinate system.

Parameters are:

a = 1.47; b = .5;

c = .67

Image 1: Ode view.

Image 2: IMap view with blue attractor about (0,0).

View/Sys/Gal: Ode "y'' = 12*x" in "Classic."
Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,2.500)
VFld: (y,12*t)

This example comes from "Differential Equations," by Kaj L. Nielsen, p. 23.

The problem is to show that:

y(x) = 2*x^3 + A*x + B

is a solution of the ode:

y'' = 12*x,

which is easy enough to show.

To solve the ode using Ode Factory enter

y'' = 12*x

in the "y(x) form" field and click "Update Sys" to get:

dx/dt = y,        (1)

dy/dt = 12*t.        (2)

Integrating (2) gives:

y(t) = 6*t^2 + A

Using y(t) in (1) and integrating, gives:

x(t) = 2*t^3 + A*t + B,

To get a particular solution, set A = -2 and B = 0 which corresponds to initial conditions:

x(0) = 0, y(0) = -2

The solution curves are:

x(t) = 2*t(t^2-1) and

y(t) = 2*(3t^2-1)

Image 1: x(t).

Image 2: y(t).

We see that x(t) = 0 at t = -1, 0, 1 and y(t) = 0 at t = sqrt(1/3) = +-0.577. We can plot x vs t and y vs t to verify the solutions. To see that y(t) crosses the t axis at 0.577, click on the crossing point.

View/Sys/Gal: EMap "y'' = x+k*y, k = -2" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y, t+k*x), k = -2

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = t+k*x

in the (t,x,y) coordinate system.

The 1st order 2D system corresponds to the 2nd order ode

y'' = x+k*y

in the (x,y,y') coordinate system.

Parameters are:

k = -2

Image 1: Ode view.

Image 2: EMap view with CT2.

View/Sys/Gal: Ode "y''' = 2*y'' - y' + 2*y +4*x +5*e^(2*x) + 20*cos(x)" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y,z, 2*z - y + 2*x +4*t +5*e^(2*t) + 20*cos(t))

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = z,

dz/dt = 2*z - y + 2*x +4*t +5*e^(2*t) + 20*cos(t)

in the (t,x,y,z) coordinate system.

The 1st order 3D system corresponds to the 3rd order ode

y''' = 2*y'' - y' + 2*y +4*x +5*e^(2*x) + 20*cos(x)

in the (x,y,y',y'') coordinate system.

Image 1: x(1) for y = 2.

View/Sys/Gal: Ode "y'''' = a*y''' +b*y'' + c*y' + d*y" in "Classic."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld: (y,z,w, a*w +b*z + c*y + d*x), a = 1; b = 2; c = 3; d = 4

This system of odes is defined by the equations:

dx/dt = y,

dy/dt = z,

dz/dt = w,

dw/dt = a*w +b*z + c*y + d*x

in the (t,x,y,z,w) coordinate system.

The 1st order 4D system corresponds to the 4th order ode

y'''' = a*y''' +b*y'' + c*y' + d*y

in the (x,y,y',y'',y''') coordinate system.

Parameters are:

a = 1; b = 2;

c = 3; d = 4

Image 1: Trajectories in the (x.y) plain for (z,w) = (0,0).